Difference between revisions of "PDE Geometric Analysis seminar"

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= PDE and Geometric Analysis Seminar =
+
The seminar's format will be a combination of online and in-person; we will make sure to update you as soon as we have more details available. First talk is tentatively scheduled for September 20th ! 
  
The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
+
Some of the seminars will be held online. When that would be the case we would use the following zoom link
  
== Seminar Schedule Spring 2011 ==
+
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09
{| cellpadding="8"
+
 
!align="left" | date 
+
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]
!align="left" | speaker
+
Passcode: 180680
!align="left" | title
+
 
!align="left" | host(s)
+
 
|-
+
 
|Jan 24
+
[[Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.]]
|Bing Wang (Princeton)
+
 
|[[#Bing Wang (Princeton)|
+
 
''The Kaehler Ricci flow on Fano manifold '']]
+
===[[Previous PDE/GA seminars]]===
|Viaclovsky
+
 
|-
+
===[[Fall 2022-Spring 2023 | Schedule for Fall 2022-Spring 2023]]===
|Mar 15 (TUESDAY) at 4pm in B139 (joint wit Analysis)
+
 
|Francois Hamel (Marseille)
+
 
|[[#Francois Hamel (Marseille)|
+
 
''Optimization of eigenvalues of non-symmetric elliptic operators'']]
+
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==
|Zlatos
+
 
|-
+
=Fall 2022=   
|Mar 28
+
 
|Juraj Foldes (Vanderbilt)
+
 
|[[#Juraj Foldes (Vanderbilt)|
+
 
''Symmetry properties of parabolic problems and their applications'']]
+
'''September 12, 2022'''
|Zlatos
+
 
|-
+
[[No Seminar]]
|Apr 11
+
 
|Alexey Cheskidov (UIC)
+
 
|[[#Alexey Cheskidov (UIC)|
+
 
''Navier-Stokes and Euler equations: a unified approach to the problem of blow-up'']]
+
 
|Kiselev
+
 
|-
+
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar
|Date TBA
+
 
|Mikhail Feldman (UW Madison)
+
[[Andrej Zlatos]] (UCSD). Host: Hung Tran.
|''TBA''
+
 
|Local speaker
+
Format: in-person. Time: 4-5PM, VV B139.
|-
+
 
|Date TBA
+
Title: Homogenization in front propagation models
|Sigurd Angenent (UW Madison)
+
 
|''TBA''
+
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit.  I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients.  These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.
|Local speaker
+
 
|-
+
 
|}
+
 
== Seminar Schedule Fall 2010 ==
+
 
{| cellpadding="8"
+
'''September 26, 2022 '''
!align="left" | date 
+
 
!align="left" | speaker
+
[[Haotian Wu]] (The University of Sydney, Australia).  Host: Sigurd Angenent.
!align="left" | title
+
 
!align="left" | host(s)
+
Format: in person, Time: 3:30pm-4:30pm VV 901 
|-
+
 
|Sept 13
+
<u>Title:</u>  ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''
|Fausto Ferrari (Bologna)
+
 
|[[#Fausto Ferrari (Bologna)|
+
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).
''Semilinear PDEs and some symmetry properties of stable solutions'']]
+
 
|Feldman
+
 
|-
+
 
|Sept 27
+
 
|Arshak Petrosyan (Purdue)
+
'''October 3, 2022'''
|[[#Arshak Petrosyan (Purdue)|
+
 
''Nonuniqueness in a free boundary problem from combustion'']]
+
[[No Seminar]]
|Feldman
+
 
|-
+
Format:  , Time: 
|Oct 7, Thursday, 4:30 pm, Room: 901 Van Vleck.  '''Special day, time & room.'''
+
 
|Changyou Wang (U. of Kentucky)
+
Title:
|[[#Changyou Wang (U. of Kentucky)|
+
 
''Phase transition for higher dimensional wells'']]
+
Abstract:
|Feldman
+
 
|-
+
 
|Oct 11
+
 
|Philippe LeFloch (Paris VI)
+
 
|[[#Philippe LeFloch (Paris VI)|
+
'''October 10, 2022'''
''Kinetic relations for undercompressive shock waves and propagating phase boundaries'']]
+
 
|Feldman
+
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.
|-
+
 
|Oct 29 Friday 2:30pm, Room: B115 Van Vleck.   '''Special day, time & room.'''
+
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).
|[http://www.ima.umn.edu/~imitrea/ Irina Mitrea] (IMA)
+
 
|[[#Irina Mitrea |
+
Speaker: Sasha Kiselev (Duke)
''Boundary Value Problems for Higher Order Differential Operators'']]
+
 
|[https://www.math.wisc.edu/~wimaw/ WiMaW]
+
Title: The flow of polynomial roots under differentiation
|-
+
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work
|-
+
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.
|Nov 1
+
 
|Panagiota Daskalopoulos (Columbia U)
+
 
|[[#Panagiota Daskalopoulos (Columbia U)|
+
 
''Ancient solutions to geometric flows'']]
+
'''October 17, 2022'''
|Feldman
+
 
|-
+
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.
|Nov 8
+
 
|Maria Gualdani (UT Austin)
+
Format: in person, Time: 3:30pm-4:30pm VV 901 
|[[#Maria Gualdani (UT Austin)|
+
 
''A nonlinear diffusion model in mean-field games'']]
+
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.
|Feldman
+
 
|-
+
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures.
|Nov 18 Thursday 1:20pm  Room: 901 Van Vleck '''Special day & time.'''
+
 
|Hiroshi Matano (Tokyo University)
+
 
|[[#Hiroshi Matano (Tokyo University)|
+
 
''Traveling waves in a sawtoothed cylinder and their homogenization limit'']]
+
 
|Angenent & Rabinowitz
+
'''October 24, 2022'''
|-
+
 
|Nov 29
+
[[No seminar.]]
|Ian Tice (Brown University)
+
 
|[[#Ian Tice (Brown University)|
+
Format:  , Time: 
''Global well-posedness and decay for the viscous surface wave
+
 
problem without surface tension'']]
+
Title:
|Feldman
+
 
|-
+
Abstract:
|Dec. 8 Wed 2:25pm, Room: 901 Van Vleck. '''Special day, time & room.'''
+
 
|Hoai Minh Nguyen (NYU-Courant Institute)
+
 
|[[#Hoai Minh Nguyen (NYU-Courant Institute)|
+
 
''Cloaking via change of variables for the Helmholtz equation'']]
+
 
|Feldman
+
 
|-
+
'''October 31, 2022 '''
|}
+
 
 +
[[Yuan Gao]] (Purdue). Host: Hung Tran.
 +
 
 +
Format: in person, Time: 3:30pm-4:30pm VV 901 
 +
 
 +
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures
 +
 
 +
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both  self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.
 +
 
 +
 
 +
 
 +
'''November 7, 2022 '''
 +
 
 +
[[Beomjun Choi]] (Postech)
 +
 
 +
Format:  in person, Time: 3:30pm-4:30pm VV 901
 +
 
 +
Title:  Liouville theorem for surfaces translating by powers of Gauss curvature
 +
 
 +
Abstract:  We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.
 +
 
 +
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.
 +
 
 +
 
 +
 
 +
 
 +
'''November 14, 2022 '''
 +
 
 +
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.
 +
 
 +
Format: in person, Time: 3:30pm-4:30pm VV 901 
 +
 
 +
Title: Sticky Particles with Sticky Boundary
 +
 
 +
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.
 +
 
 +
 
 +
 
 +
 
 +
 
 +
'''November 21, 2022 '''
 +
 
 +
[[Jason Murphy]] (Missouri S&T)
 +
 
 +
Format: online 
 +
 
 +
Join Zoom Meeting
 +
https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09
 +
Meeting ID: 948 7748 3456
 +
Passcode: 303105
 +
 
 +
Time: 3:30 PM -4:30 PM
 +
 
 +
Title: Sharp scattering results for the 3d cubic NLS
 +
 
 +
Abstract:  I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential.  After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials.  The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.
 +
 
 +
 +
 
 +
 
 +
 
 +
 
 +
'''November 28, 2022 '''
 +
 
 +
[[No Seminar]]- Thanksgiving
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
'''December 5, 2022 '''
 +
 
 +
[[James Rowan]] (UC Berkeley)
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
 
 +
'''December 12, 2022 '''
 +
 
 +
[[Calumn Rickard ]] UC Davis
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
'''March 27, 2023'''
 +
 
 +
[[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran.
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
 
 +
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===
 +
 
 +
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==
 +
 
 +
=Spring 2022=
 +
 
 +
 
 +
 
 +
 
 +
'''January  31th, 2022.'''
 +
 
 +
[[No Seminar]]
 +
 
 +
 
 +
'''February 7th, 2022.'''
 +
 
 +
[[Jonah Duncan]] from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM
 +
 
 +
Title: Estimates and regularity for the k-Yamabe equation
 +
 
 +
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.
 +
 
 +
 
 +
 
 +
 
 +
'''February 14th, 2022.'''
 +
 
 +
[[Sigurd Angenent]]; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM
 +
 
 +
Title: MCF after the Velázquez&mdash;Stolarski example.
 +
 
 +
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$.
 +
Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. 
 +
In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez&mdash;Stolarski solution to positive times and show that it also has uniformly bounded mean curvature.  In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.
 +
 
 +
 
 +
 
 +
'''February  21th, 2022.'''
 +
 
 +
[[Birgit Schoerkhuber]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09
 +
 
 +
Time: 11:00 AM
 +
 
 +
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations
 +
 
 +
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions.  Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck).
 +
 
 +
 
 +
 
 +
 
 +
'''February  28th, 2021.'''
 +
 
 +
[[Michael Hott]]; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09
 +
 
 +
Time:3:30PM-4:30PM
 +
 
 +
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC
 +
 
 +
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.
 +
 
 +
 
 +
'''March 7th, 2022.'''
 +
 
 +
[[ No Seminar]];
 +
 
 +
 
 +
'''March 14th, 2022.'''
 +
 
 +
[[Spring recess - No Seminar]];
 +
 
 +
 
 +
 
 +
 
 +
'''March 21th, 2022.'''
 +
 
 +
[[ No Seminar]];
 +
 
 +
 
 +
 
 +
'''March 28th, 2022.'''
 +
 
 +
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM
 +
 
 +
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09
 +
 
 +
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]
 +
Passcode: 180680
 +
 
 +
Title: Determinants, commuting flows, and recent progress on
 +
completely integrable systems
 +
 
 +
Abstract:  I will survey a number of recent developments in the theory
 +
of completely integrable nonlinear dispersive PDE.  These include a
 +
priori bounds, the orbital stability of multisoliton solutions,
 +
well-posedness at optimal regularity, and the existence of dynamics
 +
for Gibbs distributed initial data.  I will describe the basic objects
 +
that tie together these disparate results, as well as the diverse
 +
ideas required for each problem.
 +
 
 +
 
 +
'''April 4th, 2022.'''
 +
 
 +
[[Marcelo Disconzi]]; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM
 +
 
 +
Title: General-relativistic viscous fluids.
 +
 
 +
 +
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has
 +
 
 +
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.
 +
 
 +
 
 +
 
 +
'''April 11th, 2022.'''
 +
 
 +
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM
 +
 
 +
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09
 +
 
 +
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]
 +
Passcode: 180680
 +
 
 +
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations
 +
 
 +
Abstract:  In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo
 +
 
 +
 
 +
 
 +
'''April 18th, 2022.'''
 +
 
 +
[[Loc Nguyen]] (UNCC); Format: in-person seminar, Room:  901, Time: 3:30PM-4:30PM. Host: Hung Tran.
 +
 
 +
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.
 +
 
 +
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.
 +
 
 +
 
 +
 
 +
'''April 25th, 2022.'''
 +
 
 +
[[No seminar]]
 +
 
 +
 
 +
 
 +
'''May 2nd, 2022.'''
 +
 
 +
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM
 +
 
 +
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness
 +
 
 +
Abstract: In this talk, I will present some recent results obtained in
 +
collaboration with V. Patel (Oxford) in connection with a system
 +
describing a heat-conducting incompressible fluid. I will introduce the
 +
notion of a dissipative weak solution of the system and highlight the
 +
connections and differences to the existing approaches in the
 +
literature. One of the advantages of the proposed approach is that the
 +
solution satisfies a weak-strong uniqueness principle, which guarantees
 +
that the weak solution will coincide with the strong solution, as long
 +
as the latter exists; moreover, the solutions are constructed via a
 +
finite element approximation, leading (almost, not quite) to the first
 +
convergence result for the full system including viscous dissipation.
 +
 
 +
 
 +
 
 +
=Fall 2021=
 +
 
 +
'''September 20th, 2021.'''
 +
 
 +
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room:  901, Time: 3:30PM-4:30PM
 +
 
 +
 
 +
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts
 +
 
 +
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).
 +
 
 +
 
 +
 
 +
'''September 27th, 2021.'''
 +
 
 +
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room:  901, Time: 3:30PM-4:30PM
 +
 
 +
Title: Volume-preserving crystalline and anisotropic mean curvature flow
 +
 
 +
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).
 +
 
 +
 
 +
'''October 4th, 2021.'''
 +
 
 +
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room:  901, Time: 3:30PM-4:30PM
 +
 
 +
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries
 +
 
 +
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.
 +
 
 +
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).
 +
 
 +
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.
 +
 
 +
 
 +
'''October 11th, 2021.'''
 +
 
 +
[[No seminar]]
 +
 
 +
 
 +
'''October 18th, 2021.'''
 +
 
 +
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM
 +
 
 +
Title: Well-posedness of logarithmic spiral vortex sheets.
 +
 
 +
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.
 +
 
 +
 
 +
'''October 25th, 2021.'''
 +
 
 +
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm
 +
 
 +
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature
 +
 
 +
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.
 +
 
 +
 
 +
'''November 1th, 2021.'''
 +
 
 +
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room:  901, Time: 3:30PM-4:30PM
 +
 
 +
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity
 +
 
 +
Abstract:  This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.
 +
 
 +
 
 +
 
 +
'''November 8th, 2021.'''
 +
 
 +
[[ Albert Ai]] (UW Madison);
 +
 
 +
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation
 +
 
 +
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.
 +
 
 +
 
 +
 
 +
'''November 15th, 2021.'''
 +
 
 +
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]]
 +
 
 +
[[Please observe the time change! ]]
 +
 
 +
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP
 +
 
 +
 
 +
Title: Global wellposedness for the energy-critical Zakharov system below the ground state
 +
 
 +
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.
 +
 
 +
 
 +
 
 +
'''November 22th, 2021.'''
 +
 
 +
[[No seminar]]
 +
 
 +
 
 +
'''November 29th, 2021.'''
 +
 
 +
[[No seminar]]
 +
 
 +
 
 +
'''December  6th, 2021.'''
 +
 
 +
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room:  901, Time: 3:30PM-4:30PM. Host: Hung Tran.
 +
 
 +
Title: Quantitative homogenization of Hamilton-Jacobi equations
 +
 
 +
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).
 +
 
 +
 
 +
'''December  13th, 2021.'''
 +
 
 +
 
 +
[[No seminar ]]
 +
 
 +
 
 +
 
 +
== PDE GA Seminar Schedule Fall 2020-Spring 2021 ==
 +
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. 
 +
 
 +
'''Week 1 (9/1/2020-9/5/2020)'''
 +
 
 +
1. Paul Rabinowitz - The calculus of variations and phase transition problems.
 +

https://www.youtube.com/watch?v=vs3rd8RPosA
 +
 
 +
2. Frank Merle - On the implosion of a three dimensional compressible fluid.
 +
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be 
 +
 
 +
'''Week 2 (9/6/2020-9/12/2020)'''
 +
 
 +
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.
 +
https://www.youtube.com/watch?v=4ndtUh38AU0
 +
 
 +
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI
 +
 
 +
 
 +
 
 +
'''Week 3 (9/13/2020-9/19/2020)'''
 +
 
 +
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ
 +
 
 +
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE
 +
 
 +
 
 +
 
 +
'''Week 4 (9/20/2020-9/26/2020)'''
 +
 
 +
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be
 +
 
 +
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM
 +
 
 +
 
 +
 
 +
'''Week 5 (9/27/2020-10/03/2020)'''
 +
 
 +
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo
 +
 
 +
2.  Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c
 +
 
 +
 
 +
'''Week 6 (10/04/2020-10/10/2020)'''
 +
 
 +
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E
 +
 
 +
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing
 +
http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html
 +
 
 +
 
 +
'''Week 7 (10/11/2020-10/17/2020)'''
 +
 
 +
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s
 +
 
 +
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg
 +
 
 +
 
 +
'''Week 8 (10/18/2020-10/24/2020)'''
 +
 
 +
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg
 +
 
 +
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ
 +
 
 +
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.
 +
 
 +
 
 +
'''Week 9 (10/25/2020-10/31/2020)'''
 +
 
 +
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE
 +
 
 +
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764
 +
 
 +
 
 +
 
 +
'''Week 10 (11/1/2020-11/7/2020)'''
 +
 
 +
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be
 +
 
 +
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html
 +
 
 +
 
 +
 
 +
'''Week 11 (11/8/2020-11/14/2020)'''
 +
 
 +
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc
 +
 
 +
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0
 +
 
 +
 
 +
'''Week 12 (11/15/2020-11/21/2020)'''
 +
 
 +
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY
 +
 
 +
2.  Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk
 +
 
 +
 
 +
'''Week 13 (11/22/2020-11/28/2020)'''
 +
 
 +
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be
 +
 
 +
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8
 +
 
 +
'''Week 14 (11/29/2020-12/5/2020)'''
 +
 
 +
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations,
 +
https://youtu.be/xfAKGc0IEUw
 +
 
 +
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc
 +
 
 +
 
 +
 
 +
'''Week 15 (12/6/2020-12/12/2020)'''
 +
 
 +
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be
 +
 
 +
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU
 +
 
 +
 
 +
'''Spring 2021'''
 +
 
 +
'''Week 1 (1/31/2021- 2/6/2021)'''
 +
 
 +
1. Emmanuel Grenier -  instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be
 +
 
 +
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84
 +
 
 +
 
 +
'''Week 2 ( 2/7/2021- 2/13/2021)'''
 +
 
 +
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek
 +
 
 +
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE
 +
 
 +
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.
 +
 
 +
'''Week 3 ( 2/14/2021- 2/20/2021)'''
 +
 
 +
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s
 +
 
 +
2. Hao Jia -  nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg
 +
 
 +
 
 +
 
 +
'''Week 4 ( 2/21/2021- 2/27/2021)'''
 +
 
 +
1. Anne-Laure Dalibard -  Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309
  
== Abstracts ==
+
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68
===Fausto Ferrari (Bologna)===
 
''Semilinear PDEs and some symmetry properties of stable solutions''
 
  
I will deal with stable solutions of semilinear elliptic PDE's
+
'''Week 5 ( 2/28/2021- 3/6/2021)'''
and some of their symmetry's properties. Moreover, I will introduce some weighted Poincaré inequalities obtained by combining the notion of stable solution with the definition of weak solution.
 
  
===Arshak Petrosyan (Purdue)===
+
1. Inwon Kim -  A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317
''Nonuniqueness in a free boundary problem from combustion''
 
  
We consider a parabolic free boundary problem with a fixed gradient condition
+
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k
which serves as a simplified model for the propagation of premixed equidiffusional
 
flames. We give a rigorous justification of an example due to J.L. V ́azquez that
 
the initial data in the form of two circular humps leads to the nonuniqueness of limit
 
solutions if the supports of the humps touch at the time of their maximal expansion.
 
  
This is a joint work with Aaron Yip.
+
'''Week 6 (3/7/2021-3/13/2021)'''
  
 +
1. Ondřej Kreml -  Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html
  
===Changyou Wang (U. of Kentucky)===
+
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c
''Phase transition for higher dimensional wells''
 
  
For a potential function <math>F</math> that has two global minimum sets consisting of two compact connected
 
Riemannian submanifolds in <math style="vertical-align=100%" >\mathbb{R}^k</math>, we consider the singular perturbation problem:
 
  
Minimizing <math>\int \left(|\nabla u|^2+\frac{1}{\epsilon^2} F(u)\right)</math> under given Dirichlet boundary data.
+
'''Week 7 (3/14/2021-3/20/2021)'''
  
I will discuss a recent joint work with F.H.Lin and X.B.Pan on the asymptotic,  as  the parameter <math>\epsilon</math>
+
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM
tends to zero, in terms of the area of minimal hypersurface interfaces, the minimal connecting energy, and
 
the energy of minimizing harmonic maps into the phase manifolds under both Dirichlet and partially free boundary
 
data. Our results in particular addressed the static case of the so-called Keller-Rubinstein-Sternberg problem.
 
  
===Philippe LeFloch (Paris VI)===
+
2. Alexis Vasseur -  Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html
''Kinetic relations for undercompressive shock waves and propagating phase boundaries''
 
  
I will discuss the existence and properties of shock wave solutions to nonlinear hyperbolic systems that are small-scale dependent and, especially, contain undercompressive shock waves or propagating phase boundaries. Regularization-sensitive patterns often arise in continuum physics, especially in complex fluid flows. The so-called kinetic relation is introduced to characterize the correct dynamics of these nonclassical waves, and is tied to a higher-order regularization induced by a more complete model that takes into account additional small-scale physics.  In the present lecture, I will especially explain the techniques of Riemann problems, Glimm-type scheme, and total variation functionals adapted to nonclassical shock waves.
 
  
  
 +
'''Week 8 (3/21/2021- 3/27/2021)'''
  
===Irina Mitrea===
+
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs
''Boundary Value Problems for Higher Order Differential Operators''
 
  
As is well known, many phenomena in engineering and mathematical physics
+
2. François Golse -  Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A
can be modeled by means of boundary value problems for a certain elliptic
 
differential operator L in a domain D.
 
  
When L is a differential operator of second order a variety of tools
 
are available for dealing with such problems including boundary integral
 
methods,
 
variational methods, harmonic measure techniques, and methods based on
 
classical
 
harmonic analysis. The situation when the differential operator has higher order
 
(as is the case for instance with anisotropic plate bending when one
 
deals with
 
fourth order) stands in sharp contrast with this as only fewer options
 
could be
 
successfully implemented. Alberto Calderon, one of the founders of the
 
modern theory
 
of Singular Integral Operators, has advocated in the seventies the use
 
of layer potentials
 
for the treatment of higher order elliptic boundary value problems.
 
While the
 
layer potential method has proved to be tremendously successful in the
 
treatment
 
of second order problems, this approach is insufficiently developed to deal
 
with the intricacies of the theory of higher order operators. In fact,
 
it is largely
 
absent from the literature dealing with such problems.
 
  
In this talk I will discuss recent progress in developing a multiple
 
layer potential
 
approach for the treatment of boundary value problems associated with
 
higher order elliptic differential operators. This is done in a very
 
general class
 
of domains which is in the nature of best possible from the point of
 
view of
 
geometric measure theory.
 
  
 +
'''Week 9 (3/28/2021- 4/3/2021)'''
  
===Panagiota Daskalopoulos (Columbia U)===
+
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM
''Ancient solutions to geometric flows''
 
  
We will discuss the clasification of ancient solutions to nonlinear geometric flows.  
+
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9
It is well known that ancient solutions  appear as blow up limits  at a finite time 
 
singularity of the  flow.
 
Special emphasis will be given to the 2-dimensional Ricci flow.
 
In this case we will show that ancient  compact solution
 
is either the Einstein (trivial)  or one of the King-Rosenau solutions.  
 
  
===Maria Gualdani (UT Austin)===
 
''A nonlinear diffusion model in mean-field games''
 
  
We present an overview of mean-field games theory and show
+
'''Week 10 (4/4/2021- 4/10/2021)'''
recent results on a free boundary value problem, which models
 
price formation dynamics.
 
In such model, the price is formed through a game among infinite number
 
of agents.
 
Existence and regularity results, as well as linear stability, will be shown.
 
  
===Hiroshi Matano (Tokyo University)===
+
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235
''Traveling waves in a sawtoothed cylinder and their homogenization limit''
 
  
My talk is concerned with a curvature-dependent motion of plane
+
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8
curves in a two-dimensional cylinder with spatially undulating
 
boundary. In other words, the boundary has many bumps and we
 
assume that the bumps are aligned in a spatially recurrent manner.
 
  
The goal is to study how the average speed of the traveling wave
+
'''Week 11(4/11/2021- 4/17/2021)'''
depends on the geometry of the domain boundary.  More specifically,
 
we consider the homogenization problem as the boundary undulation
 
becomes finer and finer, and determine the homogenization limit
 
of the average speed and the limit profile of the traveling waves.
 
Quite surprisingly, this homogenized speed depends only on the
 
maximal opening angles of the domain boundary and no other
 
geometrical features are relevant.
 
  
Next we consider the special case where the boundary undulation
+
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo
is quasi-periodic with ''m'' independent frequencies.  We show that
 
the rate of convergence to the homogenization limit depends on
 
this number ''m''.
 
  
This is joint work with Bendong Lou and Ken-Ichi Nakamura.
+
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html
  
===Ian Tice (Brown University)===
+
'''Week 12(4/18/2021- 4/24/2021)'''
''Global well-posedness and decay for the viscous surface wave
 
problem without surface tension''
 
  
We study the incompressible, gravity-driven Navier-Stokes
+
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0
equations in three dimensional domains with free upper boundaries and
 
fixed lower boundaries, in both the horizontally periodic and
 
non-periodic settings. The effect of surface tension is not included.
 
We employ a novel two-tier nonlinear energy method that couples the
 
boundedness of certain high-regularity norms to the algebraic decay of
 
lower-regularity norms. The algebraic decay allows us to balance the
 
growth of the highest order derivatives of the free surface function,
 
which then allows us to derive a priori estimates for solutions. We
 
then prove local well-posedness in our energy space, which yields global
 
well-posedness and decay.  The novel LWP theory is established through
 
the study of the linear Stokes problem in moving domains.  This is joint
 
work with Yan Guo.
 
  
 +
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo
  
===Hoai Minh Nguyen (NYU-Courant Institute)===
+
'''Week 13(4/25/2021- 5/1/2021)'''
''Cloaking via change of variables for the Helmholtz equation''
 
  
A region of space is cloaked for a class of measurements if observers
+
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI
are not only unaware of its contents, but also unaware of the presence
 
of the cloak using such measurements. One approach to cloaking is the
 
change of variables scheme introduced by Greenleaf, Lassas, and
 
Uhlmann for electrical impedance tomography and by Pendry, Schurig,
 
and Smith for the Maxwell equation.  They used a singular change of
 
variables which blows up a point into the cloaked region. To avoid
 
this singularity, various regularized schemes have been proposed. In
 
this talk I present results related to cloaking via change of
 
variables for the Helmholtz equation using the natural regularized
 
scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the
 
authors used a transformation which blows up a small ball instead of a
 
point into the cloaked region. I will discuss the degree of
 
invisibility for a finite range or the full range of frequencies, and
 
the possibility of achieving perfect cloaking. If time permits, I will
 
mention some results related to the wave equation.
 
  
===Bing Wang (Princeton)===
+
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html
''The Kaehler Ricci flow on Fano manifold ''
 
  
We show the convergence of the Kaehler Ricci flow on every 2-dimensional Fano manifold which admits big <math>\alpha_{\nu, 1}</math>
 
or <math>\alpha_{\nu, 2}</math> (Tian's invariants).    Our method also works for 2-dimensional Fano orbifolds.
 
Since Tian's invariants can be calculated by algebraic geometry method,  our convergence theorem implies that one can find new Kaehler Einstein metrics
 
on orbifolds by calculating Tian's invariants.
 
An essential part of the proof is to confirm the Hamilton-Tian conjecture in complex dimension 2.
 
  
===Francois Hamel (Marseille)===
 
''Optimization of eigenvalues of non-symmetric elliptic operators''
 
  
The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of <math>R^n</math>. To each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new symmetrization technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.
+
{| cellpadding="8"
 +
!style="width:20%" align="left" | date 
 +
!align="left" | speaker
 +
!align="left" | title
 +
!style="width:20%" align="left" | host(s)
 +
|-  
 +
|}
  
===Juraj Foldes (Vanderbilt)===
+
== Abstracts ==
''Symmetry properties of parabolic problems and their applications''
 
  
Positive solutions of nonlinear parabolic problems can have a very complex behavior. However, assuming certain symmetry conditions, it is possible to prove that the solutions converge to the space of symmetric functions. We show that this property is 'stable'; more specifically if the symmetry conditions are replaced by asymptotically symmetric ones, the solutions still approach the space of symmetric functions. As an application, we show new results on convergence of solutions to a single equilibrium.
+
=== ===
  
===Alexey Cheskidov (UIC)===
+
Title:
''Navier-Stokes and Euler equations: a unified approach to the problem of blow-up''
 
  
The problems of blow-up for Navier-Stokes and Euler equations
+
Abstract:
have been extensively studied for decades using different techniques.
 
Motivated by Kolmogorov's theory of turbulence, we present a new unified
 
approach to the blow-up problem for the equations of incompressible
 
fluid motion. In particular, we present a new regularity criterion which
 
is weaker than the Beale-Kato-Majda condition in the inviscid case, and
 
weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case.
 

Latest revision as of 13:02, 20 November 2022

The seminar's format will be a combination of online and in-person; we will make sure to update you as soon as we have more details available. First talk is tentatively scheduled for September 20th !

Some of the seminars will be held online. When that would be the case we would use the following zoom link

https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09

Meeting ID: 963 [1353|5468 1353] Passcode: 180680


Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.


Previous PDE/GA seminars

Schedule for Fall 2022-Spring 2023

PDE GA Seminar Schedule Fall 2022-Spring 2023

Fall 2022

September 12, 2022

No Seminar



September 20, 2022 (Tuesday) joint PDE and Analysis Seminar

Andrej Zlatos (UCSD). Host: Hung Tran.

Format: in-person. Time: 4-5PM, VV B139.

Title: Homogenization in front propagation models

Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit.  I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients.  These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.



September 26, 2022

Haotian Wu (The University of Sydney, Australia). Host: Sigurd Angenent.

Format: in person, Time: 3:30pm-4:30pm VV 901

Title: Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

Abstract: The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).



October 3, 2022

No Seminar

Format: , Time:

Title:

Abstract:



October 10, 2022

Alexander Kiselev (Duke). Host: Sergey Denisov.

Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).

Speaker: Sasha Kiselev (Duke)

Title: The flow of polynomial roots under differentiation Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.


October 17, 2022

Nicolas Garca Trillos (Stats, UW Madison). Host: Hung Tran.

Format: in person, Time: 3:30pm-4:30pm VV 901

Title: Analysis of adversarial robustness and of other problems in modern machine learning.

Abstract: Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures.



October 24, 2022

No seminar.

Format: , Time:

Title:

Abstract:



October 31, 2022

Yuan Gao (Purdue). Host: Hung Tran.

Format: in person, Time: 3:30pm-4:30pm VV 901

Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures

Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both  self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.


November 7, 2022

Beomjun Choi (Postech)

Format: in person, Time: 3:30pm-4:30pm VV 901

Title: Liouville theorem for surfaces translating by powers of Gauss curvature

Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.

For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.



November 14, 2022

Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.

Format: in person, Time: 3:30pm-4:30pm VV 901

Title: Sticky Particles with Sticky Boundary

Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.



November 21, 2022

Jason Murphy (Missouri S&T)

Format: online

Join Zoom Meeting https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09 Meeting ID: 948 7748 3456 Passcode: 303105

Time: 3:30 PM -4:30 PM

Title: Sharp scattering results for the 3d cubic NLS

Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.




November 28, 2022

No Seminar- Thanksgiving




December 5, 2022

James Rowan (UC Berkeley)

Format: , Time:

Title:

Abstract:



December 12, 2022

Calumn Rickard UC Davis

Format: , Time:

Title:

Abstract:


March 27, 2023

Matt Jacobs (Purdue). Host: Hung Tran.

Format: , Time:

Title:

Abstract:



Schedule for Fall 2021-Spring 2022

PDE GA Seminar Schedule Fall 2021-Spring 2022

Spring 2022

January 31th, 2022.

No Seminar


February 7th, 2022.

Jonah Duncan from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM

Title: Estimates and regularity for the k-Yamabe equation

Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.



February 14th, 2022.

Sigurd Angenent; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM

Title: MCF after the Velázquez—Stolarski example.

Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$. Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez—Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.


February 21th, 2022.

Birgit Schoerkhuber; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09

Time: 11:00 AM

Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations

Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck).



February 28th, 2021.

Michael Hott; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09

Time:3:30PM-4:30PM

Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC

Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.


March 7th, 2022.

No Seminar;


March 14th, 2022.

Spring recess - No Seminar;



March 21th, 2022.

No Seminar;


March 28th, 2022.

Monica Visan; Format: online seminar via Zoom, Time: 3:30PM-4:30PM

https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09

Meeting ID: 963 [1353|5468 1353] Passcode: 180680

Title: Determinants, commuting flows, and recent progress on completely integrable systems

Abstract: I will survey a number of recent developments in the theory of completely integrable nonlinear dispersive PDE. These include a priori bounds, the orbital stability of multisoliton solutions, well-posedness at optimal regularity, and the existence of dynamics for Gibbs distributed initial data. I will describe the basic objects that tie together these disparate results, as well as the diverse ideas required for each problem.


April 4th, 2022.

Marcelo Disconzi; Format: online seminar via Zoom/ in person, Room: 901, Time: 3:30PM

Title: General-relativistic viscous fluids.


Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has

intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.


April 11th, 2022.

Dallas Albitron; Format: online seminar via Zoom, Time: 3:30PM-4:30PM

https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09

Meeting ID: 963 [1353|5468 1353] Passcode: 180680

Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations

Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo


April 18th, 2022.

Loc Nguyen (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.

Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.

Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.


April 25th, 2022.

No seminar


May 2nd, 2022.

Alexei Gazca; Format: online seminar via Zoom, Time:3:30PM -4:30PM

Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness

Abstract: In this talk, I will present some recent results obtained in collaboration with V. Patel (Oxford) in connection with a system describing a heat-conducting incompressible fluid. I will introduce the notion of a dissipative weak solution of the system and highlight the connections and differences to the existing approaches in the literature. One of the advantages of the proposed approach is that the solution satisfies a weak-strong uniqueness principle, which guarantees that the weak solution will coincide with the strong solution, as long as the latter exists; moreover, the solutions are constructed via a finite element approximation, leading (almost, not quite) to the first convergence result for the full system including viscous dissipation.


Fall 2021

September 20th, 2021.

Simion Schulz (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM


Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts

Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).


September 27th, 2021.

Dohyun Kwon (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM

Title: Volume-preserving crystalline and anisotropic mean curvature flow

Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).


October 4th, 2021.

Antoine Remind-Tiedrez (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM

Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries

Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.

These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).

In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.


October 11th, 2021.

No seminar


October 18th, 2021.

Wojciech Ozanski (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM

Title: Well-posedness of logarithmic spiral vortex sheets.

Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.


October 25th, 2021.

Maxwell Stolarski (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm

Title: Mean Curvature Flow Singularities with Bounded Mean Curvature

Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.


November 1th, 2021.

Lizhe Wan (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM

Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity

Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.


November 8th, 2021.

Albert Ai (UW Madison);

Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation

Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.


November 15th, 2021.

Sebastien Herr (Bielefeld University); Format: online seminar via Zoom, Time:10 AM

Please observe the time change!

Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP


Title: Global wellposedness for the energy-critical Zakharov system below the ground state

Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.


November 22th, 2021.

No seminar


November 29th, 2021.

No seminar


December 6th, 2021.

William Cooperman (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.

Title: Quantitative homogenization of Hamilton-Jacobi equations

Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).


December 13th, 2021.


No seminar


PDE GA Seminar Schedule Fall 2020-Spring 2021

Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks. 

Week 1 (9/1/2020-9/5/2020)

1. Paul Rabinowitz - The calculus of variations and phase transition problems. 
https://www.youtube.com/watch?v=vs3rd8RPosA

2. Frank Merle - On the implosion of a three dimensional compressible fluid. https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be 

Week 2 (9/6/2020-9/12/2020)

1. Yoshikazu Giga - On large time behavior of growth by birth and spread. https://www.youtube.com/watch?v=4ndtUh38AU0

2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI


Week 3 (9/13/2020-9/19/2020)

1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ

2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE


Week 4 (9/20/2020-9/26/2020)

1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be

2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM


Week 5 (9/27/2020-10/03/2020)

1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo

2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c


Week 6 (10/04/2020-10/10/2020)

1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E

2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html


Week 7 (10/11/2020-10/17/2020)

1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s

2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg


Week 8 (10/18/2020-10/24/2020)

1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg

2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ

Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.


Week 9 (10/25/2020-10/31/2020)

1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE

2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764


Week 10 (11/1/2020-11/7/2020)

1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be

2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html


Week 11 (11/8/2020-11/14/2020)

1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc

2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0


Week 12 (11/15/2020-11/21/2020)

1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY

2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk


Week 13 (11/22/2020-11/28/2020)

1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be

2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8

Week 14 (11/29/2020-12/5/2020)

1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, https://youtu.be/xfAKGc0IEUw

2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc


Week 15 (12/6/2020-12/12/2020)

1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be

2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU


Spring 2021

Week 1 (1/31/2021- 2/6/2021)

1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be

2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84


Week 2 ( 2/7/2021- 2/13/2021)

1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek

2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE

Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.

Week 3 ( 2/14/2021- 2/20/2021)

1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s

2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg


Week 4 ( 2/21/2021- 2/27/2021)

1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309

2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68

Week 5 ( 2/28/2021- 3/6/2021)

1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317

2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k

Week 6 (3/7/2021-3/13/2021)

1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html

2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c


Week 7 (3/14/2021-3/20/2021)

1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM

2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html


Week 8 (3/21/2021- 3/27/2021)

1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs

2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A


Week 9 (3/28/2021- 4/3/2021)

1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM

2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9


Week 10 (4/4/2021- 4/10/2021)

1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235

2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8

Week 11(4/11/2021- 4/17/2021)

1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo

2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html

Week 12(4/18/2021- 4/24/2021)

1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0

2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo

Week 13(4/25/2021- 5/1/2021)

1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI

2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html


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Abstracts

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