Difference between revisions of "PDE Geometric Analysis seminar"

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(PDE GA Seminar Schedule Fall 2019-Spring 2020)
 
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The seminar will be held  in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
+
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]]===
 
===[[Previous PDE/GA seminars]]===
 +
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===
 +
 +
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==
 +
  
== Seminar Schedule Spring 2012 ==
 
 
{| cellpadding="8"
 
{| cellpadding="8"
!align="left" | date   
+
!style="width:20%" align="left" | date   
 
!align="left" | speaker
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | title
!align="left" | host(s)
+
!style="width:20%" align="left" | host(s)
 +
|- 
 +
|Sep 9
 +
| Scott Smith (UW Madison)
 +
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]
 +
| Kim and Tran
 +
|- 
 +
|Sep 14-15
 +
|
 +
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html  ]]
 +
 +
|- 
 +
|Sep 23
 +
| Son Tu (UW Madison)
 +
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]
 +
| Kim and Tran
 +
|- 
 +
|Sep 28-29, VV901
 +
|  https://www.ki-net.umd.edu/content/conf?event_id=993
 +
|  |  Recent progress in analytical aspects of kinetic equations and related fluid models 
 +
 +
|- 
 +
|Oct 7
 +
| Jin Woo Jang (Postech)
 +
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]
 +
| Kim
 +
|- 
 +
|Oct 14
 +
| Stefania Patrizi (UT Austin)
 +
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]
 +
| Tran
 +
|- 
 +
|Oct 21
 +
| Claude Bardos (Université Paris Denis Diderot, France)
 +
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]
 +
| Li
 +
|- 
 +
|Oct 25-27, VV901
 +
| https://www.ki-net.umd.edu/content/conf?event_id=1015
 +
||  Forward and Inverse Problems in Kinetic Theory
 +
| Li
 +
|-
 +
|Oct 28
 +
| Albert Ai (UW Madison)
 +
|[[#Albert Ai | TBA ]]
 +
| Ifrim
 +
|- 
 +
|Nov 4
 +
| Yunbai Cao (UW Madison)
 +
|[[#Yunbai Cao | TBA ]]
 +
| Kim and Tran
 +
|- 
 +
|Nov 11
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|Nov 18
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 
|-
 
|-
|Feb 6
+
|Nov 25
|Yao Yao (UCLA)
+
| Mathew Langford (UT Knoxville)
|[[#Yao Yao (UCLA)|
+
|[[#Speaker | TBA ]]
  Degenerate diffusion with nonlocal aggregation: behavior of solutions]]
+
| Angenent
|Kiselev
+
|- 
|-
+
|- 
|March 12
+
|Feb 17
| Xuan Hien Nguyen (Iowa State)
+
| Yannick Sire (JHU)
|[[#Xuan Hien Nguyen (Iowa State)|
+
|[[#Yannick Sire (JHU) | TBA ]]
Gluing constructions for solitons and self-shrinkers under mean curvature flow]]
+
| Tran
|Angenent
+
|-  
|-
+
|Feb 24
|March 21(Wednesday!), Room 901 Van Vleck
+
| Speaker (Institute)
|Nestor Guillen (UCLA)
+
|[[#Speaker | TBA ]]
|[[#Nestor Guillen (UCLA)|
+
| Host
The local geometry of maps with c-convex potentials]]
+
|-
|Feldman
+
|March 2
|-
+
| Theodora Bourni (UT Knoxville)
|March 26
+
|[[#Speaker | TBA ]]
|Vlad Vicol (University of Chicago)
+
| Angenent
|[[#Vlad Vicol (U Chicago)|
+
|-
Shape dependent maximum principles and applications]]
+
|March 9
|Kiselev
+
| Ian Tice (CMU)
|-
+
|[[#Ian Tice| TBA ]]
|April 9
+
| Kim
|Charles Smart (MIT)  
+
|- 
|[[#Charles Smart (MIT)|
+
|March 16
TBA]]
+
| No seminar (spring break)
|Seeger
+
|[[#Speaker | TBA ]]
|-
+
| Host
|April 16
+
|- 
|Jiahong Wu (Oklahoma)
+
|March 23
|[[#Jiahong Wu (Oklahoma State)|
+
| Jared Speck (Vanderbilt)
The 2D Boussinesq equations with partial dissipation]]
+
|[[#Jared Speck | TBA ]]
|Kiselev
+
| SCHRECKER
|-
+
|-
|May 14
+
|March 30
|Jacob Glenn-Levin (UT Austin)
+
| Speaker (Institute)
|[[#Jacob Glenn-Levin (UT Austin)|
+
|[[#Speaker | TBA ]]
  TBA]]
+
| Host
|Kiselev
+
|-
 +
|April 6
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|-
 +
|April 13
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|-
 +
|April 20
 +
| Hyunju Kwon (IAS)
 +
|[[#Hyunju Kwon | TBA ]]
 +
| Kim
 +
|-
 +
|April 27
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|-  
 +
|May 18-21
 +
| Madison Workshop in PDE 2020
 +
|[[#Speaker | TBA ]]
 +
| Tran
 
|}
 
|}
  
==Abstracts==
+
== Abstracts ==
 +
 
 +
===Scott Smith===
  
===Yao Yao (UCLA)===
+
Title: Recent progress on singular, quasi-linear stochastic PDE
''Degenerate diffusion with nonlocal aggregation: behavior of solutions''
 
  
The Patlak-Keller-Segel (PKS) equation models the collective motion of
+
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing .  These equations are ill-posed in the traditional sense of distribution theoryThey require flexibility in the notion of solution as well as new a priori boundsDrawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.
cells which are attracted by a self-emitted chemical substanceWhile the
 
global well-posedness and finite-time blow up criteria are well known, the
 
asymptotic behaviors of solutions are not completely clearIn this talk I
 
will present some results on the asymptotic behavior of solutions when
 
there is global existence. The key tools used in the paper are
 
maximum-principle type arguments as well as estimates on mass concentration
 
of solutions. This is a joint work with Inwon Kim.
 
  
===Xuan Hien Nguyen (Iowa State)===
 
  
''Gluing constructions for solitons and self-shrinkers under mean curvature flow''
+
===Son Tu===
  
In the 1990s, Kapouleas and Traizet constructed new examples of minimal surfaces by desingularizing the intersection of existing ones with Scherk surfaces. Using this idea, one can find new examples of self-translating solutions for the mean curvature flow asymptotic at infinity to a finite family of grim reaper cylinders in general position. Recently, it has been shown that it is possible to desingularize the intersection of a sphere and a plane to obtain a family of self-shrinkers under mean curvature flow. I will discuss the main steps and difficulties for these gluing constructions, as well as open problems.
+
Title: State-Constraint static Hamilton-Jacobi equations in nested domains
  
===Nestor Guillen (UCLA)===
+
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases,  the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).
  
We consider the Monge-Kantorovich problem, which consists in
 
transporting a given measure into another "target" measure in a way
 
that minimizes the total cost of moving each unit of mass to its new
 
location. When the transport cost is given by the square of the
 
distance between two points, the optimal map is given by a convex
 
potential which solves the Monge-Ampère equation, in general, the
 
solution is given by what is called a c-convex potential. In recent
 
work with Jun Kitagawa, we prove local Holder estimates of optimal
 
transport maps for more general cost functions satisfying a
 
"synthetic" MTW condition, in particular, the proof does not really
 
use the C^4 assumption made in all previous works. A similar result
 
was recently obtained by Figalli, Kim and McCann using different
 
methods and assuming strict convexity of the target.
 
  
===Charles Smart (MIT)===
+
===Jin Woo Jang===
  
''PDE methods for the Abelian sandpile''
+
Title: On a Cauchy problem for the Landau-Boltzmann equation
  
Abstract: The Abelian sandpile growth model is a deterministic
+
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.
diffusion process for chips placed on the $d$-dimensional integer
 
lattice. One of the most striking features of the sandpile is that it
 
appears to produce terminal configurations converging to a peculiar
 
lattice. One of the most striking features of the sandpile is that it
 
appears to produce terminal configurations converging to a peculiar
 
fractal limit when begun from increasingly large stacks of chips at
 
the origin. This behavior defied explanation for many years until
 
viscosity solution theory offered a new perspective.  This is joint
 
work with Lionel Levine and Wesley Pegden.
 
  
===Vlad Vicol (University of Chicago)===
 
  
Title: Shape dependent maximum principles and applications
+
===Stefania Patrizi===
  
Abstract: We present a non-linear lower bound for the fractional Laplacian, when
+
Title:  
evaluated at extrema of a function. Applications to the global well-posedness of active
+
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity
scalar equations arising in fluid dynamics are discussed. This is joint work with P.
 
Constantin.
 
  
 +
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic  limit of  solutions of  Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is  joint work with E. Valdinoci.
  
===Jiahong Wu (Oklahoma State)===
 
  
"The 2D Boussinesq equations with partial dissipation"
+
===Claude Bardos===
 +
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture
  
The Boussinesq equations concerned here model geophysical flows such
+
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.
as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq
 
equations serve as a lower-dimensional model of the 3D hydrodynamics
 
equations. In fact, the 2D Boussinesq equations retain some key features
 
of the 3D Euler and the Navier-Stokes equations such as the vortex stretching
 
mechanism.  The global regularity problem on the 2D Boussinesq equations
 
with partial dissipation has attracted considerable attention in the last few years.
 
In this talk we will summarize recent results on various cases of partial dissipation,
 
present the work of Cao and Wu on the 2D Boussinesq equations with vertical
 
dissipation and vertical thermal diffusion, and explain  the work of Chae and Wu on
 
the critical Boussinesq equations with a logarithmically singular velocity.
 
  
===Jacob Glenn-Levin (UT Austin)===
+
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.
  
TBA
+
Eventually the above results are compared with  several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.

Latest revision as of 13:45, 15 October 2019

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 2020-Spring 2021

PDE GA Seminar Schedule Fall 2019-Spring 2020

date speaker title host(s)
Sep 9 Scott Smith (UW Madison) Recent progress on singular, quasi-linear stochastic PDE Kim and Tran
Sep 14-15 AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html
Sep 23 Son Tu (UW Madison) State-Constraint static Hamilton-Jacobi equations in nested domains Kim and Tran
Sep 28-29, VV901 https://www.ki-net.umd.edu/content/conf?event_id=993 Recent progress in analytical aspects of kinetic equations and related fluid models
Oct 7 Jin Woo Jang (Postech) On a Cauchy problem for the Landau-Boltzmann equation Kim
Oct 14 Stefania Patrizi (UT Austin) Dislocations dynamics: from microscopic models to macroscopic crystal plasticity Tran
Oct 21 Claude Bardos (Université Paris Denis Diderot, France) From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture Li
Oct 25-27, VV901 https://www.ki-net.umd.edu/content/conf?event_id=1015 Forward and Inverse Problems in Kinetic Theory Li
Oct 28 Albert Ai (UW Madison) TBA Ifrim
Nov 4 Yunbai Cao (UW Madison) TBA Kim and Tran
Nov 11 Speaker (Institute) TBA Host
Nov 18 Speaker (Institute) TBA Host
Nov 25 Mathew Langford (UT Knoxville) TBA Angenent
Feb 17 Yannick Sire (JHU) TBA Tran
Feb 24 Speaker (Institute) TBA Host
March 2 Theodora Bourni (UT Knoxville) TBA Angenent
March 9 Ian Tice (CMU) TBA Kim
March 16 No seminar (spring break) TBA Host
March 23 Jared Speck (Vanderbilt) TBA SCHRECKER
March 30 Speaker (Institute) TBA Host
April 6 Speaker (Institute) TBA Host
April 13 Speaker (Institute) TBA Host
April 20 Hyunju Kwon (IAS) TBA Kim
April 27 Speaker (Institute) TBA Host
May 18-21 Madison Workshop in PDE 2020 TBA Tran

Abstracts

Scott Smith

Title: Recent progress on singular, quasi-linear stochastic PDE

Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.


Son Tu

Title: State-Constraint static Hamilton-Jacobi equations in nested domains

Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).


Jin Woo Jang

Title: On a Cauchy problem for the Landau-Boltzmann equation

Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.


Stefania Patrizi

Title: Dislocations dynamics: from microscopic models to macroscopic crystal plasticity

Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.


Claude Bardos

Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture

Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.

Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.

Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.