https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Kiselev&feedformat=atomUW-Math Wiki - User contributions [en]2022-12-02T10:54:32ZUser contributionsMediaWiki 1.35.6https://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6908PDE Geometric Analysis seminar2014-04-20T23:29:26Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 10<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Free Boundary Problem related to Euler-Poisson system. ]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
Global classical solution and long time behavior of macroscopic flocking models. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
Parabolic equations in time-varying domains. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 31<br />
|[http://www.mth.kcl.ac.uk/staff/a_pushnitski.html Alexander Pushnitski (King's College London)]<br />
|[[#Alexander Pushnitski (King's College) |<br />
An inverse spectral problem for Hankel operators. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
Compressible Navier-Stokes-Fourier system with temperature dependent dissipation. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Fall 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22 (joint with Analysis Seminar)<br />
|Steven Hofmann (U. of Missouri)<br />
|[[#Steven Hofmann (U. of Missouri) |<br />
TBA]]<br />
|Seeger<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Free Boundary Problem related to Euler-Poisson system''<br />
<br />
One dimensional analysis of Euler-Poisson system shows that when incoming <br />
supersonic flow is fixed, transonic shock can be represented as a monotone <br />
function of exit pressure. From this observation, we expect well-posedness <br />
of transonic shock problem for Euler-Poisson system when exit pressure is <br />
prescribed in a proper range. In this talk, I will present recent progress <br />
on transonic shock problem for Euler-Poisson system, which is formulated <br />
as a free boundary problem with mixed type PDE system. <br />
This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU) <br />
and Jingjing Xiao(CUHK).<br />
<br />
===Changhui Tan (University of Maryland)===<br />
''Global classical solution and long time behavior of macroscopic flocking models''<br />
<br />
Abstract: Self-organized behaviors are very common in nature and human societies.<br />
One widely discussed example is the flocking phenomenon which describes<br />
animal groups emerging towards the same direction. Several models such<br />
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing<br />
flocking behaviors. In this talk, we will discuss macroscopic representation<br />
of flocking models. These systems can be interpreted as compressible Eulerian <br />
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time <br />
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set<br />
of initial conditions will lead to a finite time break down of the system. This<br />
is a joint work with Eitan Tadmor.<br />
<br />
===Hongjie Dong (Brown University)===<br />
''Parabolic equations in time-varying domains''<br />
<br />
Abstract: I will present a recent result on the Dirichlet boundary value<br />
problem for parabolic equations in time-varying domains. The equations are<br />
in either divergence or non-divergence form with boundary blowup low-order<br />
coefficients. The domains satisfy an exterior measure condition.<br />
<br />
===Hao Jia (University of Chicago)===<br />
''Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions.''<br />
<br />
Abstract: We consider the long term dynamics of radial solution to the<br />
above mentioned equation. For general potential, the equation can have a<br />
unique positive ground state and a number of excited states. One can expect<br />
that some solutions might stay for very long time near excited states<br />
before settling down to an excited state of lower energy or the ground<br />
state. Thus the detailed dynamics can be extremely complicated. However<br />
using the ``channel of energy" inequality discovered by T.Duyckaerts,<br />
C.Kenig and F.Merle, we can show for generic potential, any radial solution<br />
is asymptotically the sum of a free radiation and a steady state as time<br />
goes to infinity. This provides another example of the power of ``channel<br />
of energy" inequality and the method of profile decompositions. I will<br />
explain the basic tools in some detail. Joint work with Baoping Liu and<br />
Guixiang Xu.<br />
<br />
===Alexander Pushnitski (King's College)===<br />
''An inverse spectral problem for Hankel operators''<br />
<br />
Abstract:<br />
I will discuss an inverse spectral problem for a certain class of Hankel<br />
operators. The problem appeared in the recent work by P.Gerard and S.Grellier as a<br />
step towards description of evolution in a model integrable non-dispersive<br />
equation. Several features of this inverse problem make it strikingly (and somewhat<br />
mysteriously) similar to an inverse problem for Sturm-Liouville operators. I will<br />
describe the available results for Hankel operators, emphasizing this similarity.<br />
This is joint work with Patrick Gerard (Orsay).<br />
<br />
===Ronghua Pan (Georgia Tech)===<br />
''Compressible Navier-Stokes-Fourier system with temperature dependent dissipation''<br />
<br />
Abstract: From its physical origin such as Chapman-Enskog or Sutherland, the viscosity and<br />
heat conductivity coefficients in compressible fluids depend on absolute temperature<br />
through power laws. The mathematical theory on the well-posedness and regularity on this<br />
setting is widely open. I will report some recent progress on this direction,<br />
with emphasis on the lower bound of temperature, and global existence of<br />
solutions in one or multiple dimensions. The relation between thermodynamics laws<br />
and Navier-Stokes-Fourier system will also be discussed. This talk is based on joint works<br />
with Junxiong Jia and Weizhe Zhang.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6812PDE Geometric Analysis seminar2014-03-28T16:46:21Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 10<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Free Boundary Problem related to Euler-Poisson system. ]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
Global classical solution and long time behavior of macroscopic flocking models. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
Parabolic equations in time-varying domains. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 31<br />
|[http://www.mth.kcl.ac.uk/staff/a_pushnitski.html Alexander Pushnitski (King's College London)]<br />
|[[#Alexander Pushnitski (King's College) |<br />
An inverse spectral problem for Hankel operators. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Fall 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22 (joint with Analysis Seminar)<br />
|Steven Hofmann (U. of Missouri)<br />
|[[#Steven Hofmann (U. of Missouri) |<br />
TBA]]<br />
|Seeger<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Free Boundary Problem related to Euler-Poisson system''<br />
<br />
One dimensional analysis of Euler-Poisson system shows that when incoming <br />
supersonic flow is fixed, transonic shock can be represented as a monotone <br />
function of exit pressure. From this observation, we expect well-posedness <br />
of transonic shock problem for Euler-Poisson system when exit pressure is <br />
prescribed in a proper range. In this talk, I will present recent progress <br />
on transonic shock problem for Euler-Poisson system, which is formulated <br />
as a free boundary problem with mixed type PDE system. <br />
This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU) <br />
and Jingjing Xiao(CUHK).<br />
<br />
===Changhui Tan (University of Maryland)===<br />
''Global classical solution and long time behavior of macroscopic flocking models''<br />
<br />
Abstract: Self-organized behaviors are very common in nature and human societies.<br />
One widely discussed example is the flocking phenomenon which describes<br />
animal groups emerging towards the same direction. Several models such<br />
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing<br />
flocking behaviors. In this talk, we will discuss macroscopic representation<br />
of flocking models. These systems can be interpreted as compressible Eulerian <br />
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time <br />
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set<br />
of initial conditions will lead to a finite time break down of the system. This<br />
is a joint work with Eitan Tadmor.<br />
<br />
===Hongjie Dong (Brown University)===<br />
''Parabolic equations in time-varying domains''<br />
<br />
Abstract: I will present a recent result on the Dirichlet boundary value<br />
problem for parabolic equations in time-varying domains. The equations are<br />
in either divergence or non-divergence form with boundary blowup low-order<br />
coefficients. The domains satisfy an exterior measure condition.<br />
<br />
===Hao Jia (University of Chicago)===<br />
''Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions.''<br />
<br />
Abstract: We consider the long term dynamics of radial solution to the<br />
above mentioned equation. For general potential, the equation can have a<br />
unique positive ground state and a number of excited states. One can expect<br />
that some solutions might stay for very long time near excited states<br />
before settling down to an excited state of lower energy or the ground<br />
state. Thus the detailed dynamics can be extremely complicated. However<br />
using the ``channel of energy" inequality discovered by T.Duyckaerts,<br />
C.Kenig and F.Merle, we can show for generic potential, any radial solution<br />
is asymptotically the sum of a free radiation and a steady state as time<br />
goes to infinity. This provides another example of the power of ``channel<br />
of energy" inequality and the method of profile decompositions. I will<br />
explain the basic tools in some detail. Joint work with Baoping Liu and<br />
Guixiang Xu.<br />
<br />
===Alexander Pushnitski (King's College)===<br />
''An inverse spectral problem for Hankel operators''<br />
<br />
Abstract:<br />
I will discuss an inverse spectral problem for a certain class of Hankel<br />
operators. The problem appeared in the recent work by P.Gerard and S.Grellier as a<br />
step towards description of evolution in a model integrable non-dispersive<br />
equation. Several features of this inverse problem make it strikingly (and somewhat<br />
mysteriously) similar to an inverse problem for Sturm-Liouville operators. I will<br />
describe the available results for Hankel operators, emphasizing this similarity.<br />
This is joint work with Patrick Gerard (Orsay).</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6770PDE Geometric Analysis seminar2014-03-13T15:51:38Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 10<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Free Boundary Problem related to Euler-Poisson system. ]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
Global classical solution and long time behavior of macroscopic flocking models. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
Parabolic equations in time-varying domains. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 31<br />
|[http://www.mth.kcl.ac.uk/staff/a_pushnitski.html Alexander Pushnitski (King's College London)]<br />
|[[#Alexander Pushnitski (King's College) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Fall 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22 (joint with Analysis Seminar)<br />
|Steven Hofmann (U. of Missouri)<br />
|[[#Steven Hofmann (U. of Missouri) |<br />
TBA]]<br />
|Seeger<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Free Boundary Problem related to Euler-Poisson system''<br />
<br />
One dimensional analysis of Euler-Poisson system shows that when incoming <br />
supersonic flow is fixed, transonic shock can be represented as a monotone <br />
function of exit pressure. From this observation, we expect well-posedness <br />
of transonic shock problem for Euler-Poisson system when exit pressure is <br />
prescribed in a proper range. In this talk, I will present recent progress <br />
on transonic shock problem for Euler-Poisson system, which is formulated <br />
as a free boundary problem with mixed type PDE system. <br />
This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU) <br />
and Jingjing Xiao(CUHK).<br />
<br />
===Changhui Tan (University of Maryland)===<br />
''Global classical solution and long time behavior of macroscopic flocking models''<br />
<br />
Abstract: Self-organized behaviors are very common in nature and human societies.<br />
One widely discussed example is the flocking phenomenon which describes<br />
animal groups emerging towards the same direction. Several models such<br />
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing<br />
flocking behaviors. In this talk, we will discuss macroscopic representation<br />
of flocking models. These systems can be interpreted as compressible Eulerian <br />
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time <br />
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set<br />
of initial conditions will lead to a finite time break down of the system. This<br />
is a joint work with Eitan Tadmor.<br />
<br />
===Hongjie Dong (Brown University)===<br />
''Parabolic equations in time-varying domains''<br />
<br />
Abstract: I will present a recent result on the Dirichlet boundary value<br />
problem for parabolic equations in time-varying domains. The equations are<br />
in either divergence or non-divergence form with boundary blowup low-order<br />
coefficients. The domains satisfy an exterior measure condition.<br />
<br />
===Hao Jia (University of Chicago)===<br />
''Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions.''<br />
<br />
Abstract: We consider the long term dynamics of radial solution to the<br />
above mentioned equation. For general potential, the equation can have a<br />
unique positive ground state and a number of excited states. One can expect<br />
that some solutions might stay for very long time near excited states<br />
before settling down to an excited state of lower energy or the ground<br />
state. Thus the detailed dynamics can be extremely complicated. However<br />
using the ``channel of energy" inequality discovered by T.Duyckaerts,<br />
C.Kenig and F.Merle, we can show for generic potential, any radial solution<br />
is asymptotically the sum of a free radiation and a steady state as time<br />
goes to infinity. This provides another example of the power of ``channel<br />
of energy" inequality and the method of profile decompositions. I will<br />
explain the basic tools in some detail. Joint work with Baoping Liu and<br />
Guixiang Xu.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6766PDE Geometric Analysis seminar2014-03-10T15:11:54Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 10<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Free Boundary Problem related to Euler-Poisson system. ]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
Global classical solution and long time behavior of macroscopic flocking models. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
Parabolic equations in time-varying domains. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 31<br />
|[http://www.mth.kcl.ac.uk/staff/a_pushnitski.html Alexander Pushnitski (King's College London)]<br />
|[[#Alexander Pushnitski (King's College) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Fall 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 22 (joint with Analysis Seminar)<br />
|Steven Hofmann (U. of Missouri)<br />
|[[#Steven Hofmann (U. of Missouri) |<br />
TBA]]<br />
|Seeger<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Free Boundary Problem related to Euler-Poisson system''<br />
<br />
One dimensional analysis of Euler-Poisson system shows that when incoming <br />
supersonic flow is fixed, transonic shock can be represented as a monotone <br />
function of exit pressure. From this observation, we expect well-posedness <br />
of transonic shock problem for Euler-Poisson system when exit pressure is <br />
prescribed in a proper range. In this talk, I will present recent progress <br />
on transonic shock problem for Euler-Poisson system, which is formulated <br />
as a free boundary problem with mixed type PDE system. <br />
This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU) <br />
and Jingjing Xiao(CUHK).<br />
<br />
===Changhui Tan (University of Maryland)===<br />
''Global classical solution and long time behavior of macroscopic flocking models''<br />
<br />
Abstract: Self-organized behaviors are very common in nature and human societies.<br />
One widely discussed example is the flocking phenomenon which describes<br />
animal groups emerging towards the same direction. Several models such<br />
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing<br />
flocking behaviors. In this talk, we will discuss macroscopic representation<br />
of flocking models. These systems can be interpreted as compressible Eulerian <br />
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time <br />
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set<br />
of initial conditions will lead to a finite time break down of the system. This<br />
is a joint work with Eitan Tadmor.<br />
<br />
===Hongjie Dong (Brown University)===<br />
''Parabolic equations in time-varying domains''<br />
<br />
Abstract: I will present a recent result on the Dirichlet boundary value<br />
problem for parabolic equations in time-varying domains. The equations are<br />
in either divergence or non-divergence form with boundary blowup low-order<br />
coefficients. The domains satisfy an exterior measure condition.<br />
<br />
===Hao Jia (University of Chicago)===<br />
''Long time dynamics of energy critical defocusing wave equation with<br />
radial potential in 3+1 dimensions.''<br />
<br />
Abstract: We consider the long term dynamics of radial solution to the<br />
above mentioned equation. For general potential, the equation can have a<br />
unique positive ground state and a number of excited states. One can expect<br />
that some solutions might stay for very long time near excited states<br />
before settling down to an excited state of lower energy or the ground<br />
state. Thus the detailed dynamics can be extremely complicated. However<br />
using the ``channel of energy" inequality discovered by T.Duyckaerts,<br />
C.Kenig and F.Merle, we can show for generic potential, any radial solution<br />
is asymptotically the sum of a free radiation and a steady state as time<br />
goes to infinity. This provides another example of the power of ``channel<br />
of energy" inequality and the method of profile decompositions. I will<br />
explain the basic tools in some detail. Joint work with Baoping Liu and<br />
Guixiang Xu.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6649PDE Geometric Analysis seminar2014-02-14T17:35:49Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 10<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Free Boundary Problem related to Euler-Poisson system. ]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
Global classical solution and long time behavior of macroscopic flocking models. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
Parabolic equations in time-varying domains. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 31<br />
|[http://www.mth.kcl.ac.uk/staff/a_pushnitski.html Alexander Pushnitski (King's College London)]<br />
|[[#Alexander Pushnitski (King's College) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Free Boundary Problem related to Euler-Poisson system''<br />
<br />
One dimensional analysis of Euler-Poisson system shows that when incoming <br />
supersonic flow is fixed, transonic shock can be represented as a monotone <br />
function of exit pressure. From this observation, we expect well-posedness <br />
of transonic shock problem for Euler-Poisson system when exit pressure is <br />
prescribed in a proper range. In this talk, I will present recent progress <br />
on transonic shock problem for Euler-Poisson system, which is formulated <br />
as a free boundary problem with mixed type PDE system. <br />
This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU) <br />
and Jingjing Xiao(CUHK).<br />
<br />
===Changhui Tan (University of Maryland)===<br />
''Global classical solution and long time behavior of macroscopic flocking models''<br />
<br />
Abstract: Self-organized behaviors are very common in nature and human societies.<br />
One widely discussed example is the flocking phenomenon which describes<br />
animal groups emerging towards the same direction. Several models such<br />
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing<br />
flocking behaviors. In this talk, we will discuss macroscopic representation<br />
of flocking models. These systems can be interpreted as compressible Eulerian <br />
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time <br />
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set<br />
of initial conditions will lead to a finite time break down of the system. This<br />
is a joint work with Eitan Tadmor.<br />
<br />
===Hongjie Dong (Brown University)===<br />
''Parabolic equations in time-varying domains''<br />
<br />
Abstract: I will present a recent result on the Dirichlet boundary value<br />
problem for parabolic equations in time-varying domains. The equations are<br />
in either divergence or non-divergence form with boundary blowup low-order<br />
coefficients. The domains satisfy an exterior measure condition.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6632PDE Geometric Analysis seminar2014-02-12T18:12:48Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 10<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Free Boundary Problem related to Euler-Poisson system. ]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
Global classical solution and long time behavior of macroscopic flocking models. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 31<br />
|[http://www.mth.kcl.ac.uk/staff/a_pushnitski.html Alexander Pushnitski (King's College London)]<br />
|[[#Alexander Pushnitski (King's College) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Free Boundary Problem related to Euler-Poisson system''<br />
<br />
One dimensional analysis of Euler-Poisson system shows that when incoming <br />
supersonic flow is fixed, transonic shock can be represented as a monotone <br />
function of exit pressure. From this observation, we expect well-posedness <br />
of transonic shock problem for Euler-Poisson system when exit pressure is <br />
prescribed in a proper range. In this talk, I will present recent progress <br />
on transonic shock problem for Euler-Poisson system, which is formulated <br />
as a free boundary problem with mixed type PDE system. <br />
This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU) <br />
and Jingjing Xiao(CUHK).<br />
<br />
===Changhui Tan (University of Maryland)===<br />
''Global classical solution and long time behavior of macroscopic flocking models''<br />
<br />
Abstract: Self-organized behaviors are very common in nature and human societies.<br />
One widely discussed example is the flocking phenomenon which describes<br />
animal groups emerging towards the same direction. Several models such<br />
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing<br />
flocking behaviors. In this talk, we will discuss macroscopic representation<br />
of flocking models. These systems can be interpreted as compressible Eulerian <br />
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time <br />
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set<br />
of initial conditions will lead to a finite time break down of the system. This<br />
is a joint work with Eitan Tadmor.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6568PDE Geometric Analysis seminar2014-02-05T22:56:14Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
Global classical solution and long time behavior of macroscopic flocking models. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.<br />
<br />
===Changhui Tan (University of Maryland)===<br />
''Global classical solution and long time behavior of macroscopic flocking models''<br />
<br />
Abstract: Self-organized behaviors are very common in nature and human societies.<br />
One widely discussed example is the flocking phenomenon which describes<br />
animal groups emerging towards the same direction. Several models such<br />
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing<br />
flocking behaviors. In this talk, we will discuss macroscopic representation<br />
of flocking models. These systems can be interpreted as compressible Eulerian <br />
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time <br />
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set<br />
of initial conditions will lead to a finite time break down of the system. This<br />
is a joint work with Eitan Tadmor.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6567PDE Geometric Analysis seminar2014-02-05T22:55:38Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
Global classical solution and long time behavior of macroscopic flocking models. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.<br />
<br />
===Changhui Tan (University of Maryland)===<br />
''Global classical solution and long time behavior of macroscopic flocking models''<br />
<br />
Abstract: Self-organized behaviors are very common in nature and human societies.<br />
One widely discussed example is the flocking phenomenon which describes<br />
animal groups emerging towards the same direction. Several models such<br />
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing<br />
flocking behaviors. In this talk, we will discuss macroscopic representation<br />
of flocking models. These systems can be interpreted as compressible Eulerian <br />
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time <br />
flocking behavior of the system, when initial profi�le satisfi�es a threshold condition. On the other hand, another set<br />
of initial conditions will lead to a �finite time break down of the system. This<br />
is a joint work with Eitan Tadmor.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6553PDE Geometric Analysis seminar2014-02-05T16:21:36Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
On the global classical solution of the Keller-Segel-Navier -Stokes system and its asymptotic behavior. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|-<br />
|December 16<br />
|Antonio Ache(Princeton)<br />
|[[#Antonio Ache(Princeton) |<br />
Ricci Curvature and the manifold learning problem. NOTE: Due to final exams, this seminar will be held in B231. ]]<br />
|Viaclovsky<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Jean-Michel Roquejoffre (Toulouse) |<br />
Front propagation in the presence of integral diffusion. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|February 24<br />
|[http://www2.math.umd.edu/~jefftan/ Changhui Tan (Maryland)]<br />
|[[#Changhui Tan (Maryland) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.<br />
<br />
===Antonio Ache(Princeton)===<br />
''Ricci Curvature and the manifold learning problem''<br />
<br />
Abstract: In the first half of this talk we will review several notions of coarse or weak<br />
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm<br />
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as <br />
motivation for developing a method to estimate the Ricci curvature of a an embedded<br />
submaifold of Euclidean space from a point cloud which has applications to the Manifold<br />
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"<br />
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is<br />
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space<br />
from point clouds. This is joint work with Micah Warren.<br />
<br />
===Jean-Michel Roquejoffre (Toulouse)===<br />
''Front propagation in the presence of integral diffusion''<br />
<br />
Abstract: In many reaction-diffusion equations, where diffusion is<br />
given by a second order elliptic operator, the solutions<br />
will exhibit spatial transitions whose velocity is asymptotically<br />
linear in time. The situation can be different when the diffusion is of the<br />
integral type, the most basic example being the fractional Laplacian:<br />
the velocity can be time-exponential. We will explain why, and<br />
discuss several situations where this type of fast propagation<br />
occurs.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6227PDE Geometric Analysis seminar2013-11-04T19:02:11Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.<br />
<br />
===Roman Shterenberg(UAB)===<br />
''Recent progress in multidimensional periodic and almost-periodic spectral<br />
problems''<br />
<br />
Abstract: We present a review of the results in multidimensional periodic<br />
and almost-periodic spectral problems. We discuss some recent progress and<br />
old/new ideas used in the constructions. The talk is mostly based on the<br />
joint works with Yu. Karpeshina and L. Parnovski.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6159PDE Geometric Analysis seminar2013-10-28T00:07:57Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|Myeongju Chae (Hankyong National University visiting UW)<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6158PDE Geometric Analysis seminar2013-10-28T00:07:22Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 25<br />
|[ Myeongju Chae (Hankyong National University visiting UW)]<br />
|[[#Myeongju Chae (Hankyong National University) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
myeongju Chae <br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6129PDE Geometric Analysis seminar2013-10-20T05:33:56Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|March 10<br />
|[http://math.uchicago.edu/~jiahao/ Hao Jia (University of Chicago)]<br />
|[[#Hao Jia (University of Chicago) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6124PDE Geometric Analysis seminar2013-10-19T20:04:09Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|December 2<br />
|Xiaojie Wang<br />
|[[#Xiaojie Wang (Stony Brook University) |<br />
Uniqueness of Ricci flow solutions on noncompact manifolds. ]]<br />
|Wang<br />
|-<br />
|}<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 14 at 4pm in B139 (TUESDAY), joint with Analysis<br />
|[http://www.math.univ-toulouse.fr/~roque/ Jean-Michel Roquejoffre (Toulouse)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Zlatos<br />
|-<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 21<br />
|[http://people.math.gatech.edu/~panrh/ Ronghua Pan (Georgia Tech)]<br />
|[[#Ronghua Pan (Georgia Tech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.<br />
<br />
===Xiaojie Wang(Stony Brook)===<br />
''Uniqueness of Ricci flow solutions on noncompact manifolds''<br />
<br />
Abstract:<br />
Ricci flow is an important evolution equation of Riemannian metrics.<br />
Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=6052PDE Geometric Analysis seminar2013-10-06T13:43:58Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 9<br />
|Greg Drugan (U. of Washington)<br />
|[[#Greg Drugan (U. of Washington) |<br />
Construction of immersed self-shrinkers]]<br />
|Angenent<br />
|-<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
Potentially Singular Solutions of the 3D Incompressible Euler Equations. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Greg Drugan (U. of Washington)===<br />
''Construction of immersed self-shrinkers''<br />
<br />
Abstract: We describe a procedure for constructing immersed<br />
self-shrinking solutions to mean curvature flow. <br />
The self-shrinkers we construct have a rotational symmetry, and<br />
the construction involves a detailed study of geodesics in the<br />
upper-half plane with a conformal metric.<br />
This is a joint work with Stephen Kleene.<br />
<br />
===Guo Luo (Caltech)===<br />
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''<br />
<br />
Abstract:<br />
Whether the 3D incompressible Euler equations can develop a singularity in <br />
finite time from smooth initial data is one of the most challenging problems in <br />
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this <br />
long-standing open question from a numerical point of view, by presenting a class of <br />
potentially singular solutions to the Euler equations computed in axisymmetric <br />
geometries. The solutions satisfy a periodic boundary condition along the axial direction <br />
and no-flow boundary condition on the solid wall. The equations are discretized in space <br />
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially <br />
designed adaptive (moving) meshes that are dynamically adjusted to the evolving <br />
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the <br />
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and <br />
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a <br />
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and <br />
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity <br />
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup <br />
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and <br />
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also <br />
suggests that the blowing-up solution develops a self-similar structure near the point of <br />
the singularity, as the singularity time is approached.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=5777PDE Geometric Analysis seminar2013-09-04T17:05:49Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Spring 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|-<br />
|February 4<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Transonic shocks for Euler-Poisson system and related problems]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 18<br />
|Mike Cullen (Met. Office, UK)<br />
|[[ #Mike Cullen (Met. Office, UK)|<br />
Modelling the uncertainty in predicting large-scale atmospheric circulations.]]<br />
|Feldman<br />
|-<br />
|-<br />
|March 18<br />
|Mohammad Ghomi(Math. Georgia Tech)<br />
|[[ #Mohammad Ghomi(Math. Georgia Tech)|<br />
Tangent lines, inflections, and vertices of closed curves.]]<br />
|Angenent<br />
|-<br />
|-<br />
|April 8<br />
|[http://www.maths.ox.ac.uk/contact/details/xiang Wei Xiang (Oxford)]<br />
|[[ #Wei Xiang (Oxford)|<br />
Shock Diffraction Problem to the<br />
Two Dimensional Nonlinear Wave System and Potential Flow Equation.]]<br />
|Feldman<br />
|-<br />
|-<br />
|Thursday April 25, Time: 3:30, Room: 901 Van Vleck (Note: Special Day, Room)<br />
|[http://math.wvu.edu/~adriant/CV1.html Adrian Tudorascu (West Virginia University)]<br />
|[[ #Adrian Tudorascu (West Virginia University)|One-dimensional pressureless <br />
Euler/Euler-Poisson systems with/without viscosity<br />
.]]<br />
|Feldman<br />
|-<br />
|-<br />
|May 6<br />
|[http://www.icmat.es/dcordoba Diego Cordoba (Madrid)]<br />
|[[#Diego Cordoba (Madrid) |<br />
Interface dynamics for incompressible fluids. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.<br />
<br />
<br />
===Mike Cullen (Met. Office, UK)===<br />
''Modelling the uncertainty in predicting large-scale atmospheric circulations''<br />
<br />
Abstract: This talk describes work on quantifying the uncertainty in climate models by using data assimilation techniques; <br />
and in estimating the properties of large-scale atmospheric flows by using asymptotic limit solutions.<br />
<br />
===Mohammad Ghomi(Math. Georgia Tech)===<br />
<br />
>> Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow with surgery, is based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's ``tennis ball theorem".<br />
<br />
===Wei Xiang (Oxford)===<br />
<br />
Abstract: The vertical shock which initially separates two<br />
piecewise constant Riemann data, passes the wedge from left to right,<br />
then shock diffraction phenomena will occur and the incident shock<br />
becomes a transonic shock. Here we study this problem on nonlinear<br />
wave system as well as on potential flow equations. The existence and<br />
the optimal regularity across sonic circle of the solutions to this<br />
problem is established. The comparison of these two systems is<br />
discussed, and some related open problems are proposed.<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
<br />
Abstract: <br />
This chalk & board presentation will start with an interesting motivation for these systems (the simplest of which was shown to model adhesion dynamics/ballistic aggregation by Zeldovich in 1970 in a paper published in "Astronomy & Astrophysics"). A general existence result for one-dimensional pressureless Euler/Euler-Poisson systems with/without viscosity will be discussed. This is achieved via entropy solutions for some appropriate scalar conservation laws; I will show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. I will also discuss the stability and uniqueness of solutions, which is obtained via a contraction principle in the Wasserstein metric. This is based on joint work with T. Nguyen (U. Akron).<br />
<br />
===Diego Cordoba (Madrid)===<br />
<br />
Abstract: We consider the evolution of an interface generated between two immiscible, <br />
incompressible and irrotational fluids. Specifically we study the Muskat equation (the <br />
interface between oil and water in sand) and water wave equation (interface between water <br />
and vacuum). For both equations we will study well-posedness and the existence of smooth <br />
initial data for which the smoothness of the interface breaks down in finite time. We <br />
will also discuss some open problems.<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|April 7<br />
|[http://pi.math.virginia.edu/~zg7c/ Zoran Grujic (University of Virginia)]<br />
|[[#Zoran Grujic (University of Virginia) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=5776PDE Geometric Analysis seminar2013-09-04T17:02:58Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Spring 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|-<br />
|February 4<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Transonic shocks for Euler-Poisson system and related problems]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 18<br />
|Mike Cullen (Met. Office, UK)<br />
|[[ #Mike Cullen (Met. Office, UK)|<br />
Modelling the uncertainty in predicting large-scale atmospheric circulations.]]<br />
|Feldman<br />
|-<br />
|-<br />
|March 18<br />
|Mohammad Ghomi(Math. Georgia Tech)<br />
|[[ #Mohammad Ghomi(Math. Georgia Tech)|<br />
Tangent lines, inflections, and vertices of closed curves.]]<br />
|Angenent<br />
|-<br />
|-<br />
|April 8<br />
|[http://www.maths.ox.ac.uk/contact/details/xiang Wei Xiang (Oxford)]<br />
|[[ #Wei Xiang (Oxford)|<br />
Shock Diffraction Problem to the<br />
Two Dimensional Nonlinear Wave System and Potential Flow Equation.]]<br />
|Feldman<br />
|-<br />
|-<br />
|Thursday April 25, Time: 3:30, Room: 901 Van Vleck (Note: Special Day, Room)<br />
|[http://math.wvu.edu/~adriant/CV1.html Adrian Tudorascu (West Virginia University)]<br />
|[[ #Adrian Tudorascu (West Virginia University)|One-dimensional pressureless <br />
Euler/Euler-Poisson systems with/without viscosity<br />
.]]<br />
|Feldman<br />
|-<br />
|-<br />
|May 6<br />
|[http://www.icmat.es/dcordoba Diego Cordoba (Madrid)]<br />
|[[#Diego Cordoba (Madrid) |<br />
Interface dynamics for incompressible fluids. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.<br />
<br />
<br />
===Mike Cullen (Met. Office, UK)===<br />
''Modelling the uncertainty in predicting large-scale atmospheric circulations''<br />
<br />
Abstract: This talk describes work on quantifying the uncertainty in climate models by using data assimilation techniques; <br />
and in estimating the properties of large-scale atmospheric flows by using asymptotic limit solutions.<br />
<br />
===Mohammad Ghomi(Math. Georgia Tech)===<br />
<br />
>> Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow with surgery, is based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's ``tennis ball theorem".<br />
<br />
===Wei Xiang (Oxford)===<br />
<br />
Abstract: The vertical shock which initially separates two<br />
piecewise constant Riemann data, passes the wedge from left to right,<br />
then shock diffraction phenomena will occur and the incident shock<br />
becomes a transonic shock. Here we study this problem on nonlinear<br />
wave system as well as on potential flow equations. The existence and<br />
the optimal regularity across sonic circle of the solutions to this<br />
problem is established. The comparison of these two systems is<br />
discussed, and some related open problems are proposed.<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
<br />
Abstract: <br />
This chalk & board presentation will start with an interesting motivation for these systems (the simplest of which was shown to model adhesion dynamics/ballistic aggregation by Zeldovich in 1970 in a paper published in "Astronomy & Astrophysics"). A general existence result for one-dimensional pressureless Euler/Euler-Poisson systems with/without viscosity will be discussed. This is achieved via entropy solutions for some appropriate scalar conservation laws; I will show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. I will also discuss the stability and uniqueness of solutions, which is obtained via a contraction principle in the Wasserstein metric. This is based on joint work with T. Nguyen (U. Akron).<br />
<br />
===Diego Cordoba (Madrid)===<br />
<br />
Abstract: We consider the evolution of an interface generated between two immiscible, <br />
incompressible and irrotational fluids. Specifically we study the Muskat equation (the <br />
interface between oil and water in sand) and water wave equation (interface between water <br />
and vacuum). For both equations we will study well-posedness and the existence of smooth <br />
initial data for which the smoothness of the interface breaks down in finite time. We <br />
will also discuss some open problems.<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|-<br />
|November 18<br />
|[http://people.cas.uab.edu/~shterenb/ Roman Shterenberg (UAB)]<br />
|[[#Roman Shterenberg (UAB) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=5688PDE Geometric Analysis seminar2013-08-27T23:45:06Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Spring 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|-<br />
|February 4<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Transonic shocks for Euler-Poisson system and related problems]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 18<br />
|Mike Cullen (Met. Office, UK)<br />
|[[ #Mike Cullen (Met. Office, UK)|<br />
Modelling the uncertainty in predicting large-scale atmospheric circulations.]]<br />
|Feldman<br />
|-<br />
|-<br />
|March 18<br />
|Mohammad Ghomi(Math. Georgia Tech)<br />
|[[ #Mohammad Ghomi(Math. Georgia Tech)|<br />
Tangent lines, inflections, and vertices of closed curves.]]<br />
|Angenent<br />
|-<br />
|-<br />
|April 8<br />
|[http://www.maths.ox.ac.uk/contact/details/xiang Wei Xiang (Oxford)]<br />
|[[ #Wei Xiang (Oxford)|<br />
Shock Diffraction Problem to the<br />
Two Dimensional Nonlinear Wave System and Potential Flow Equation.]]<br />
|Feldman<br />
|-<br />
|-<br />
|Thursday April 25, Time: 3:30, Room: 901 Van Vleck (Note: Special Day, Room)<br />
|[http://math.wvu.edu/~adriant/CV1.html Adrian Tudorascu (West Virginia University)]<br />
|[[ #Adrian Tudorascu (West Virginia University)|One-dimensional pressureless <br />
Euler/Euler-Poisson systems with/without viscosity<br />
.]]<br />
|Feldman<br />
|-<br />
|-<br />
|May 6<br />
|[http://www.icmat.es/dcordoba Diego Cordoba (Madrid)]<br />
|[[#Diego Cordoba (Madrid) |<br />
Interface dynamics for incompressible fluids. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.<br />
<br />
<br />
===Mike Cullen (Met. Office, UK)===<br />
''Modelling the uncertainty in predicting large-scale atmospheric circulations''<br />
<br />
Abstract: This talk describes work on quantifying the uncertainty in climate models by using data assimilation techniques; <br />
and in estimating the properties of large-scale atmospheric flows by using asymptotic limit solutions.<br />
<br />
===Mohammad Ghomi(Math. Georgia Tech)===<br />
<br />
>> Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow with surgery, is based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's ``tennis ball theorem".<br />
<br />
===Wei Xiang (Oxford)===<br />
<br />
Abstract: The vertical shock which initially separates two<br />
piecewise constant Riemann data, passes the wedge from left to right,<br />
then shock diffraction phenomena will occur and the incident shock<br />
becomes a transonic shock. Here we study this problem on nonlinear<br />
wave system as well as on potential flow equations. The existence and<br />
the optimal regularity across sonic circle of the solutions to this<br />
problem is established. The comparison of these two systems is<br />
discussed, and some related open problems are proposed.<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
<br />
Abstract: <br />
This chalk & board presentation will start with an interesting motivation for these systems (the simplest of which was shown to model adhesion dynamics/ballistic aggregation by Zeldovich in 1970 in a paper published in "Astronomy & Astrophysics"). A general existence result for one-dimensional pressureless Euler/Euler-Poisson systems with/without viscosity will be discussed. This is achieved via entropy solutions for some appropriate scalar conservation laws; I will show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. I will also discuss the stability and uniqueness of solutions, which is obtained via a contraction principle in the Wasserstein metric. This is based on joint work with T. Nguyen (U. Akron).<br />
<br />
===Diego Cordoba (Madrid)===<br />
<br />
Abstract: We consider the evolution of an interface generated between two immiscible, <br />
incompressible and irrotational fluids. Specifically we study the Muskat equation (the <br />
interface between oil and water in sand) and water wave equation (interface between water <br />
and vacuum). For both equations we will study well-posedness and the existence of smooth <br />
initial data for which the smoothness of the interface breaks down in finite time. We <br />
will also discuss some open problems.<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|October 7<br />
|[http://users.cms.caltech.edu/~gluo/ Guo Luo (Caltech)]<br />
|[[#Guo Luo (Caltech) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=5466PDE Geometric Analysis seminar2013-08-02T19:26:14Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Spring 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|-<br />
|February 4<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Transonic shocks for Euler-Poisson system and related problems]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 18<br />
|Mike Cullen (Met. Office, UK)<br />
|[[ #Mike Cullen (Met. Office, UK)|<br />
Modelling the uncertainty in predicting large-scale atmospheric circulations.]]<br />
|Feldman<br />
|-<br />
|-<br />
|March 18<br />
|Mohammad Ghomi(Math. Georgia Tech)<br />
|[[ #Mohammad Ghomi(Math. Georgia Tech)|<br />
Tangent lines, inflections, and vertices of closed curves.]]<br />
|Angenent<br />
|-<br />
|-<br />
|April 8<br />
|[http://www.maths.ox.ac.uk/contact/details/xiang Wei Xiang (Oxford)]<br />
|[[ #Wei Xiang (Oxford)|<br />
Shock Diffraction Problem to the<br />
Two Dimensional Nonlinear Wave System and Potential Flow Equation.]]<br />
|Feldman<br />
|-<br />
|-<br />
|Thursday April 25, Time: 3:30, Room: 901 Van Vleck (Note: Special Day, Room)<br />
|[http://math.wvu.edu/~adriant/CV1.html Adrian Tudorascu (West Virginia University)]<br />
|[[ #Adrian Tudorascu (West Virginia University)|One-dimensional pressureless <br />
Euler/Euler-Poisson systems with/without viscosity<br />
.]]<br />
|Feldman<br />
|-<br />
|-<br />
|May 6<br />
|[http://www.icmat.es/dcordoba Diego Cordoba (Madrid)]<br />
|[[#Diego Cordoba (Madrid) |<br />
Interface dynamics for incompressible fluids. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.<br />
<br />
<br />
===Mike Cullen (Met. Office, UK)===<br />
''Modelling the uncertainty in predicting large-scale atmospheric circulations''<br />
<br />
Abstract: This talk describes work on quantifying the uncertainty in climate models by using data assimilation techniques; <br />
and in estimating the properties of large-scale atmospheric flows by using asymptotic limit solutions.<br />
<br />
===Mohammad Ghomi(Math. Georgia Tech)===<br />
<br />
>> Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow with surgery, is based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's ``tennis ball theorem".<br />
<br />
===Wei Xiang (Oxford)===<br />
<br />
Abstract: The vertical shock which initially separates two<br />
piecewise constant Riemann data, passes the wedge from left to right,<br />
then shock diffraction phenomena will occur and the incident shock<br />
becomes a transonic shock. Here we study this problem on nonlinear<br />
wave system as well as on potential flow equations. The existence and<br />
the optimal regularity across sonic circle of the solutions to this<br />
problem is established. The comparison of these two systems is<br />
discussed, and some related open problems are proposed.<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
<br />
Abstract: <br />
This chalk & board presentation will start with an interesting motivation for these systems (the simplest of which was shown to model adhesion dynamics/ballistic aggregation by Zeldovich in 1970 in a paper published in "Astronomy & Astrophysics"). A general existence result for one-dimensional pressureless Euler/Euler-Poisson systems with/without viscosity will be discussed. This is achieved via entropy solutions for some appropriate scalar conservation laws; I will show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. I will also discuss the stability and uniqueness of solutions, which is obtained via a contraction principle in the Wasserstein metric. This is based on joint work with T. Nguyen (U. Akron).<br />
<br />
===Diego Cordoba (Madrid)===<br />
<br />
Abstract: We consider the evolution of an interface generated between two immiscible, <br />
incompressible and irrotational fluids. Specifically we study the Muskat equation (the <br />
interface between oil and water in sand) and water wave equation (interface between water <br />
and vacuum). For both equations we will study well-posedness and the existence of smooth <br />
initial data for which the smoothness of the interface breaks down in finite time. We <br />
will also discuss some open problems.<br />
<br />
= Seminar Schedule Fall 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|}<br />
<br />
<br />
= Seminar Schedule Spring 2014 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|March 3<br />
|[http://www.dam.brown.edu/people/hdong Hongjie Dong (Brown)]<br />
|[[#Hongjie Dong (Brown) |<br />
TBA. ]]<br />
|Kiselev<br />
|-<br />
|}</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=5281PDE Geometric Analysis seminar2013-04-16T16:03:05Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Spring 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|-<br />
|February 4<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Transonic shocks for Euler-Poisson system and related problems]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 18<br />
|Mike Cullen (Met. Office, UK)<br />
|[[ #Mike Cullen (Met. Office, UK)|<br />
Modelling the uncertainty in predicting large-scale atmospheric circulations.]]<br />
|Feldman<br />
|-<br />
|-<br />
|March 18<br />
|Mohammad Ghomi(Math. Georgia Tech)<br />
|[[ #Mohammad Ghomi(Math. Georgia Tech)|<br />
Tangent lines, inflections, and vertices of closed curves.]]<br />
|Angenent<br />
|-<br />
|-<br />
|April 8<br />
|[http://www.maths.ox.ac.uk/contact/details/xiang Wei Xiang (Oxford)]<br />
|[[ #Wei Xiang (Oxford)|<br />
Shock Diffraction Problem to the<br />
Two Dimensional Nonlinear Wave System and Potential Flow Equation.]]<br />
|Feldman<br />
|-<br />
|-<br />
|May 6<br />
|[http://www.icmat.es/dcordoba Diego Cordoba (Madrid)]<br />
|[[#Diego Cordoba (Madrid) |<br />
Interface dynamics for incompressible fluids. ]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.<br />
<br />
<br />
===Mike Cullen (Met. Office, UK)===<br />
''Modelling the uncertainty in predicting large-scale atmospheric circulations''<br />
<br />
Abstract: This talk describes work on quantifying the uncertainty in climate models by using data assimilation techniques; <br />
and in estimating the properties of large-scale atmospheric flows by using asymptotic limit solutions.<br />
<br />
===Mohammad Ghomi(Math. Georgia Tech)===<br />
<br />
>> Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow with surgery, is based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's ``tennis ball theorem".<br />
<br />
===Wei Xiang (Oxford)===<br />
<br />
Abstract: The vertical shock which initially separates two<br />
piecewise constant Riemann data, passes the wedge from left to right,<br />
then shock diffraction phenomena will occur and the incident shock<br />
becomes a transonic shock. Here we study this problem on nonlinear<br />
wave system as well as on potential flow equations. The existence and<br />
the optimal regularity across sonic circle of the solutions to this<br />
problem is established. The comparison of these two systems is<br />
discussed, and some related open problems are proposed.<br />
<br />
===Diego Cordoba (Madrid)===<br />
<br />
Abstract: We consider the evolution of an interface generated between two immiscible, <br />
incompressible and irrotational fluids. Specifically we study the Muskat equation (the <br />
interface between oil and water in sand) and water wave equation (interface between water <br />
and vacuum). For both equations we will study well-posedness and the existence of smooth <br />
initial data for which the smoothness of the interface breaks down in finite time. We <br />
will also discuss some open problems.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=5254PDE Geometric Analysis seminar2013-04-12T19:17:08Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Spring 2013 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|-<br />
|February 4<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Transonic shocks for Euler-Poisson system and related problems]]<br />
|Feldman<br />
|-<br />
|-<br />
|February 18<br />
|Mike Cullen (Met. Office, UK)<br />
|[[ #Mike Cullen (Met. Office, UK)|<br />
Modelling the uncertainty in predicting large-scale atmospheric circulations.]]<br />
|Feldman<br />
|-<br />
|-<br />
|March 18<br />
|Mohammad Ghomi(Math. Georgia Tech)<br />
|[[ #Mohammad Ghomi(Math. Georgia Tech)|<br />
Tangent lines, inflections, and vertices of closed curves.]]<br />
|Angenent<br />
|-<br />
|-<br />
|April 8<br />
|[http://www.maths.ox.ac.uk/contact/details/xiang Wei Xiang (Oxford)]<br />
|[[ #Wei Xiang (Oxford)|<br />
Shock Diffraction Problem to the<br />
Two Dimensional Nonlinear Wave System and Potential Flow Equation.]]<br />
|Feldman<br />
|-<br />
|-<br />
|May 6<br />
|[http://www.icmat.es/dcordoba Diego Cordoba (Madrid)]<br />
|[[#Diego Cordoba (Madrid) |<br />
TBA]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.<br />
<br />
<br />
===Mike Cullen (Met. Office, UK)===<br />
''Modelling the uncertainty in predicting large-scale atmospheric circulations''<br />
<br />
Abstract: This talk describes work on quantifying the uncertainty in climate models by using data assimilation techniques; <br />
and in estimating the properties of large-scale atmospheric flows by using asymptotic limit solutions.<br />
<br />
===Mohammad Ghomi(Math. Georgia Tech)===<br />
<br />
>> Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow with surgery, is based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's ``tennis ball theorem".<br />
<br />
===Wei Xiang (Oxford)===<br />
<br />
Abstract: The vertical shock which initially separates two<br />
piecewise constant Riemann data, passes the wedge from left to right,<br />
then shock diffraction phenomena will occur and the incident shock<br />
becomes a transonic shock. Here we study this problem on nonlinear<br />
wave system as well as on potential flow equations. The existence and<br />
the optimal regularity across sonic circle of the solutions to this<br />
problem is established. The comparison of these two systems is<br />
discussed, and some related open problems are proposed.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=4901PDE Geometric Analysis seminar2013-01-22T17:18:33Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|-<br />
|November 26<br />
|[http://www.math.wisc.edu/~kchoi/ Kyudong Choi (UW Madison)]<br />
|[[#Kyudong Choi (UW Madison)|<br />
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations]]<br />
|local<br />
|-<br />
|-<br />
|December 10<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
Confinement for nonlocal interaction equation with repulsive-attractive kernels]]<br />
|local<br />
|-<br />
|-<br />
|February 4<br />
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]<br />
|[[#Myoungjean Bae (POSTECH) |<br />
Transonic shocks for Euler-Poisson system and related problems]]<br />
|Feldman<br />
|-<br />
|-<br />
|May 5<br />
|[http://www.icmat.es/dcordoba Diego Cordoba (Madrid)]<br />
|[[#Diego Cordoba (Madrid) |<br />
TBA]]<br />
|Kiselev<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
<br />
<br />
===Kyudong Choi (UW Madison)===<br />
''Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations''<br />
<br />
We study weak solutions of the 3D Navier-Stokes equations with L^2 initial data. We <br />
prove that k-th derivative of weak solutions is locally integrable in space-time for any <br />
real k such that 1 < k < 3. Up to now, only the second derivative was known to be locally <br />
integrable by standard parabolic regularization. We also present sharp estimates of those <br />
quantities in weak-L^{4/(k+1)} locally. These estimates depend only on the L^2 norm of <br />
the initial data and on the domain of integration. Moreover, they are valid even for k >= <br />
3 as long as we have a smooth solution. The proof uses a standard approximation of <br />
Navier-Stokes from Leray and a blow-up techniques. The local study is based on De Giorgi <br />
techniques with a new pressure decomposition. To handle the non-locality of fractional <br />
Laplacians, Hardy space and Maximal functions are introduced. This is joint work with A. <br />
Vasseur.<br />
<br />
<br />
===Yao Yao (UW Madison)===<br />
''Confinement for nonlocal interaction equation with repulsive-attractive kernels''<br />
<br />
In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.<br />
<br />
<br />
<br />
<br />
===Myoungjean Bae (POSTECH)===<br />
''Transonic shocks for Euler-Poisson system and related problems''<br />
<br />
Abstract: Euler-Poisson system models various physical phenomena<br />
including the propagation of electrons in submicron semiconductor<br />
devices and plasmas, and the biological transport of ions for channel<br />
proteins. I will explain difference between Euler system and<br />
Euler-Poisson system and mathematical difficulties arising due to this<br />
difference. And, recent results about subsonic flow and transonic flow<br />
for Euler-Poisson system will be presented. This talk is based on<br />
collaboration with Ben Duan and Chunjing Xie.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=4742PDE Geometric Analysis seminar2012-11-26T19:26:34Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|-<br />
|November 26<br />
|[http://www.math.wisc.edu/~kchoi/ Kyudong Choi (UW Madison)]<br />
|[[#Kyudong Choi (UW Madison)|<br />
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations]]<br />
|local<br />
|-<br />
|-<br />
|December 10<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
Confinement for nonlocal interaction equation with repulsive-attractive kernels]]<br />
|local<br />
|-<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
<br />
<br />
===Kyudong Choi (UW Madison)===<br />
''Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations''<br />
<br />
We study weak solutions of the 3D Navier-Stokes equations with L^2 initial data. We <br />
prove that k-th derivative of weak solutions is locally integrable in space-time for any <br />
real k such that 1 < k < 3. Up to now, only the second derivative was known to be locally <br />
integrable by standard parabolic regularization. We also present sharp estimates of those <br />
quantities in weak-L^{4/(k+1)} locally. These estimates depend only on the L^2 norm of <br />
the initial data and on the domain of integration. Moreover, they are valid even for k >= <br />
3 as long as we have a smooth solution. The proof uses a standard approximation of <br />
Navier-Stokes from Leray and a blow-up techniques. The local study is based on De Giorgi <br />
techniques with a new pressure decomposition. To handle the non-locality of fractional <br />
Laplacians, Hardy space and Maximal functions are introduced. This is joint work with A. <br />
Vasseur.<br />
<br />
<br />
===Yao Yao (UW Madison)===<br />
''Confinement for nonlocal interaction equation with repulsive-attractive kernels''<br />
<br />
In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=4627PDE Geometric Analysis seminar2012-10-30T16:59:29Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|-<br />
|November 12<br />
|[http://www.math.wisc.edu/~kchoi/ Kyudong Choi (UW Madison)]<br />
|[[#Kyudong Choi (UW Madison)|<br />
Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations]]<br />
|local<br />
|-<br />
|-<br />
|November 19<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
Confinement for nonlocal interaction equation with repulsive-attractive kernels]]<br />
|local<br />
|-<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
===Kyudong Choi (UW Madison)===<br />
''Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations''<br />
<br />
We study weak solutions of the 3D Navier-Stokes equations with L^2 initial data. We <br />
prove that k-th derivative of weak solutions is locally integrable in space-time for any <br />
real k such that 1 < k < 3. Up to now, only the second derivative was known to be locally <br />
integrable by standard parabolic regularization. We also present sharp estimates of those <br />
quantities in weak-L^{4/(k+1)} locally. These estimates depend only on the L^2 norm of <br />
the initial data and on the domain of integration. Moreover, they are valid even for k >= <br />
3 as long as we have a smooth solution. The proof uses a standard approximation of <br />
Navier-Stokes from Leray and a blow-up techniques. The local study is based on De Giorgi <br />
techniques with a new pressure decomposition. To handle the non-locality of fractional <br />
Laplacians, Hardy space and Maximal functions are introduced. This is joint work with A. <br />
Vasseur.<br />
<br />
===Yao Yao (UW Madison)===<br />
''Confinement for nonlocal interaction equation with repulsive-attractive kernels''<br />
<br />
In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=4611PDE Geometric Analysis seminar2012-10-26T19:37:25Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|-<br />
|November 12<br />
|[http://www.math.wisc.edu/~kchoi/ Kyudong Choi (UW Madison)]<br />
|[[#Kyudong Choi (UW Madison)|<br />
TBA]]<br />
|local<br />
|-<br />
|-<br />
|November 19<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
Confinement for nonlocal interaction equation with repulsive-attractive kernels]]<br />
|local<br />
|-<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
===Yao Yao (UW Madison)===<br />
''Confinement for nonlocal interaction equation with repulsive-attractive kernels''<br />
<br />
In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=4609PDE Geometric Analysis seminar2012-10-26T19:15:13Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|November 19<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
Confinement for nonlocal interaction equation with repulsive-attractive kernels]]<br />
|local<br />
|-<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
===Yao Yao (UW Madison)===<br />
''Confinement for nonlocal interaction equation with repulsive-attractive kernels''<br />
<br />
In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=4467PDE Geometric Analysis seminar2012-10-02T19:45:15Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|November 20<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
Confinement for nonlocal interaction equation with repulsive-attractive kernels]]<br />
|local<br />
|-<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
===Yao Yao (UW Madison)===<br />
''Confinement for nonlocal interaction equation with repulsive-attractive kernels''<br />
<br />
In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=4466PDE Geometric Analysis seminar2012-10-02T19:44:34Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|November 20<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
"Confinement for nonlocal interaction equation with repulsive-attractive kernels"]]<br />
|local<br />
|-<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
===Yao Yao (UW Madison)===<br />
"Confinement for nonlocal interaction equation with repulsive-attractive kernels"<br />
<br />
In this talk, we consider sets of particles interacting pairwise via a potential W, which is repulsive at short distances, but attractive at long distances. The main question we consider is whether an initially compactly supported configuration remains compactly supported for all times, regardless of the number of particles. We improved the sufficient conditions on the potential W for the above confinement property to hold. This is a joint work with Jose Carrillo and Daniel Balague.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=4465PDE Geometric Analysis seminar2012-10-02T19:22:11Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|November 20<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
TBA]]<br />
|local<br />
|-<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
===Yao Yao (UW Madison)===<br />
TBA</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=4464PDE Geometric Analysis seminar2012-10-02T19:21:45Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
= Seminar Schedule Fall 2012 =<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~bwang/ Bing Wang (UW Madison)]<br />
|[[#Bing Wang (UW Madison)|<br />
On the regularity of limit space]]<br />
|local<br />
|-<br />
|-<br />
|October 15<br />
|[http://www.math.umn.edu/~polacik/ Peter Polacik (University of Minnesota)]<br />
|[[#Peter Polacik (University of Minnesota)|<br />
Exponential separation between positive and sign-changing solutions and its applications]]<br />
|Zlatos<br />
|-<br />
|September 17<br />
|[http://www.math.wisc.edu/~yaoyao/ Yao Yao (UW Madison)]<br />
|[[#Yao Yao (UW Madison)|<br />
TBA]]<br />
|local<br />
|-<br />
|-<br />
|}<br />
<br />
= Abstracts =<br />
===Bing Wang (UW Madison)===<br />
''On the regularity of limit space''<br />
<br />
This is a joint work with Gang Tian. <br />
In this talk, we will discuss how to improve regularity of the limit space by Ricci flow.<br />
We study the structure of the limit space of a sequence of almost Einstein<br />
manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such<br />
manifolds are the initial manifolds of some normalized Ricci flows whose scalar<br />
curvatures are almost constants over space-time in the L1-sense, Ricci curvatures<br />
are bounded from below at the initial time. Under the non-collapsed condition,<br />
we show that the limit space of a sequence of almost Einstein manifolds has most<br />
properties which is known for the limit space of Einstein manifolds. As applications,<br />
we can apply our structure results to study the properties of K¨ahler manifolds.<br />
<br />
<br />
===Peter Polacik (University of Minnesota)===<br />
'' Exponential separation between positive and sign-changing solutions and its applications''<br />
<br />
In linear nonautonomous second-order parabolic equations, the exponential separation refers to the exponential decay of any sign-changing solution relative to any positive solution. In this lecture, after summarizing key results on exponential separation, we show how it can be effectively used in studies of some nonlinear parabolic problems.<br />
<br />
===Yao Yao (UW Madison)===<br />
TBA</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=3812PDE Geometric Analysis seminar2012-04-21T04:23:46Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
== Seminar Schedule Spring 2012 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (UCLA)|<br />
Degenerate diffusion with nonlocal aggregation: behavior of solutions]]<br />
|Kiselev<br />
|-<br />
|March 12<br />
| Xuan Hien Nguyen (Iowa State)<br />
|[[#Xuan Hien Nguyen (Iowa State)|<br />
Gluing constructions for solitons and self-shrinkers under mean curvature flow]]<br />
|Angenent<br />
|-<br />
|March 21(Wednesday!), Room 901 Van Vleck<br />
|Nestor Guillen (UCLA)<br />
|[[#Nestor Guillen (UCLA)|<br />
The local geometry of maps with c-convex potentials]]<br />
|Feldman<br />
|-<br />
|March 26<br />
|Vlad Vicol (University of Chicago)<br />
|[[#Vlad Vicol (U Chicago)|<br />
Shape dependent maximum principles and applications]]<br />
|Kiselev<br />
|-<br />
|April 9<br />
|Charles Smart (MIT) <br />
|[[#Charles Smart (MIT)|<br />
PDE methods for the Abelian sandpile<br />
]]<br />
|Seeger<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jiahong Wu (Oklahoma State)|<br />
The 2D Boussinesq equations with partial dissipation]]<br />
|Kiselev<br />
|-<br />
|April 23<br />
|Joana Oliveira dos Santos Amorim (Universite Paris Dauphine)<br />
|[[#Joana Oliveira dos Santos Amorim (Universite Paris Dauphine)|<br />
A geometric look on Aubry-Mather theory and a theorem of Birkhoff]]<br />
|Bolotin<br />
|-<br />
|April 27 (Colloquium. Friday at 4pm, in Van Vleck B239)<br />
|Gui-Qiang Chen (Oxford) <br />
|[[#Gui-Qiang Chen (Oxford) |<br />
Nonlinear Partial Differential Equations of Mixed Type ]]<br />
|Feldman<br />
|-<br />
|May 14<br />
|Jacob Glenn-Levin (UT Austin)<br />
|[[#Jacob Glenn-Levin (UT Austin)|<br />
Incompressible Boussinesq equations in borderline Besov spaces]]<br />
|Kiselev<br />
|}<br />
<br />
==Abstracts==<br />
<br />
===Yao Yao (UCLA)===<br />
''Degenerate diffusion with nonlocal aggregation: behavior of solutions''<br />
<br />
The Patlak-Keller-Segel (PKS) equation models the collective motion of<br />
cells which are attracted by a self-emitted chemical substance. While the<br />
global well-posedness and finite-time blow up criteria are well known, the<br />
asymptotic behaviors of solutions are not completely clear. In this talk I<br />
will present some results on the asymptotic behavior of solutions when<br />
there is global existence. The key tools used in the paper are<br />
maximum-principle type arguments as well as estimates on mass concentration<br />
of solutions. This is a joint work with Inwon Kim.<br />
<br />
===Xuan Hien Nguyen (Iowa State)===<br />
<br />
''Gluing constructions for solitons and self-shrinkers under mean curvature flow''<br />
<br />
In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.<br />
<br />
===Nestor Guillen (UCLA)===<br />
<br />
We consider the Monge-Kantorovich problem, which consists in<br />
transporting a given measure into another "target" measure in a way<br />
that minimizes the total cost of moving each unit of mass to its new<br />
location. When the transport cost is given by the square of the<br />
distance between two points, the optimal map is given by a convex<br />
potential which solves the Monge-Ampère equation, in general, the<br />
solution is given by what is called a c-convex potential. In recent<br />
work with Jun Kitagawa, we prove local Holder estimates of optimal<br />
transport maps for more general cost functions satisfying a<br />
"synthetic" MTW condition, in particular, the proof does not really<br />
use the C^4 assumption made in all previous works. A similar result<br />
was recently obtained by Figalli, Kim and McCann using different<br />
methods and assuming strict convexity of the target.<br />
<br />
===Charles Smart (MIT)===<br />
<br />
''PDE methods for the Abelian sandpile''<br />
<br />
Abstract: The Abelian sandpile growth model is a deterministic<br />
diffusion process for chips placed on the $d$-dimensional integer<br />
lattice. One of the most striking features of the sandpile is that it<br />
appears to produce terminal configurations converging to a peculiar<br />
lattice. One of the most striking features of the sandpile is that it<br />
appears to produce terminal configurations converging to a peculiar<br />
fractal limit when begun from increasingly large stacks of chips at<br />
the origin. This behavior defied explanation for many years until<br />
viscosity solution theory offered a new perspective. This is joint<br />
work with Lionel Levine and Wesley Pegden.<br />
<br />
===Vlad Vicol (University of Chicago)===<br />
<br />
Title: Shape dependent maximum principles and applications<br />
<br />
Abstract: We present a non-linear lower bound for the fractional Laplacian, when<br />
evaluated at extrema of a function. Applications to the global well-posedness of active<br />
scalar equations arising in fluid dynamics are discussed. This is joint work with P.<br />
Constantin.<br />
<br />
<br />
===Jiahong Wu (Oklahoma State)===<br />
<br />
"The 2D Boussinesq equations with partial dissipation"<br />
<br />
The Boussinesq equations concerned here model geophysical flows such<br />
as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq<br />
equations serve as a lower-dimensional model of the 3D hydrodynamics<br />
equations. In fact, the 2D Boussinesq equations retain some key features<br />
of the 3D Euler and the Navier-Stokes equations such as the vortex stretching<br />
mechanism. The global regularity problem on the 2D Boussinesq equations<br />
with partial dissipation has attracted considerable attention in the last few years.<br />
In this talk we will summarize recent results on various cases of partial dissipation,<br />
present the work of Cao and Wu on the 2D Boussinesq equations with vertical<br />
dissipation and vertical thermal diffusion, and explain the work of Chae and Wu on<br />
the critical Boussinesq equations with a logarithmically singular velocity.<br />
<br />
<br />
===Joana Oliveira dos Santos Amorim (Universit\'e Paris Dauphine)===<br />
<br />
"A geometric look on Aubry-Mather theory and a theorem of Birkhoff"<br />
<br />
Given a Tonelli Hamiltonian $H:T^*M \lto \Rm$ in the cotangent bundle of a compact manifold $M$, <br />
we can study its dynamic using the Aubry and Ma\~n\'e sets defined by Mather. <br />
In this talk we will explain their importance and give a new geometric definition <br />
which allows us to understand their property of symplectic invariance. <br />
Moreover, using this geometric definition, we will show that an exact <br />
Lipchitz Lagrangian manifold isotopic to a graph which is invariant <br />
by the flow of a Tonelli Hamiltonian is itself a graph. <br />
This result, in its smooth form, was a conjecture of Birkhoff.<br />
<br />
<br />
===Gui-Qiang Chen (Oxford) ===<br />
<br />
"Nonlinear Partial Differential Equations of Mixed Type"<br />
<br />
Many nonlinear partial differential equations arising in mechanics and geometry naturally are of mixed hyperbolic-elliptic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear partial differential equations of mixed type. Important examples include<br />
shock reflection-diffraction problems in fluid mechanics (the Euler equations), isometric embedding problems in in differential geometry (the Gauss-Codazzi-Ricci equations),<br />
among many others. In this talk we will present natural connections of nonlinear partial differential equations with these longstanding problems and will discuss some recent developments in the analysis of these nonlinear equations through the examples with emphasis on identifying/developing mathematical approaches, ideas, and techniques to deal with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.<br />
This talk will be based mainly on the joint work correspondingly with Mikhail Feldman, Marshall Slemrod, as well as Myoungjean Bae and Dehua Wang. <br />
<br />
<br />
===Jacob Glenn-Levin (UT Austin)===<br />
<br />
We consider the Boussinesq equations, which may be thought of as inhomogeneous, <br />
incompressible Euler equations, where the inhomogeneous term is a scalar quantity, <br />
typically density or temperature, governed by a convection-diffusion equation. I will <br />
discuss local- and global-in-time well-posedness results for the incompressible 2D <br />
Boussinesq equations, assuming the density equation has nonzero diffusion and that the <br />
initial data belongs in a Besov-type space.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=Applied_and_Computational_Mathematics&diff=3766Applied and Computational Mathematics2012-04-19T17:14:16Z<p>Kiselev: </p>
<hr />
<div>__NOTOC__<br />
[[Image:jet.jpg|link=http://www.math.wisc.edu/~jeanluc|frame|jet striking an inclined plane]]<br />
<br />
= '''Applied Mathematics at UW-Madison''' =<br />
<br />
Welcome to the Applied Mathematics Group at the University of Wisconsin, Madison. Our faculty members, postdoctoral fellows, and students are involved in a variety of research projects, including fluid dynamics, partial and stochastic differential equations, scientific computing, biology, biochemistry, and topology.<br />
<br />
<br><br />
<br />
== News and opportunities ==<br />
<br />
* Funding opportunity for a '''graduate student''' to study mathematics of fluids - regularity and mixing, more for information check http://www.math.wisc.edu/~kiselev/graduate.html (contact [http://www.math.wisc.edu/~kiselev Sasha Kiselev], supported by [http://nsf.gov NSF]). <!-- Added by kiselev 2012-04-19 --><br />
<br />
* Funding opportunity for a '''graduate student''' to study chemotaxis and applications in mathematical biology, more for information check http://www.math.wisc.edu/~kiselev/graduate.html (contact [http://www.math.wisc.edu/~kiselev Sasha Kiselev], supported by [http://nsf.gov NSF]). <!-- Added by kiselev 2012-04-19 --><br />
<br />
* '''[http://maeresearch.ucsd.edu/spagnolie/ Saverio Spagnolie]''' has accepted a position as a tenure-track assistant professor in our department. Saverio will join us this Fall. Welcome to the group, Saverio! <!-- Added by jeanluc 2012-03-15 --><br />
<br />
* '''Bokai Yan''' (PhD student with Shi Jin) graduated in Fall 2011 and is now a postdoc at UCLA. <!-- Added by jeanluc 2012-02-05 --><br />
<br />
* Funding opportunity for a '''graduate student''' to study '''persistence and multistability in biological networks''' (contact [http://www.math.wisc.edu/~craciun Gheorghe Craciun], supported by [http://nih.gov NIH]). <!-- Added by craciun 2011-09-01 --><br />
<br />
* Funding opportunity for a '''graduate student''' to study '''mathematical analysis of mass spectrometry data and proteomics''' (contact [http://www.math.wisc.edu/~craciun Gheorghe Craciun], supported by [http://nsf.gov NSF]). <!-- Added by craciun 2011-09-01 --><br />
<br />
* '''Li Wang''' (PhD student with Leslie Smith) graduated and has a job at [http://www.epic.com/ Epic]. <!-- Added by jeanluc 2011-09-01 --><br />
<br />
* Funding opportunity for a '''graduate student''' to study '''waves in geophysical flows and tropical cyclogenesis''' (contact [http://www.math.wisc.edu/~lsmith Leslie Smith], supported by [http://nsf.gov NSF]). <!-- Added by jeanluc 2011-09-01 --><br />
<br />
* Funding opportunity for a '''graduate student''' to study '''nonlinear critical layers and exact coherent states in turbulent shear flows''' (contact [http://www.math.wisc.edu/~waleffe Fabian Waleffe], supported by [http://nsf.gov NSF]). <!-- Added by Wally 2011-09-02 --><br />
<br />
<br><br />
<br />
== Seminars ==<br />
<br />
''organized by Applied Math''<br />
<br />
* [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied and Computational Math Seminar] (Fridays at 2:25pm, VV 901)<br />
* [http://www.math.wisc.edu/wiki/index.php/Applied/GPS GPS Applied Math Seminar] (Mondays at 2:25pm, B211 VV)<br />
* Joint Math/Atmospheric & Oceanic Sciences Informal Seminar (Thursdays at 3:45 pm, AOS 811)<br />
<br />
<br />
''other seminar series of interest''<br />
<br />
* [http://www.math.wisc.edu/wiki/index.php/Colloquia Mathematics Colloquium] (Fridays at 4:00pm, VV B239)<br />
* [http://sprott.physics.wisc.edu/Chaos-Complexity/ Chaos and Complex Systems Seminar] (Tuesdays at 12:05pm, 4274 Chamberlin Hall)<br />
* [http://www.engr.wisc.edu/news/events/index.phtml?start=2011-09-02&range=3650&search=Rheology RRC Lecture] (Fridays at 12:05pm, 1800 Engineering Hall)<br />
* [http://www.physics.wisc.edu/twap/view.php?name=PDC Physics Department Colloquium] (Fridays at 3:30 pm; 2241 Chamberlin Hall)<br />
<br />
<br><br />
<br />
== Tenured and tenure-track faculty ==<br />
<br />
[http://www.math.wisc.edu/~anderson/ David Anderson:] (Duke, 2005) probability and stochastic processes, computational methods for stochastic processes, mathematical/systems biology.<br />
<br />
[http://www.math.wisc.edu/~angenent/ Sigurd Angenent:] (Leiden, 1986) partial differential equations.<br />
<br />
[http://www.math.wisc.edu/~assadi/ Amir Assadi:] (Princeton, 1978) computational & mathematical models in molecular biology & neuroscience.<br />
<br />
[http://www.math.wisc.edu/~boston/ Nigel Boston:] (Harvard, 1987) algebraic number theory, group theory, arithmetic geometry, computational algebra, coding theory, cryptography, and other applications of algebra to electrical engineering. <br />
<br />
[http://www.math.wisc.edu/~craciun/ Gheorghe Craciun:] (Ohio State, 2002) mathematical biology, biochemical networks, biological interaction networks.<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich:] (Tel Aviv, 2006) Representation theory of groups, algebraic geometry, applications to signal Processing, structural biology, mathematical physics.<br />
<br />
[http://www.math.wisc.edu/~jin/ Shi Jin:] (Arizona, 1991) applied & computational mathematics.<br />
<br />
[http://www.math.wisc.edu/~kiselev/ Alex (Sasha) Kiselev:] (CalTech, 1997) partial differential equations, Fourier analysis<br />
and applications in fluid mechanics, combustion, mathematical biology and Schr&ouml;dinger operators.<br />
<br />
[http://www.math.wisc.edu/~maribeff/ Gloria Mari-Beffa:] (Minnesota, 1991) differential geometry, applied math.<br />
<br />
[http://www.math.wisc.edu/~milewski/ Paul Milewski:] (MIT, 1993) applied mathematics, fluid dynamics.<br />
<br />
[http://www.math.wisc.edu/~mitchell/ Julie Mitchell:] (Berkeley, 1998) computational mathematics, structural biology.<br />
<br />
[http://www.math.wisc.edu/~rossmani/ James Rossmanith:] (Washington, 2002) computational mathematics, hyperbolic conservation laws, plasma physics.<br />
<br />
[http://www.math.wisc.edu/~lsmith/ Leslie Smith:] (MIT, 1988) applied mathematics. Waves and coherent structures in oceanic and atmospheric flows. <br />
<br />
[http://maeresearch.ucsd.edu/spagnolie/ Saverio Spagnolie:] (Courant, 2008) fluid dynamics, biological locomotion, computational mathematics.<br />
<br />
[http://www.math.wisc.edu/~stech/ Sam Stechmann:] (Courant, 2008) fluid dynamics, atmospheric science, computational mathematics.<br />
<br />
[http://www.math.wisc.edu/~jeanluc/ Jean-Luc Thiffeault:] (Texas, 1998) fluid dynamics, mixing, biological swimming and mixing, topological dynamics.<br />
<br />
[http://www.math.wisc.edu/~waleffe/ Fabian Waleffe:] (MIT, 1989) applied and computational mathematics. Fluid dynamics, hydrodynamic instabilities. Turbulence and unstable coherent flows.<br />
<br />
[http://www.math.wisc.edu/~zlatos/ Andrej Zlatos:] (Caltech, 2003) partial differential equations, combustion, fluid dynamics, Schrödinger operators, orthogonal polynomials<br />
<br />
<br><br />
<br />
== Postdoctoral fellows ==<br />
<br />
<!-- [http://www.math.wisc.edu/~dwei/ Dongming Wei:] (Maryland, 2007) nonlinear partial differential equations, applied analysis, and numerical computation. --><br />
<br />
[http://www.math.wisc.edu/~hernande Gerardo Hernandez-Duenas:] (Michigan, 2011)<br />
<br />
<br><br />
<br />
== Current Graduate Students ==<br />
<br />
Adel Ardalan: Student of Amir Assadi.<br />
<br />
[http://www.math.wisc.edu/~blackman/ Claire Blackman:] Student of Jean-Luc Thiffeault.<br />
<br />
[http://www.math.wisc.edu/~boonkasa/ Anekewit (Tete) Boonkasame:] Student of Paul Milewski.<br />
<br />
Yongtao Cheng: Student of James Rossmanith.<br />
<br />
[http://vv811a.math.wisc.edu/index.html/index.php/component/content/article/40 Hesam Dashti:] Student of Amir Assadi.<br />
<br />
[http://www.math.wisc.edu/~qdeng/ Qiang Deng:] Student of Leslie Smith.<br />
<br />
[http://vv811a.math.wisc.edu/index.html/index.php/component/content/article/33 Alireza Fotuhi:] Student of Amir Assadi.<br />
<br />
[http://www.math.wisc.edu/~jefferis/ Leland Jefferis:] Student of Shi Jin.<br />
<br />
[http://www.math.wisc.edu/~ejohnson/ E. Alec Johnson:] Student of James Rossmanith.<br />
<br />
[http://vv811a.math.wisc.edu/index.html/index.php/component/content/article/15 Mohammad Khabbazian:] Student of Amir Assadi.<br />
<br />
[http://www.math.wisc.edu/~koyama/ Masanori (Maso) Koyama:] Student of David Anderson.<br />
<br />
[http://www.math.wisc.edu/~leili/ Lei Li:] Student of Shi Jin.<br />
<br />
[http://www.math.wisc.edu/~qinli/ Qin Li:] Student of Shi Jin.<br />
<br />
Peter Mueller: Student of Jean-Luc Thiffeault.<br />
<br />
[http://www.math.wisc.edu/~pqi/ Peng Qi:] Student of Shi Jin.<br />
<br />
[http://vv811a.math.wisc.edu/index.html/index.php/component/content/article/16 Arash Sangari:] Student of Amir Assadi.<br />
<br />
[http://www.math.wisc.edu/~seal/ David Seal:] Student of James Rossmanith.<br />
<br />
Ebru Selin Selen: Student of Amir Assadi.<br />
<br />
[http://www.math.wisc.edu/~matz/ Sarah Tumasz:] Student of Jean-Luc Thiffeault.<br />
<br />
Li Wang: Student of Leslie Smith.<br />
<br />
[http://www.math.wisc.edu/~wangli/ Li (Aug) Wang:] Student of Shi Jin.<br />
<br />
Zhan Wang: Student of Paul Milewski.<br />
<br />
Qian You: Student of Sigurd Angenent.<br />
<br />
[http://www.math.wisc.edu/~zhou/ Zhennan Zhou:] Student of Shi Jin.<br />
<br />
<!-- Past students: --><br />
<!-- [http://www.math.wisc.edu/~hu/ Jingwei Hu:] Student of Shi Jin. --><br />
<!-- [http://www.math.wisc.edu/~yan/ Bokai Yan:] Student of Shi Jin. --><br />
<br />
<br><br />
<br />
== Graduate course offerings ==<br />
<br />
=== Spring 2012 ===<br />
* Math 714: [http://www.math.wisc.edu/math-714-scientific-computing Methods of Computational Math I] (S. Stechmann)<br />
<br />
=== [http://www.math.wisc.edu/graduate/gcourses_fall Fall 2012] ===<br />
<br />
* Math 606: Mathematical Methods for Structural Biology (Julie Mitchell)<br />
* Math 632: Introduction to Stochastic Processes (David Anderson)<br />
* Math 703: Methods of Applied Mathematics 1 (Leslie Smith)<br />
* Math 705: Mathematical Fluid Dynamics (Saverio Spagnolie)<br />
* Math 714: Methods of Computational Math I (Shi Jin)<br />
* Math 801: Topics in Applied Mathematics -- Mathematical Aspects of Mixing (Jean Luc Thiffeault)<br />
* Math 842: Topics in Applied Algebra for EE/Math/CS students (Shamgar Gurevich)<br />
<br />
<!--<br />
=== Fall 2011 ===<br />
<br />
* Math 605: [http://www.math.wisc.edu/math-727-calculus-variations-0 Stochastic Methods for Biology] (D. Anderson)<br />
* Math 703: [http://www.math.wisc.edu/math-703-methods-applied-mathematics-i Methods of Applied Mathematics II] (L. Smith)<br />
* Math 707: [http://www.math.wisc.edu/math707-ema700-theory-elasticity Theory of Elasticity] (F. Waleffe)<br />
* Math 714: [http://www.math.wisc.edu/math-714-scientific-computing Methods of Computational Math I] (J. Mitchell)<br />
* Math 801: [http://www.math.wisc.edu/801-waves-fluids Comp Math Applied to Biology] (A. Assadi)<br />
* Math 837: [http://www.math.wisc.edu/math-837-topics-numerical-analysis Topics in Numerical Analysis] (S. Jin)<br />
--><br />
<br />
<!--<br />
Spring 2011:<br />
* Math 609: [https://www.math.wisc.edu/609-mathematical-methods-systems-biology Mathematical Methods for Systems Biology] (G. Craciun)<br />
* Math 704: [https://www.math.wisc.edu/704-methods-applied-mathematics-2 Methods of Applied Mathematics II] (S. Stechmann)<br />
* Math/CS 715: [https://www.math.wisc.edu/715-methods-computational-math-ii Methods of Computational Math II] (S. Jin)<br />
* Math 801: [https://www.math.wisc.edu/math-801-hydrodynamic-instabilities-chaos-and-turbulence Hydrodynamic Instabilities, Chaos and Turbulence] (F. Waleffe)<br />
* Math 826: [https://www.math.wisc.edu/826-Functional-Analysis Partial Differential Equations in Fluids and Biology] (A. Kiselev)<br />
* Math/CS 837: [https://www.math.wisc.edu/837-Numerical-Analysis Numerical Methods for Hyperbolic PDEs] (J. Rossmanith)<br />
--><br />
<br />
<br><br />
<br />
----<br />
Return to the [http://www.math.wisc.edu/wiki/index.php Mathematics Department Wiki Page]<br />
<br />
[http://www3.clustrmaps.com/stats/maps-no_clusters/www.math.wisc.edu-wiki-index.php-Applied-thumb.jpg Locations of visitors to this page] ([http://www3.clustrmaps.com/user/195f39ef Clustermaps])</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=3644PDE Geometric Analysis seminar2012-03-13T16:51:35Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
== Seminar Schedule Spring 2012 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (UCLA)|<br />
Degenerate diffusion with nonlocal aggregation: behavior of solutions]]<br />
|Kiselev<br />
|-<br />
|March 12<br />
| Xuan Hien Nguyen (Iowa State)<br />
|[[#Xuan Hien Nguyen (Iowa State)|<br />
Gluing constructions for solitons and self-shrinkers under mean curvature flow]]<br />
|Angenent<br />
|-<br />
|March 19<br />
|Nestor Guillen (UCLA)<br />
|[[#Nestor Guillen (UCLA)|<br />
The local geometry of maps with c-convex potentials]]<br />
|Feldman<br />
|-<br />
|March 26<br />
|Vlad Vicol (University of Chicago)<br />
|[[#Vlad Vicol (U Chicago)|<br />
Shape dependent maximum principles and applications]]<br />
|Kiselev<br />
|-<br />
|April 9<br />
|Charles Smart (MIT) <br />
|[[#Charles Smart (MIT)|<br />
TBA]]<br />
|Seeger<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jiahong Wu (Oklahoma State)|<br />
The 2D Boussinesq equations with partial dissipation]]<br />
|Kiselev<br />
|-<br />
|May 14<br />
|Jacob Glenn-Levin (UT Austin)<br />
|[[#Jacob Glenn-Levin (UT Austin)|<br />
TBA]]<br />
|Kiselev<br />
|}<br />
<br />
==Abstracts==<br />
<br />
===Yao Yao (UCLA)===<br />
''Degenerate diffusion with nonlocal aggregation: behavior of solutions''<br />
<br />
The Patlak-Keller-Segel (PKS) equation models the collective motion of<br />
cells which are attracted by a self-emitted chemical substance. While the<br />
global well-posedness and finite-time blow up criteria are well known, the<br />
asymptotic behaviors of solutions are not completely clear. In this talk I<br />
will present some results on the asymptotic behavior of solutions when<br />
there is global existence. The key tools used in the paper are<br />
maximum-principle type arguments as well as estimates on mass concentration<br />
of solutions. This is a joint work with Inwon Kim.<br />
<br />
===Xuan Hien Nguyen (Iowa State)===<br />
<br />
''Gluing constructions for solitons and self-shrinkers under mean curvature flow''<br />
<br />
In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.<br />
<br />
===Nestor Guillen (UCLA)===<br />
<br />
We consider the Monge-Kantorovich problem, which consists in<br />
transporting a given measure into another "target" measure in a way<br />
that minimizes the total cost of moving each unit of mass to its new<br />
location. When the transport cost is given by the square of the<br />
distance between two points, the optimal map is given by a convex<br />
potential which solves the Monge-Ampère equation, in general, the<br />
solution is given by what is called a c-convex potential. In recent<br />
work with Jun Kitagawa, we prove local Holder estimates of optimal<br />
transport maps for more general cost functions satisfying a<br />
"synthetic" MTW condition, in particular, the proof does not really<br />
use the C^4 assumption made in all previous works. A similar result<br />
was recently obtained by Figalli, Kim and McCann using different<br />
methods and assuming strict convexity of the target.<br />
<br />
===Charles Smart (MIT)===<br />
<br />
TBA<br />
<br />
===Vlad Vicol (University of Chicago)===<br />
<br />
Title: Shape dependent maximum principles and applications<br />
<br />
Abstract: We present a non-linear lower bound for the fractional Laplacian, when<br />
evaluated at extrema of a function. Applications to the global well-posedness of active<br />
scalar equations arising in fluid dynamics are discussed. This is joint work with P.<br />
Constantin.<br />
<br />
<br />
===Jiahong Wu (Oklahoma State)===<br />
<br />
"The 2D Boussinesq equations with partial dissipation"<br />
<br />
The Boussinesq equations concerned here model geophysical flows such<br />
as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq<br />
equations serve as a lower-dimensional model of the 3D hydrodynamics<br />
equations. In fact, the 2D Boussinesq equations retain some key features<br />
of the 3D Euler and the Navier-Stokes equations such as the vortex stretching<br />
mechanism. The global regularity problem on the 2D Boussinesq equations<br />
with partial dissipation has attracted considerable attention in the last few years.<br />
In this talk we will summarize recent results on various cases of partial dissipation,<br />
present the work of Cao and Wu on the 2D Boussinesq equations with vertical<br />
dissipation and vertical thermal diffusion, and explain the work of Chae and Wu on<br />
the critical Boussinesq equations with a logarithmically singular velocity.<br />
<br />
===Jacob Glenn-Levin (UT Austin)===<br />
<br />
TBA</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=3643PDE Geometric Analysis seminar2012-03-13T16:50:58Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
== Seminar Schedule Spring 2012 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (UCLA)|<br />
Degenerate diffusion with nonlocal aggregation: behavior of solutions]]<br />
|Kiselev<br />
|-<br />
|March 12<br />
| Xuan Hien Nguyen (Iowa State)<br />
|[[#Xuan Hien Nguyen (Iowa State)|<br />
Gluing constructions for solitons and self-shrinkers under mean curvature flow]]<br />
|Angenent<br />
|-<br />
|March 19<br />
|Nestor Guillen (UCLA)<br />
|[[#Nestor Guillen (UCLA)|<br />
The local geometry of maps with c-convex potentials]]<br />
|Feldman<br />
|-<br />
|March 26<br />
|Vlad Vicol (University of Chicago)<br />
|[[#Vlad Vicol (U Chicago)|<br />
Shape dependent maximum principles and applications]]<br />
|Kiselev<br />
|-<br />
|April 9<br />
|Charles Smart (MIT) <br />
|[[#Charles Smart (MIT)|<br />
TBA]]<br />
|Seeger<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jiahong Wu (Oklahoma State)|<br />
The 2D Boussinesq equations with partial dissipation]]<br />
|Kiselev<br />
-<br />
|May 14<br />
|Jacob Glenn-Levin (UT Austin)<br />
|[[#Jacob Glenn-Levin (UT Austin)|<br />
TBA]]<br />
|Kiselev<br />
|}<br />
<br />
==Abstracts==<br />
<br />
===Yao Yao (UCLA)===<br />
''Degenerate diffusion with nonlocal aggregation: behavior of solutions''<br />
<br />
The Patlak-Keller-Segel (PKS) equation models the collective motion of<br />
cells which are attracted by a self-emitted chemical substance. While the<br />
global well-posedness and finite-time blow up criteria are well known, the<br />
asymptotic behaviors of solutions are not completely clear. In this talk I<br />
will present some results on the asymptotic behavior of solutions when<br />
there is global existence. The key tools used in the paper are<br />
maximum-principle type arguments as well as estimates on mass concentration<br />
of solutions. This is a joint work with Inwon Kim.<br />
<br />
===Xuan Hien Nguyen (Iowa State)===<br />
<br />
''Gluing constructions for solitons and self-shrinkers under mean curvature flow''<br />
<br />
In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.<br />
<br />
===Nestor Guillen (UCLA)===<br />
<br />
We consider the Monge-Kantorovich problem, which consists in<br />
transporting a given measure into another "target" measure in a way<br />
that minimizes the total cost of moving each unit of mass to its new<br />
location. When the transport cost is given by the square of the<br />
distance between two points, the optimal map is given by a convex<br />
potential which solves the Monge-Ampère equation, in general, the<br />
solution is given by what is called a c-convex potential. In recent<br />
work with Jun Kitagawa, we prove local Holder estimates of optimal<br />
transport maps for more general cost functions satisfying a<br />
"synthetic" MTW condition, in particular, the proof does not really<br />
use the C^4 assumption made in all previous works. A similar result<br />
was recently obtained by Figalli, Kim and McCann using different<br />
methods and assuming strict convexity of the target.<br />
<br />
===Charles Smart (MIT)===<br />
<br />
TBA<br />
<br />
===Vlad Vicol (University of Chicago)===<br />
<br />
Title: Shape dependent maximum principles and applications<br />
<br />
Abstract: We present a non-linear lower bound for the fractional Laplacian, when<br />
evaluated at extrema of a function. Applications to the global well-posedness of active<br />
scalar equations arising in fluid dynamics are discussed. This is joint work with P.<br />
Constantin.<br />
<br />
<br />
===Jiahong Wu (Oklahoma State)===<br />
<br />
"The 2D Boussinesq equations with partial dissipation"<br />
<br />
The Boussinesq equations concerned here model geophysical flows such<br />
as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq<br />
equations serve as a lower-dimensional model of the 3D hydrodynamics<br />
equations. In fact, the 2D Boussinesq equations retain some key features<br />
of the 3D Euler and the Navier-Stokes equations such as the vortex stretching<br />
mechanism. The global regularity problem on the 2D Boussinesq equations<br />
with partial dissipation has attracted considerable attention in the last few years.<br />
In this talk we will summarize recent results on various cases of partial dissipation,<br />
present the work of Cao and Wu on the 2D Boussinesq equations with vertical<br />
dissipation and vertical thermal diffusion, and explain the work of Chae and Wu on<br />
the critical Boussinesq equations with a logarithmically singular velocity.<br />
<br />
===Jacob Glenn-Levin (UT Austin)===<br />
<br />
TBA</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=3617PDE Geometric Analysis seminar2012-03-07T00:47:44Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
== Seminar Schedule Spring 2012 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (UCLA)|<br />
Degenerate diffusion with nonlocal aggregation: behavior of solutions]]<br />
|Kiselev<br />
|-<br />
|March 12<br />
| Xuan Hien Nguyen (Iowa State)<br />
|[[#Xuan Hien Nguyen (Iowa State)|<br />
Gluing constructions for solitons and self-shrinkers under mean curvature flow]]<br />
|Angenent<br />
|-<br />
|March 19<br />
|Nestor Guillen (UCLA)<br />
|[[#Nestor Guillen (UCLA)|<br />
TBA]]<br />
|Feldman<br />
|-<br />
|March 26<br />
|Vlad Vicol (University of Chicago)<br />
|[[#Vlad Vicol (U Chicago)|<br />
Shape dependent maximum principles and applications]]<br />
|Kiselev<br />
|-<br />
|April 9<br />
|Charles Smart (MIT) <br />
|[[#Charles Smart (MIT)|<br />
TBA]]<br />
|Seeger<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jiahong Wu (Oklahoma State)|<br />
The 2D Boussinesq equations with partial dissipation]]<br />
|Kiselev<br />
|}<br />
<br />
==Abstracts==<br />
<br />
===Yao Yao (UCLA)===<br />
''Degenerate diffusion with nonlocal aggregation: behavior of solutions''<br />
<br />
The Patlak-Keller-Segel (PKS) equation models the collective motion of<br />
cells which are attracted by a self-emitted chemical substance. While the<br />
global well-posedness and finite-time blow up criteria are well known, the<br />
asymptotic behaviors of solutions are not completely clear. In this talk I<br />
will present some results on the asymptotic behavior of solutions when<br />
there is global existence. The key tools used in the paper are<br />
maximum-principle type arguments as well as estimates on mass concentration<br />
of solutions. This is a joint work with Inwon Kim.<br />
<br />
===Xuan Hien Nguyen (Iowa State)===<br />
<br />
''Gluing constructions for solitons and self-shrinkers under mean curvature flow''<br />
<br />
In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.<br />
<br />
===Nestor Guillen (UCLA)===<br />
<br />
TBA<br />
<br />
===Charles Smart (MIT)===<br />
<br />
TBA<br />
<br />
===Vlad Vicol (University of Chicago)===<br />
<br />
Title: Shape dependent maximum principles and applications<br />
<br />
Abstract: We present a non-linear lower bound for the fractional Laplacian, when<br />
evaluated at extrema of a function. Applications to the global well-posedness of active<br />
scalar equations arising in fluid dynamics are discussed. This is joint work with P.<br />
Constantin.<br />
<br />
<br />
===Jiahong Wu (Oklahoma State)===<br />
<br />
"The 2D Boussinesq equations with partial dissipation"<br />
<br />
The Boussinesq equations concerned here model geophysical flows such<br />
as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq<br />
equations serve as a lower-dimensional model of the 3D hydrodynamics<br />
equations. In fact, the 2D Boussinesq equations retain some key features<br />
of the 3D Euler and the Navier-Stokes equations such as the vortex stretching<br />
mechanism. The global regularity problem on the 2D Boussinesq equations<br />
with partial dissipation has attracted considerable attention in the last few years.<br />
In this talk we will summarize recent results on various cases of partial dissipation,<br />
present the work of Cao and Wu on the 2D Boussinesq equations with vertical<br />
dissipation and vertical thermal diffusion, and explain the work of Chae and Wu on<br />
the critical Boussinesq equations with a logarithmically singular velocity.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=3517PDE Geometric Analysis seminar2012-02-15T22:32:00Z<p>Kiselev: </p>
<hr />
<div>The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
<br />
== Seminar Schedule Spring 2012 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (UCLA)|<br />
Degenerate diffusion with nonlocal aggregation: behavior of solutions]]<br />
|Kiselev<br />
|-<br />
|March 12<br />
| Xuan Hien Nguyen (Iowa State)<br />
|[[#Xuan Hien Nguyen (Iowa State)|<br />
Gluing constructions for solitons and self-shrinkers under mean curvature flow]]<br />
|Angenent<br />
|-<br />
|March 19<br />
|Nestor Guillen (UCLA)<br />
|[[#Nestor Guillen (UCLA)|<br />
TBA]]<br />
|Feldman<br />
|-<br />
|March 26<br />
|Vlad Vicol (University of Chicago)<br />
|[[#Vlad Vicol (U Chicago)|<br />
TBA]]<br />
|Kiselev<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jianhong Wu (Oklahoma)|<br />
TBA]]<br />
|Kiselev<br />
|}<br />
<br />
==Abstracts==<br />
<br />
===Yao Yao (UCLA)===<br />
''Degenerate diffusion with nonlocal aggregation: behavior of solutions''<br />
<br />
The Patlak-Keller-Segel (PKS) equation models the collective motion of<br />
cells which are attracted by a self-emitted chemical substance. While the<br />
global well-posedness and finite-time blow up criteria are well known, the<br />
asymptotic behaviors of solutions are not completely clear. In this talk I<br />
will present some results on the asymptotic behavior of solutions when<br />
there is global existence. The key tools used in the paper are<br />
maximum-principle type arguments as well as estimates on mass concentration<br />
of solutions. This is a joint work with Inwon Kim.<br />
<br />
===Xuan Hien Nguyen (Iowa State)===<br />
<br />
''Gluing constructions for solitons and self-shrinkers under mean curvature flow''<br />
<br />
In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.<br />
<br />
===Nestor Guillen (UCLA)===<br />
<br />
TBA<br />
<br />
===Vlad Vicol (University of Chicago)===<br />
<br />
TBA<br />
<br />
===Jiahong Wu (Oklahoma State)===<br />
<br />
"The 2D Boussinesq equations with partial dissipation"<br />
<br />
The Boussinesq equations concerned here model geophysical flows such<br />
as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq<br />
equations serve as a lower-dimensional model of the 3D hydrodynamics<br />
equations. In fact, the 2D Boussinesq equations retain some key features<br />
of the 3D Euler and the Navier-Stokes equations such as the vortex stretching<br />
mechanism. The global regularity problem on the 2D Boussinesq equations<br />
with partial dissipation has attracted considerable attention in the last few years.<br />
In this talk we will summarize recent results on various cases of partial dissipation,<br />
present the work of Cao and Wu on the 2D Boussinesq equations with vertical<br />
dissipation and vertical thermal diffusion, and explain the work of Chae and Wu on<br />
the critical Boussinesq equations with a logarithmically singular velocity.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=3407PDE Geometric Analysis seminar2012-02-01T01:13:50Z<p>Kiselev: </p>
<hr />
<div>= PDE and Geometric Analysis Seminar =<br />
<br />
The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
[[Previous PDE/GA seminars]]<br />
<br />
== Seminar Schedule Fall 2011 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Oct 3<br />
|Takis Souganidis (Chicago)<br />
|[[#Takis Souganidis (Chicago)|<br />
''Stochastic homogenization of the G-equation'']]<br />
|Armstrong<br />
<br />
|-<br />
|Oct 10<br />
|Scott Armstrong (UW-Madison)<br />
|[[#Scott Armstrong (UW-Madison)|<br />
''Partial regularity for fully nonlinear elliptic equations'']]<br />
|Local speaker<br />
<br />
|-<br />
|Oct 17<br />
|Russell Schwab (Carnegie Mellon)<br />
|[[#Russell Schwab (Carnegie Mellon)|<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)'']]<br />
|Armstrong<br />
|-<br />
|October 24 ( with Geometry/Topology seminar)<br />
|[http://math.univ-lyon1.fr/~ovsienko/ Valentin Ovsienko] (University of Lyon)<br />
|[[#Valentin Ovsienko (University of Lyon)|<br />
''The pentagram map and generalized friezes of Coxeter'']]<br />
|Marí Beffa<br />
|-<br />
|-<br />
|Oct 31<br />
|Adrian Tudorascu (West Virginia University)<br />
|[[#Adrian Tudorascu (West Virginia University)|<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space'']]<br />
|Feldman<br />
|-<br />
|Nov 7<br />
|James Nolen (Duke)<br />
|[[#James Nolen (Duke)|<br />
''Normal approximation for a random elliptic PDE'']]<br />
|Armstrong<br />
|-|-<br />
|Nov 21 (Joint with [http://www.math.wisc.edu/~seeger/curr.html Analysis] seminar)<br />
|Betsy Stovall (UCLA)<br />
|[[#Betsy Stovall (UCLA)| <br />
''Scattering for the cubic Klein--Gordon equation in two dimensions'']]<br />
|Seeger<br />
|-<br />
|Dec 5<br />
|Charles Smart (MIT)<br />
|[[#Charles Smart (MIT)|<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian'']]<br />
|Armstrong<br />
|}<br />
<br />
== Seminar Schedule Spring 2012 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (UCLA)|<br />
TBA]]<br />
|Kiselev<br />
|-<br />
|March 19<br />
|Nestor Guillen (UCLA)<br />
|[[#Nestor Guillen (UCLA)|<br />
TBA]]<br />
|Feldman<br />
|-<br />
|March 26<br />
|Vlad Vicol (University of Chicago)<br />
|[[#Vlad Vicol (U Chicago)|<br />
TBA]]<br />
|Kiselev<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jianhong Wu (Oklahoma)|<br />
TBA]]<br />
|Kiselev<br />
|}<br />
<br />
==Abstracts==<br />
===Takis Souganidis (Chicago)===<br />
''Stochastic homogenization of the G-equation''<br />
<br />
The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.<br />
<br />
===Scott Armstrong (UW-Madison)===<br />
''Partial regularity for fully nonlinear elliptic equations''<br />
<br />
I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.<br />
<br />
===Russell Schwab (Carnegie Mellon)===<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)''<br />
<br />
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])<br />
<br />
===Valentin Ovsienko (University of Lyon)===<br />
''The pentagram map and generalized friezes of Coxeter''<br />
<br />
The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map.<br />
In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.<br />
<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space''<br />
<br />
Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. In spite of significant progress achieved in the analysis of the SG in dual space (i.e. the system obtained from the SG by a special change of variables), there are no existence results on the SG in physical space. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We have proved existence of such solutions in the case of discrete measures in dual space. The talk is based on joint work with M. Feldman.<br />
<br />
===James Nolen (Duke)===<br />
''Normal approximation for a random elliptic PDE''<br />
<br />
I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. This talk is about the energy dissipation rate, which is a quadratic functional of the solution and a bulk property of the random material. For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation.<br />
<br />
===Betsy Stovall (UCLA)===<br />
<br />
We will discuss recent work concerning the cubic Klein--Gordon equation<br />
u_{tt} - \Delta u + u \pm u^3 = 0<br />
in two space dimensions with real valued initial data in the energy space, u(0) in H^1, u_t(0) in L^2. We show that in the defocusing case, solutions<br />
are global and have finite L^4 norm (in space and time). In the focusing case, we characterize the dichotomy between such behavior and finite time<br />
blowup for initial data having energy less than that of the ground state. In this talk, we will pay particular attention to connections with certain<br />
questions arising in harmonic analysis. <br />
<br />
This is joint work with Rowan Killip and Monica Visan.<br />
<br />
===Charles Smart (MIT)===<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian''<br />
<br />
A classical theorem of Kirszbraun states that any Lipschitz function $f : A \to \R^m$ defined on a set $A \subseteq \R^n$ can be extended to all of $\R^n$ without increasing the Lipschitz constant. The search for a canonical such extension leads one to the notion of optimal Lipschitz extension. I will discuss joint work with Evans on the regularity of optimal Lipschitz extensions in the scalar $m = 1$ case and joint work with Sheffield on the vector valued $m > 1$ case.<br />
<br />
===Yao Yao (UCLA)===<br />
''Degenerate diffusion with nonlocal aggregation: behavior of solutions''<br />
<br />
The Patlak-Keller-Segel (PKS) equation models the collective motion of<br />
cells which are attracted by a self-emitted chemical substance. While the<br />
global well-posedness and finite-time blow up criteria are well known, the<br />
asymptotic behaviors of solutions are not completely clear. In this talk I<br />
will present some results on the asymptotic behavior of solutions when<br />
there is global existence. The key tools used in the paper are<br />
maximum-principle type arguments as well as estimates on mass concentration<br />
of solutions. This is a joint work with Inwon Kim.<br />
<br />
===Nestor Guillen (UCLA)===<br />
<br />
TBA<br />
<br />
===Vlad Vicol (University of Chicago)===<br />
<br />
TBA<br />
<br />
===Jiahong Wu (Oklahoma State)===<br />
<br />
TBA</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=3406PDE Geometric Analysis seminar2012-02-01T01:12:06Z<p>Kiselev: </p>
<hr />
<div>= PDE and Geometric Analysis Seminar =<br />
<br />
The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
[[Previous PDE/GA seminars]]<br />
<br />
== Seminar Schedule Fall 2011 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Oct 3<br />
|Takis Souganidis (Chicago)<br />
|[[#Takis Souganidis (Chicago)|<br />
''Stochastic homogenization of the G-equation'']]<br />
|Armstrong<br />
<br />
|-<br />
|Oct 10<br />
|Scott Armstrong (UW-Madison)<br />
|[[#Scott Armstrong (UW-Madison)|<br />
''Partial regularity for fully nonlinear elliptic equations'']]<br />
|Local speaker<br />
<br />
|-<br />
|Oct 17<br />
|Russell Schwab (Carnegie Mellon)<br />
|[[#Russell Schwab (Carnegie Mellon)|<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)'']]<br />
|Armstrong<br />
|-<br />
|October 24 ( with Geometry/Topology seminar)<br />
|[http://math.univ-lyon1.fr/~ovsienko/ Valentin Ovsienko] (University of Lyon)<br />
|[[#Valentin Ovsienko (University of Lyon)|<br />
''The pentagram map and generalized friezes of Coxeter'']]<br />
|Marí Beffa<br />
|-<br />
|-<br />
|Oct 31<br />
|Adrian Tudorascu (West Virginia University)<br />
|[[#Adrian Tudorascu (West Virginia University)|<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space'']]<br />
|Feldman<br />
|-<br />
|Nov 7<br />
|James Nolen (Duke)<br />
|[[#James Nolen (Duke)|<br />
''Normal approximation for a random elliptic PDE'']]<br />
|Armstrong<br />
|-|-<br />
|Nov 21 (Joint with [http://www.math.wisc.edu/~seeger/curr.html Analysis] seminar)<br />
|Betsy Stovall (UCLA)<br />
|[[#Betsy Stovall (UCLA)| <br />
''Scattering for the cubic Klein--Gordon equation in two dimensions'']]<br />
|Seeger<br />
|-<br />
|Dec 5<br />
|Charles Smart (MIT)<br />
|[[#Charles Smart (MIT)|<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian'']]<br />
|Armstrong<br />
|}<br />
<br />
== Seminar Schedule Spring 2012 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (UCLA)|<br />
TBA]]<br />
|Kiselev<br />
|-<br />
|March 19<br />
|Nestor Guillen (UCLA)<br />
|[[#Nestor Guillen (UCLA)|<br />
TBA]]<br />
|Feldman<br />
|-<br />
|March 26<br />
|Vlad Vicol (University of Chicago)<br />
|[[#Vlad Vicol (U Chicago)|<br />
TBA]]<br />
|Kiselev<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jianhong Wu (Oklahoma)|<br />
TBA]]<br />
|Kiselev<br />
|}<br />
<br />
==Abstracts==<br />
===Takis Souganidis (Chicago)===<br />
''Stochastic homogenization of the G-equation''<br />
<br />
The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.<br />
<br />
===Scott Armstrong (UW-Madison)===<br />
''Partial regularity for fully nonlinear elliptic equations''<br />
<br />
I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.<br />
<br />
===Russell Schwab (Carnegie Mellon)===<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)''<br />
<br />
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])<br />
<br />
===Valentin Ovsienko (University of Lyon)===<br />
''The pentagram map and generalized friezes of Coxeter''<br />
<br />
The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map.<br />
In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.<br />
<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space''<br />
<br />
Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. In spite of significant progress achieved in the analysis of the SG in dual space (i.e. the system obtained from the SG by a special change of variables), there are no existence results on the SG in physical space. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We have proved existence of such solutions in the case of discrete measures in dual space. The talk is based on joint work with M. Feldman.<br />
<br />
===James Nolen (Duke)===<br />
''Normal approximation for a random elliptic PDE''<br />
<br />
I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. This talk is about the energy dissipation rate, which is a quadratic functional of the solution and a bulk property of the random material. For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation.<br />
<br />
===Betsy Stovall (UCLA)===<br />
<br />
We will discuss recent work concerning the cubic Klein--Gordon equation<br />
u_{tt} - \Delta u + u \pm u^3 = 0<br />
in two space dimensions with real valued initial data in the energy space, u(0) in H^1, u_t(0) in L^2. We show that in the defocusing case, solutions<br />
are global and have finite L^4 norm (in space and time). In the focusing case, we characterize the dichotomy between such behavior and finite time<br />
blowup for initial data having energy less than that of the ground state. In this talk, we will pay particular attention to connections with certain<br />
questions arising in harmonic analysis. <br />
<br />
This is joint work with Rowan Killip and Monica Visan.<br />
<br />
===Charles Smart (MIT)===<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian''<br />
<br />
A classical theorem of Kirszbraun states that any Lipschitz function $f : A \to \R^m$ defined on a set $A \subseteq \R^n$ can be extended to all of $\R^n$ without increasing the Lipschitz constant. The search for a canonical such extension leads one to the notion of optimal Lipschitz extension. I will discuss joint work with Evans on the regularity of optimal Lipschitz extensions in the scalar $m = 1$ case and joint work with Sheffield on the vector valued $m > 1$ case.<br />
<br />
===Yao Yao (UCLA)===<br />
"Degenerate diffusion with nonlocal aggregation: behavior of solutions"<br />
<br />
The Patlak-Keller-Segel (PKS) equation models the collective motion of<br />
cells which are attracted by a self-emitted chemical substance. While the<br />
global well-posedness and finite-time blow up criteria are well known, the<br />
asymptotic behaviors of solutions are not completely clear. In this talk I<br />
will present some results on the asymptotic behavior of solutions when<br />
there is global existence. The key tools used in the paper are<br />
maximum-principle type arguments as well as estimates on mass concentration<br />
of solutions. This is a joint work with Inwon Kim.<br />
<br />
===Nestor Guillen (UCLA)===<br />
<br />
TBA</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=3405PDE Geometric Analysis seminar2012-02-01T01:10:10Z<p>Kiselev: </p>
<hr />
<div>= PDE and Geometric Analysis Seminar =<br />
<br />
The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
[[Previous PDE/GA seminars]]<br />
<br />
== Seminar Schedule Fall 2011 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Oct 3<br />
|Takis Souganidis (Chicago)<br />
|[[#Takis Souganidis (Chicago)|<br />
''Stochastic homogenization of the G-equation'']]<br />
|Armstrong<br />
<br />
|-<br />
|Oct 10<br />
|Scott Armstrong (UW-Madison)<br />
|[[#Scott Armstrong (UW-Madison)|<br />
''Partial regularity for fully nonlinear elliptic equations'']]<br />
|Local speaker<br />
<br />
|-<br />
|Oct 17<br />
|Russell Schwab (Carnegie Mellon)<br />
|[[#Russell Schwab (Carnegie Mellon)|<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)'']]<br />
|Armstrong<br />
|-<br />
|October 24 ( with Geometry/Topology seminar)<br />
|[http://math.univ-lyon1.fr/~ovsienko/ Valentin Ovsienko] (University of Lyon)<br />
|[[#Valentin Ovsienko (University of Lyon)|<br />
''The pentagram map and generalized friezes of Coxeter'']]<br />
|Marí Beffa<br />
|-<br />
|-<br />
|Oct 31<br />
|Adrian Tudorascu (West Virginia University)<br />
|[[#Adrian Tudorascu (West Virginia University)|<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space'']]<br />
|Feldman<br />
|-<br />
|Nov 7<br />
|James Nolen (Duke)<br />
|[[#James Nolen (Duke)|<br />
''Normal approximation for a random elliptic PDE'']]<br />
|Armstrong<br />
|-|-<br />
|Nov 21 (Joint with [http://www.math.wisc.edu/~seeger/curr.html Analysis] seminar)<br />
|Betsy Stovall (UCLA)<br />
|[[#Betsy Stovall (UCLA)| <br />
''Scattering for the cubic Klein--Gordon equation in two dimensions'']]<br />
|Seeger<br />
|-<br />
|Dec 5<br />
|Charles Smart (MIT)<br />
|[[#Charles Smart (MIT)|<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian'']]<br />
|Armstrong<br />
|}<br />
<br />
== Seminar Schedule Spring 2012 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (UCLA)|<br />
TBA]]<br />
|Kiselev<br />
|-<br />
|March 19<br />
|Nestor Guillen (UCLA)<br />
|[[#Nestor Guillen (UCLA)|<br />
TBA]]<br />
|Feldman<br />
|-<br />
|March 26<br />
|Vlad Vicol (University of Chicago)<br />
|[[#Vlad Vicol (U Chicago)|<br />
TBA]]<br />
|Kiselev<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jianhong Wu (Oklahoma)|<br />
TBA]]<br />
|Kiselev<br />
|}<br />
<br />
==Abstracts==<br />
===Takis Souganidis (Chicago)===<br />
''Stochastic homogenization of the G-equation''<br />
<br />
The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.<br />
<br />
===Scott Armstrong (UW-Madison)===<br />
''Partial regularity for fully nonlinear elliptic equations''<br />
<br />
I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.<br />
<br />
===Russell Schwab (Carnegie Mellon)===<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)''<br />
<br />
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])<br />
<br />
===Valentin Ovsienko (University of Lyon)===<br />
''The pentagram map and generalized friezes of Coxeter''<br />
<br />
The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map.<br />
In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.<br />
<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space''<br />
<br />
Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. In spite of significant progress achieved in the analysis of the SG in dual space (i.e. the system obtained from the SG by a special change of variables), there are no existence results on the SG in physical space. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We have proved existence of such solutions in the case of discrete measures in dual space. The talk is based on joint work with M. Feldman.<br />
<br />
===James Nolen (Duke)===<br />
''Normal approximation for a random elliptic PDE''<br />
<br />
I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. This talk is about the energy dissipation rate, which is a quadratic functional of the solution and a bulk property of the random material. For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation.<br />
<br />
===Betsy Stovall (UCLA)===<br />
<br />
We will discuss recent work concerning the cubic Klein--Gordon equation<br />
u_{tt} - \Delta u + u \pm u^3 = 0<br />
in two space dimensions with real valued initial data in the energy space, u(0) in H^1, u_t(0) in L^2. We show that in the defocusing case, solutions<br />
are global and have finite L^4 norm (in space and time). In the focusing case, we characterize the dichotomy between such behavior and finite time<br />
blowup for initial data having energy less than that of the ground state. In this talk, we will pay particular attention to connections with certain<br />
questions arising in harmonic analysis. <br />
<br />
This is joint work with Rowan Killip and Monica Visan.<br />
<br />
===Charles Smart (MIT)===<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian''<br />
<br />
A classical theorem of Kirszbraun states that any Lipschitz function $f : A \to \R^m$ defined on a set $A \subseteq \R^n$ can be extended to all of $\R^n$ without increasing the Lipschitz constant. The search for a canonical such extension leads one to the notion of optimal Lipschitz extension. I will discuss joint work with Evans on the regularity of optimal Lipschitz extensions in the scalar $m = 1$ case and joint work with Sheffield on the vector valued $m > 1$ case.<br />
<br />
<br />
<br />
===Nestor Guillen (UCLA)===<br />
<br />
TBA</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=3205PDE Geometric Analysis seminar2011-12-08T20:38:57Z<p>Kiselev: </p>
<hr />
<div>= PDE and Geometric Analysis Seminar =<br />
<br />
The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
[[Previous PDE/GA seminars]]<br />
<br />
== Seminar Schedule Fall 2011 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Oct 3<br />
|Takis Souganidis (Chicago)<br />
|[[#Takis Souganidis (Chicago)|<br />
''Stochastic homogenization of the G-equation'']]<br />
|Armstrong<br />
<br />
|-<br />
|Oct 10<br />
|Scott Armstrong (UW-Madison)<br />
|[[#Scott Armstrong (UW-Madison)|<br />
''Partial regularity for fully nonlinear elliptic equations'']]<br />
|Local speaker<br />
<br />
|-<br />
|Oct 17<br />
|Russell Schwab (Carnegie Mellon)<br />
|[[#Russell Schwab (Carnegie Mellon)|<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)'']]<br />
|Armstrong<br />
|-<br />
|October 24 ( with Geometry/Topology seminar)<br />
|[http://math.univ-lyon1.fr/~ovsienko/ Valentin Ovsienko] (University of Lyon)<br />
|[[#Valentin Ovsienko (University of Lyon)|<br />
''The pentagram map and generalized friezes of Coxeter'']]<br />
|Marí Beffa<br />
|-<br />
|-<br />
|Oct 31<br />
|Adrian Tudorascu (West Virginia University)<br />
|[[#Adrian Tudorascu (West Virginia University)|<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space'']]<br />
|Feldman<br />
|-<br />
|Nov 7<br />
|James Nolen (Duke)<br />
|[[#James Nolen (Duke)|<br />
''Normal approximation for a random elliptic PDE'']]<br />
|Armstrong<br />
|-|-<br />
|Nov 21 (Joint with [http://www.math.wisc.edu/~seeger/curr.html Analysis] seminar)<br />
|Betsy Stovall (UCLA)<br />
|[[#Betsy Stovall (UCLA)| <br />
''Scattering for the cubic Klein--Gordon equation in two dimensions'']]<br />
|Seeger<br />
|-<br />
|Dec 5<br />
|Charles Smart (MIT)<br />
|[[#Charles Smart (MIT)|<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian'']]<br />
|Armstrong<br />
|}<br />
<br />
== Seminar Schedule Spring 2012 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Feb 6<br />
|Yao Yao (UCLA)<br />
|[[#Yao Yao (MIT)|<br />
TBA]]<br />
|Kiselev<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jianhong Wu (Oklahoma)|<br />
TBA]]<br />
|Kiselev<br />
|}<br />
<br />
==Abstracts==<br />
===Takis Souganidis (Chicago)===<br />
''Stochastic homogenization of the G-equation''<br />
<br />
The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.<br />
<br />
===Scott Armstrong (UW-Madison)===<br />
''Partial regularity for fully nonlinear elliptic equations''<br />
<br />
I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.<br />
<br />
===Russell Schwab (Carnegie Mellon)===<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)''<br />
<br />
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])<br />
<br />
===Valentin Ovsienko (University of Lyon)===<br />
''The pentagram map and generalized friezes of Coxeter''<br />
<br />
The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map.<br />
In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.<br />
<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space''<br />
<br />
Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. In spite of significant progress achieved in the analysis of the SG in dual space (i.e. the system obtained from the SG by a special change of variables), there are no existence results on the SG in physical space. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We have proved existence of such solutions in the case of discrete measures in dual space. The talk is based on joint work with M. Feldman.<br />
<br />
===James Nolen (Duke)===<br />
''Normal approximation for a random elliptic PDE''<br />
<br />
I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. This talk is about the energy dissipation rate, which is a quadratic functional of the solution and a bulk property of the random material. For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation.<br />
<br />
===Betsy Stovall (UCLA)===<br />
<br />
We will discuss recent work concerning the cubic Klein--Gordon equation<br />
u_{tt} - \Delta u + u \pm u^3 = 0<br />
in two space dimensions with real valued initial data in the energy space, u(0) in H^1, u_t(0) in L^2. We show that in the defocusing case, solutions<br />
are global and have finite L^4 norm (in space and time). In the focusing case, we characterize the dichotomy between such behavior and finite time<br />
blowup for initial data having energy less than that of the ground state. In this talk, we will pay particular attention to connections with certain<br />
questions arising in harmonic analysis. <br />
<br />
This is joint work with Rowan Killip and Monica Visan.<br />
<br />
===Charles Smart (MIT)===<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian''<br />
<br />
A classical theorem of Kirszbraun states that any Lipschitz function $f : A \to \R^m$ defined on a set $A \subseteq \R^n$ can be extended to all of $\R^n$ without increasing the Lipschitz constant. The search for a canonical such extension leads one to the notion of optimal Lipschitz extension. I will discuss joint work with Evans on the regularity of optimal Lipschitz extensions in the scalar $m = 1$ case and joint work with Sheffield on the vector valued $m > 1$ case.</div>Kiselevhttps://wiki.math.wisc.edu/index.php?title=PDE_Geometric_Analysis_seminar&diff=3194PDE Geometric Analysis seminar2011-12-01T17:25:17Z<p>Kiselev: </p>
<hr />
<div>= PDE and Geometric Analysis Seminar =<br />
<br />
The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
[[Previous PDE/GA seminars]]<br />
<br />
== Seminar Schedule Fall 2011 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Oct 3<br />
|Takis Souganidis (Chicago)<br />
|[[#Takis Souganidis (Chicago)|<br />
''Stochastic homogenization of the G-equation'']]<br />
|Armstrong<br />
<br />
|-<br />
|Oct 10<br />
|Scott Armstrong (UW-Madison)<br />
|[[#Scott Armstrong (UW-Madison)|<br />
''Partial regularity for fully nonlinear elliptic equations'']]<br />
|Local speaker<br />
<br />
|-<br />
|Oct 17<br />
|Russell Schwab (Carnegie Mellon)<br />
|[[#Russell Schwab (Carnegie Mellon)|<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)'']]<br />
|Armstrong<br />
|-<br />
|October 24 ( with Geometry/Topology seminar)<br />
|[http://math.univ-lyon1.fr/~ovsienko/ Valentin Ovsienko] (University of Lyon)<br />
|[[#Valentin Ovsienko (University of Lyon)|<br />
''The pentagram map and generalized friezes of Coxeter'']]<br />
|Marí Beffa<br />
|-<br />
|-<br />
|Oct 31<br />
|Adrian Tudorascu (West Virginia University)<br />
|[[#Adrian Tudorascu (West Virginia University)|<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space'']]<br />
|Feldman<br />
|-<br />
|Nov 7<br />
|James Nolen (Duke)<br />
|[[#James Nolen (Duke)|<br />
''Normal approximation for a random elliptic PDE'']]<br />
|Armstrong<br />
|-|-<br />
|Nov 21 (Joint with [http://www.math.wisc.edu/~seeger/curr.html Analysis] seminar)<br />
|Betsy Stovall (UCLA)<br />
|[[#Betsy Stovall (UCLA)| <br />
''Scattering for the cubic Klein--Gordon equation in two dimensions'']]<br />
|Seeger<br />
|-<br />
|Dec 5<br />
|Charles Smart (MIT)<br />
|[[#Charles Smart (MIT)|<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian'']]<br />
|Armstrong<br />
|}<br />
<br />
== Seminar Schedule Spring 2012 ==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|April 16<br />
|Jiahong Wu (Oklahoma)<br />
|[[#Jianhong Wu (Oklahoma)|<br />
TBA]]<br />
|Kiselev<br />
|}<br />
<br />
==Abstracts==<br />
===Takis Souganidis (Chicago)===<br />
''Stochastic homogenization of the G-equation''<br />
<br />
The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.<br />
<br />
===Scott Armstrong (UW-Madison)===<br />
''Partial regularity for fully nonlinear elliptic equations''<br />
<br />
I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.<br />
<br />
===Russell Schwab (Carnegie Mellon)===<br />
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)''<br />
<br />
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])<br />
<br />
===Valentin Ovsienko (University of Lyon)===<br />
''The pentagram map and generalized friezes of Coxeter''<br />
<br />
The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map.<br />
In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.<br />
<br />
<br />
===Adrian Tudorascu (West Virginia University)===<br />
''Weak Lagrangian solutions for the Semi-Geostrophic system in physical space''<br />
<br />
Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. In spite of significant progress achieved in the analysis of the SG in dual space (i.e. the system obtained from the SG by a special change of variables), there are no existence results on the SG in physical space. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We have proved existence of such solutions in the case of discrete measures in dual space. The talk is based on joint work with M. Feldman.<br />
<br />
===James Nolen (Duke)===<br />
''Normal approximation for a random elliptic PDE''<br />
<br />
I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. This talk is about the energy dissipation rate, which is a quadratic functional of the solution and a bulk property of the random material. For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation.<br />
<br />
===Betsy Stovall (UCLA)===<br />
<br />
We will discuss recent work concerning the cubic Klein--Gordon equation<br />
u_{tt} - \Delta u + u \pm u^3 = 0<br />
in two space dimensions with real valued initial data in the energy space, u(0) in H^1, u_t(0) in L^2. We show that in the defocusing case, solutions<br />
are global and have finite L^4 norm (in space and time). In the focusing case, we characterize the dichotomy between such behavior and finite time<br />
blowup for initial data having energy less than that of the ground state. In this talk, we will pay particular attention to connections with certain<br />
questions arising in harmonic analysis. <br />
<br />
This is joint work with Rowan Killip and Monica Visan.<br />
<br />
===Charles Smart (MIT)===<br />
''Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian''<br />
<br />
A classical theorem of Kirszbraun states that any Lipschitz function $f : A \to \R^m$ defined on a set $A \subseteq \R^n$ can be extended to all of $\R^n$ without increasing the Lipschitz constant. The search for a canonical such extension leads one to the notion of optimal Lipschitz extension. I will discuss joint work with Evans on the regularity of optimal Lipschitz extensions in the scalar $m = 1$ case and joint work with Sheffield on the vector valued $m > 1$ case.</div>Kiselev