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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Fall 2016 | Tentative schedule for Spring 2017]]===
+
===[[Fall 2021-Spring 2022 | Tentative schedule for Fall 2021-Spring 2022]]===
  
= PDE GA Seminar Schedule Fall 2016 =
 
{| cellpadding="8"
 
!align="left" | date 
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
|-
 
|September 12
 
| Daniel Spirn (U of Minnesota)
 
|[[#Daniel Spirn  |  Dipole Trajectories in Bose-Einstein Condensates ]]
 
| Kim
 
|-
 
|September 19
 
| Donghyun Lee  (UW-Madison)
 
|[[#Donghyun Lee | The Boltzmann equation with specular boundary condition in convex domains ]]
 
| Feldman
 
|-
 
|September 26
 
| Kevin Zumbrun (Indiana)
 
|[[#Kevin Zumbrun |  A Stable Manifold Theorem for a class of degenerate evolution equations ]]
 
| Kim
 
|-
 
|October 3
 
| Will Feldman (UChicago )
 
|[[#Will Feldman | Liquid Drops on a Rough Surface  ]]
 
| Lin & Tran
 
|-
 
|October 10
 
| Ryan Hynd (UPenn)
 
|[[#Ryan Hynd |  Extremal functions for Morrey’s inequality in convex domains  ]]
 
| Feldman
 
|-
 
|October 17
 
| Gung-Min Gie (Louisville)
 
|[[#Gung-Min Gie | Boundary layer analysis of some incompressible flows  ]]
 
| Kim
 
|-
 
|October 24
 
| Tau Shean Lim (UW Madison)
 
|[[#Tau Shean Lim | Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators  ]]
 
| Kim & Tran
 
|-
 
|October 31 ('''Special time and room''': B313VV, 3PM-4PM)
 
| Tarek Elgindi ( Princeton)
 
|[[#Tarek Elgindi | Propagation of Singularities in Incompressible Fluids  ]]
 
| Lee & Kim
 
|-
 
|November 7
 
| Adrian Tudorascu (West Virginia)
 
|[[#Adrian Tudorascu | Hamilton-Jacobi equations in the Wasserstein space of probability measures  ]]
 
| Feldman
 
|-
 
|November 14
 
| Alexis Vasseur ( UT-Austin)
 
|[[#Alexis Vasseur |  Compressible Navier-Stokes equations with degenerate viscosities  ]]
 
| Feldman
 
|-
 
|November 21
 
| Minh-Binh Tran (UW Madison )
 
|[[#Minh-Binh Tran | Quantum Kinetic Problems  ]]
 
| Hung Tran
 
|-
 
|November 28
 
|  David Kaspar (Brown)
 
|[[#David Kaspar | Kinetics of shock clustering  ]]
 
|Tran
 
|-
 
|December 5 ('''Special time and room''': 3PM-4PM, B313)
 
| Brian Weber (University of Pennsylvania)
 
|[[#Brian Weber | Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds  ]]
 
|Bing Wang
 
|-
 
|December 12
 
|
 
|[[# |    ]]
 
|
 
|}
 
  
=Abstracts=
 
  
===Daniel Spirn===
+
== 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
  
Dipole Trajectories in Bose-Einstein Condensates
 
  
Bose-Einstein condensates (BEC) are a state of matter in which supercooled atoms condense into the lowest possible quantum state.  One interesting important feature of BECs are the presence of vortices that form when the condensate is stirred with lasers.  I will discuss the behavior of these vortices, which interact with both the confinement potential and other vortices.  I will also discuss a related inverse problem in which the features of the confinement can be extracted by the propagation of vortex dipoles.
 
  
===Donghyun Lee===
+
'''Week 5 (9/27/2020-10/03/2020)'''
  
The Boltzmann equation with specular reflection boundary condition in convex domains
+
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo
  
I will present a recent work (https://arxiv.org/abs/1604.04342) with Chanwoo Kim on the global-wellposedness and stability of the Boltzmann equation in general smooth convex domains.
+
2.  Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c
  
===Kevin Zumbrun===
 
  
TITLE: A Stable Manifold Theorem for a class of degenerate evolution equations
+
'''Week 6 (10/04/2020-10/10/2020)'''
  
ABSTRACT: We establish a Stable Manifold Theorem, with consequent exponential decay to equilibrium, for a class
+
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
  
of degenerate evolution equations $Au'+u=D(u,u)$ with A bounded, self-adjoint, and one-to-one, but not invertible, and
+
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
  
$D$ a bounded, symmetric bilinear map.  This is related to a number of other scenarios investigated recently for which the
 
  
associated linearized ODE $Au'+u=0$ is ill-posed with respect to the Cauchy problem.  The particular case studied here
+
'''Week 7 (10/11/2020-10/17/2020)'''
  
pertains to the steady Boltzmann equation, yielding exponential decay of large-amplitude shock and boundary layers.
+
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
  
  
===Will Feldman===
+
'''Week 8 (10/18/2020-10/24/2020)'''
  
Liquid Drops on a Rough Surface
+
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg
  
I will discuss the problem of determining the minimal energy shape of a liquid droplet resting on a rough solid surface. The shape of a liquid drop on a solid is strongly affected by the micro-structure of the surface on which it rests, where the surface inhomogeneity arises through varying chemical composition and surface roughness. I will explain a macroscopic regularity theory for the free boundary which allows to study homogenization, and more delicate properties like the size of the boundary layer induced by the surface roughness.  
+
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ
  
The talk is based on joint work with Inwon Kim. A remark for those attending the weekend conference: this talk will attempt to have as little as possible overlap with I. Kim's conference talks.  
+
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.
  
===Ryan Hynd===
 
  
Extremal functions for Morrey’s inequality in convex domains
+
'''Week 9 (10/25/2020-10/31/2020)'''
  
A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold.  
+
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE
  
===Gung-Min Gie ===
+
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
  
Boundary layer analysis of some incompressible flows
 
 
The motions of viscous and inviscid fluids are modeled respectively by the Navier-Stokes and Euler equations. Considering the Navier-Stokes equations at vanishing viscosity as a singular perturbation of the Euler equations, one major problem, still essentially open, is to verify if the Navier-Stokes solutions converge as the viscosity tends to zero to the Euler solution in the presence of physical boundary. In this talk, we study the inviscid limit and boundary layers of some simplified Naiver-Stokes equations by either imposing a certain symmetry to the flow or linearizing the model around a stationary Euler flow. For the examples, we systematically use the method of correctors proposed earlier by J. L. Lions and construct an asymptotic expansion as the sum of the Navier-Stokes solution and the corrector. The corrector, which corrects the discrepancies between the boundary values of the viscous and inviscid solutions, is in fact an (approximating) solution of the corresponding Prandtl type equations. The validity of our asymptotic expansions is then confirmed globally in the whole domain by energy estimates on the difference of the viscous solution and the proposed expansion. This is a joint work with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes.
 
  
===Tau Shean Lim===
 
  
Traveling Fronts of Reaction-Diffusion Equations with Ignition Media and Levy Operators
+
'''Week 10 (11/1/2020-11/7/2020)'''
  
We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.
+
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be
  
===Tarek M. ELgindi===
+
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
  
Propagation of Singularities in Incompressible Fluids
 
  
We will discuss some recent results on the local and global stability of certain singular solutions to the incompressible 2d Euler equation. We will begin by giving a brief overview of the classical and modern results on the 2d Euler equation--particularly related to well-posedness theory in critical spaces. Then we will present a new well-posedness class which allows for merely Lipschitz continuous velocity fields and non-decaying vorticity. This will be based upon some interesting estimates for singular integrals on spaces with L^\infty scaling. After that we will introduce a class of scale invariant solutions to the 2d Euler equation and describe some of their remarkable properties including the existence of pendulum-like quasi periodic solutions and infinite-time cusp formation in vortex patches with corners. This is a joint work with I. Jeong.
 
  
 +
'''Week 11 (11/8/2020-11/14/2020)'''
  
===Adrian Tudorascu===
+
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc
  
Hamilton-Jacobi equations in the Wasserstein space of probability measures
+
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0
  
In 2008 Gangbo, Nguyen and Tudorascu showed that certain variational solutions of the Euler-Poisson system in 1D can be regarded as optimal paths for the value-function giving the viscosity solution of some (infinite-dimensional) Hamilton-Jacobi equation whose phase-space is the Wasserstein space of Borel probability measures with finite second moment. At around the same time, Lasry, Lions, and others became interested in such Hamilton-Jacobi equations (HJE) in connection with their developing theory of Mean-Field games. A different approach (less intrinsic than ours) to the notion of viscosity solution was preferred, one that made an immediate connection between HJE in the Wasserstein space and HJE in Hilbert spaces (whose theory was well-studied and fairly well-understood). At the heart of the difference between these approaches lies the choice of the sub/supper-differential in the context of the Wasserstein space (i.e. the interpretation of ``cotangent space'' to this ``pseudo-Riemannian'' manifold) . In this talk I will start with a brief introduction to Mean-Field games and Optimal Transport, then I will discuss the challenges we encounter in the analysis of (our intrinsic) viscosity solutions of HJE in the Wasserstein space. Based on joint work with W. Gangbo.
 
  
===Alexis Vasseur===
+
'''Week 12 (11/15/2020-11/21/2020)'''
  
Compressible Navier-Stokes equations with degenerate viscosities 
+
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
  
We will discuss recent results on the construction of weak solutions for
+
2.  Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk
3D compressible Navier-Stokes equations with degenerate viscosities.
 
The method is based on the Bresch and Desjardins entropy. The main
 
contribution is to derive  MV type inequalities for the weak solutions,
 
even if it is not verified by the first level of approximation. This
 
provides existence of global solutions in time, for the compressible
 
Navier-Stokes equations,  in three dimensional space, with large initial
 
data, possibly vanishing on the vacuum.
 
  
===Minh-Binh Tran===
 
  
Quantum kinetic problems
+
'''Week 13 (11/22/2020-11/28/2020)'''
  
After the production of the first BECs, there has been an explosion of research on the kinetic theory associated to BECs. Later, Gardinier, Zoller and collaborators derived a Master Quantum Kinetic Equation for BECs and introduced the terminology ”Quantum Kinetic Theory”. In 2012, Reichl and collaborators made a breakthrough in discovering a new collision operator, which had been missing in the previous works.
+
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be
My talk is devoted to the description of our recent mathematical works on quantum kinetic theory. The talk will be based on my joint works with Alonso, Gamba (existence, uniqueness, propagation of moments), Nguyen (Maxwellian lower bound), Soffer (coupling Schrodinger–kinetic equations), Escobedo (convergence to equilibrium), Craciun (the analog between the global attractor conjecture in chemical reaction network and the convergence to equilibrium of quantum kinetic equations), Reichl (derivation).
 
  
===David Kaspar===
+
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8
  
Kinetics of shock clustering
+
{| cellpadding="8"
 +
!style="width:20%" align="left" | date 
 +
!align="left" | speaker
 +
!align="left" | title
 +
!style="width:20%" align="left" | host(s)
 +
|- 
 +
|}
  
Suppose we solve a (deterministic) scalar conservation law
+
== Abstracts ==
with random initial data.  Can we describe the probability law of the
 
solution as a stochastic process in x for fixed later time t?  The
 
answer is yes, for certain Markov initial data, and the probability
 
law factorizes as a product of kernels.  These kernels are obtained by
 
solving a mean-field kinetic equation which most closely resembles the
 
Smoluchowski coagulation equation.  We discuss prior and ongoing work
 
concerning this and related problems.
 
  
===Brian Weber===
+
=== ===
  
Degenerate-Elliptic PDE and Toric Kahler 4-manfiolds
+
Title: 
  
Understanding scalar-flat instantons is crucial for knowing how Ka ̈hler manifolds degenerate. It is known that scalar-flat Kahler 4-manifolds with two symmetries give rise to a pair of linear degenerate-elliptic Heston type equations
+
Abstract:
of the form x(fxx + fyy) + fx = 0, which were originally studied in mathematical finance. Vice- versa, solving these PDE produce scalar-flat Kahler 4-manifolds. These PDE have been studied locally, but here we describe new global results
 
and their implications, partic- ularly a classification of scalar-flat metrics on K ̈ahler 4-manifolds and applications for the study of constant scalar curvature and extremal Ka ̈hler metrics.
 

Latest revision as of 21:46, 22 November 2020

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Previous PDE/GA seminars

Tentative schedule for Fall 2021-Spring 2022

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

date speaker title host(s)

Abstracts

Title:

Abstract: