PDE Geometric Analysis seminar: Difference between revisions

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= Abstracts =
===Greg Drugan (U. of Washington)===
''Construction of immersed self-shrinkers''
Abstract: We describe a procedure for constructing immersed
self-shrinking solutions to mean curvature flow.
The self-shrinkers we construct have a rotational symmetry, and
the construction involves a detailed study of geodesics in the
upper-half plane with a conformal metric.
This is a joint work with Stephen Kleene.
===Guo Luo (Caltech)===
''Potentially Singular Solutions of the 3D Incompressible Euler Equations''
Whether the 3D incompressible Euler equations can develop a singularity in
finite time from smooth initial data is one of the most challenging problems in
mathematical fluid dynamics. This work attempts to provide an affirmative answer to this
long-standing open question from a numerical point of view, by presenting a class of
potentially singular solutions to the Euler equations computed in axisymmetric
geometries. The solutions satisfy a periodic boundary condition along the axial direction
and no-flow boundary condition on the solid wall. The equations are discretized in space
using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially
designed adaptive (moving) meshes that are dynamically adjusted to the evolving
solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the
point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and
predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a
\emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and
observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity
$\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup
(non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and
Deng-Hou-Yu, to confirm the validity of the singularity. A careful local analysis also
suggests that the blowing-up solution develops a self-similar structure near the point of
the singularity, as the singularity time is approached.
===Xiaojie Wang(Stony Brook)===
''Uniqueness of Ricci flow solutions on noncompact manifolds''
Ricci flow is an important evolution equation of Riemannian metrics.
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.
===Roman Shterenberg(UAB)===
''Recent progress in multidimensional periodic and almost-periodic spectral
Abstract:  We present a review of the results in multidimensional periodic
and almost-periodic spectral problems. We discuss some recent progress and
old/new ideas used in the constructions. The talk is mostly based on the
joint works with Yu. Karpeshina and L. Parnovski.
===Antonio Ache(Princeton)===
''Ricci Curvature and the manifold learning problem''
Abstract:  In the first half of this talk we will review several notions of coarse or weak
Ricci Curvature on metric measure spaces which include the works of Lott-Villani, Sturm
and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as
motivation for developing a method to estimate the Ricci curvature of a an embedded
submaifold of Euclidean space from a point cloud which has applications to the Manifold
Learning Problem. Our method is based on combining the notion of ``Carre du Champ"
introduced by Bakry-Emery with a result of Belkin and Niyogi which shows that it is
possible to recover the rough laplacian of embedded submanifolds of the Euclidean space
from point clouds. This is joint work with Micah Warren.
===Jean-Michel Roquejoffre (Toulouse)===
''Front propagation in the presence of integral diffusion''
Abstract:  In many reaction-diffusion equations, where diffusion is
given by a second order elliptic operator, the solutions
will exhibit spatial transitions whose velocity is asymptotically
linear in time. The situation can be different when the diffusion is of the
integral type, the most basic example being the fractional Laplacian:
the velocity can be time-exponential. We will explain why, and
discuss several situations where this type of fast propagation
===Myoungjean Bae (POSTECH)===
''Free Boundary Problem related to Euler-Poisson system''
One dimensional analysis of Euler-Poisson system shows that when incoming
supersonic flow is fixed, transonic shock can be represented as a monotone
function of exit pressure. From this observation, we expect well-posedness
of transonic shock problem for Euler-Poisson system when exit pressure is
prescribed in a proper range. In this talk, I will present recent progress
on transonic shock problem for Euler-Poisson system, which is formulated
as a free boundary problem with mixed type PDE system.
This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU)
and Jingjing Xiao(CUHK).
===Changhui Tan (University of Maryland)===
''Global classical solution and long time behavior of macroscopic flocking models''
Abstract: Self-organized behaviors are very common in nature and human societies.
One widely discussed example is the flocking phenomenon which describes
animal groups emerging towards the same direction. Several models such
as Cucker-Smale and Motsch-Tadmor are very successful in characterizing
flocking behaviors. In this talk, we will discuss macroscopic representation
of flocking models. These systems can be interpreted as compressible Eulerian
dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time
flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set
of initial conditions will lead to a finite time break down of the system. This
is a joint work with Eitan Tadmor.
===Hongjie Dong (Brown University)===
''Parabolic equations in time-varying domains''
Abstract: I will present a recent result on the Dirichlet boundary value
problem for parabolic equations in time-varying domains. The equations are
in either divergence or non-divergence form with boundary blowup low-order
coefficients. The domains satisfy an exterior measure condition.
===Hao Jia (University of Chicago)===
''Long time dynamics of energy critical defocusing wave equation with
radial potential in 3+1 dimensions.''
Abstract: We consider the long term dynamics of radial solution to the
above mentioned equation. For general potential, the equation can have a
unique positive ground state and a number of excited states. One can expect
that some solutions might stay for very long time near excited states
before settling down to an excited state of lower energy or the ground
state. Thus the detailed dynamics can be extremely complicated. However
using the ``channel of energy" inequality discovered by T.Duyckaerts,
C.Kenig and F.Merle, we can show for generic potential, any radial solution
is asymptotically the sum of a free radiation and a steady state as time
goes to infinity. This provides another example of the power of ``channel
of energy" inequality and the method of profile decompositions. I will
explain the basic tools in some detail. Joint work with Baoping Liu and
Guixiang Xu.
===Alexander Pushnitski (King's College)===
''An inverse spectral problem for Hankel operators''
I will discuss an inverse spectral problem for a certain class of Hankel
operators. The problem appeared in the recent work by P.Gerard and S.Grellier as a
step towards description of evolution in a model integrable non-dispersive
equation. Several features of this inverse problem make it strikingly (and somewhat
mysteriously) similar to an inverse problem for Sturm-Liouville operators. I will
describe the available results for Hankel operators, emphasizing this similarity.
This is joint work with Patrick Gerard (Orsay).
===Ronghua Pan (Georgia Tech)===
''Compressible Navier-Stokes-Fourier system with temperature dependent dissipation''
Abstract: From its physical origin such as Chapman-Enskog or Sutherland, the viscosity and
heat conductivity coefficients in compressible fluids depend on absolute temperature
through power laws. The mathematical theory on the well-posedness and regularity on this
setting is widely open. I will report some recent progress on this direction,
with emphasis on the lower bound of temperature, and global existence of
solutions in one or multiple dimensions. The relation between thermodynamics laws
and Navier-Stokes-Fourier system will also be discussed. This talk is based on joint works
with Junxiong Jia and Weizhe Zhang.

Revision as of 14:46, 7 September 2014

The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Previous PDE/GA seminars

Seminar Schedule Fall 2014

date speaker title host(s)
September 15 Greg Kuperberg (UC-Davis)
September 22 (joint with Analysis Seminar) Steven Hofmann (U. of Missouri)
Oct 6th, Xiangwen Zhang (Columbia University)