PDE Geometric Analysis seminar: Difference between revisions

From UW-Math Wiki
Jump to navigation Jump to search
(286 intermediate revisions by 14 users not shown)
Line 2: Line 2:


===[[Previous PDE/GA seminars]]===
===[[Previous PDE/GA seminars]]===
===[[Spring 2016 | Tentative schedule for Spring 2016]]===
===[[Fall 2018 | Tentative schedule for Fall 2018]]===






= Seminar Schedule Fall 2015 =
== PDE GA Seminar Schedule Spring 2018 ==
 
 
{| 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)
|-
 
|September 7 (Labor Day)
|- 
|  
|January 29, '''3-3:50PM,  B341 VV.'''
|[[#  |  ]]
| Dan Knopf (UT Austin)
|
|[[#Dan Knopf |  Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons]]
| Angenent
|-
|February 5,  '''3-3:50PM,  B341 VV.'''
| Andreas Seeger (UW)
|[[#Andreas Seeger |  Singular integrals and  a problem on mixing flows ]]
| Kim & Tran
|-
|February 12
| Sam Krupa (UT-Austin)
|[[#Sam Krupa |  Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]
| Lee
|-
|February 19
| Maja Taskovic (UPenn)
|[[#Maja Taskovic |  Exponential tails for the non-cutoff Boltzmann equation ]]
| Kim
|-  
|February 26
|  Ashish Kumar Pandey (UIUC)
|[[#  |  Instabilities in shallow water wave models  ]]
| Kim & Lee
|-
|March 5
| Khai Nguyen (NCSU)
|[[#Khai Nguyen | Burgers Equation with Some Nonlocal Sources ]]
| Tran
|-
|March 12
| Hongwei Gao (UCLA)
|[[#Hongwei Gao |  Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations ]]
| Tran
|-
|March 19
| Huy Nguyen (Princeton)
|[[#Huy Nguyen Compressible fluids and active potentials ]]
| Lee
|-
|-
|September 14 (special room: B115)
|March 26
| Hung Tran (Madison)
|[[#Hung Tran  | Some inverse problems in periodic homogenization of Hamilton--Jacobi equations ]]
|  
|  
|-
|[[# | Spring recess (Mar 24-Apr 1, 2018) ]]
|September 21 (special room: B115)
| Eric Baer (Madison)
||[[#Eric Baer | Optimal function spaces for continuity of the Hessian determinant as a distribution ]]
|-
|September 28
| Donghyun Lee (Madison)
|[[#Donghyun Lee  | FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT]]
|   
|   
|-
|-
|October 5
|April 2
|Hyung-Ju Hwang (Postech & Brown Univ)
| In-Jee Jeong (Princeton)
|[[#Hyung-Ju Hwang | The Fokker-Planck equation in bounded domains ]]
|[[#In-Jee Jeong Singularity formation for the 3D axisymmetric Euler equations ]]
| Kim
| Kim
|-
|-  
|October 12
|April 9
| Minh-Binh Tran (Madison)
| Jeff Calder (Minnesota)
|[[#Minh-Binh Tran | Nonlinear approximation theory for kinetic equations ]]
|[[#Jeff Calder | TBD ]]
|
|-
|October 19
| Bob Jensen (Loyola University Chicago)
||[[#Bob Jensen | Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs ]]
| Tran
| Tran
|-
|-  
|October 26
|April 21-22 (Saturday-Sunday)
|Luis Silvestre (Chicago)
| Midwest PDE seminar
|[[# Luis Silvestre  | TBA  ]]
|[[#Midwest PDE seminar |  ]]
|Kim
| Angenent, Feldman, Kim, Tran.
|-
|-  
|November 2
|April 25 (Wednesday)
| Connor Mooney (UT Austin)
| Hitoshi Ishii (Wasow lecture)
|[[# Connor Mooney | TBA  ]]
|[[#Hitoshi Ishii TBD]]
|Lin
| Tran.
|-
|November 9
| Javier Gomez-Serrano (Princeton)
||[[# Javier Gomez-Serrano | TBA ]]
|Zlatos
|-
|November 16
| Yifeng Yu (UC Irvine)
|[[# Yifeng Yu | TBA ]]
| Tran
|-
|November 23
| Nam Le (Indiana)
|[[# Nam Le | TBA ]]
|Tran
|-
|November 30
| Qin Li (Madison)
|[[# Qin Li | TBA ]]
|
|-
|December 7
| Lu Wang (Madison)
||[[# Lu Wang | TBA ]]
|
|-
|December 14
| Christophe Lacave (Paris 7)
|[[# Christophe Lacave | TBA ]]
| Zlatos
|}
|}


=Abstract=
== Abstracts ==
 
===Dan Knopf===
 
Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons
 
Abstract: We describe Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kähler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kähler solutions of Ricci flow that become asymptotically Kähler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kähler metrics under Ricci flow.
 
===Andreas Seeger===
 
Title: Singular integrals and a problem on mixing flows
 
Abstract: The talk will be about  results related to Bressan's mixing problem. We present  an inequality for the change of a  Bianchini semi-norm of characteristic functions under the  flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator  for which one proves bounds  on Hardy spaces. This is joint work with Mahir Hadžić,  Charles Smart and    Brian Street.
 
===Sam Krupa===
 
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case
 
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi.  Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.
 
 
===Maja Taskovic===
 
Title: Exponential tails for the non-cutoff Boltzmann equation


===Hung Tran===
Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function.  The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel.  For a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.


Some inverse problems in periodic homogenization of Hamilton--Jacobi equations.
We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.


Abstract: We look at the effective Hamiltonian $\overline{H}$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\overline{H}$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. I will discuss some interesting cases in both convex and nonconvex settings. Joint work with Songting Luo and Yifeng Yu.


===Ashish Kumar Pandey===


===Eric Baer===
Title: Instabilities in shallow water wave models


Optimal function spaces for continuity of the Hessian determinant as a distribution.
Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.


Abstract: In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. Inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space $B(2-2/N,N)$ of fractional order, and that all continuity results in this scale of Besov spaces are consequences of this result.  A key ingredient in the argument is the characterization of $B(2-2/N,N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ (of codimension 2).  The most elaborate part of the analysis is the construction of a counterexample to continuity in $B(2-2/N,p)$ with $p>N$.  Tools involved in this step include the choice of suitable ``atoms" having a tensor product structure and Hessian determinant of uniform sign, formation of lacunary series of rescaled atoms, and delicate estimates of terms in the resulting multilinear expressions.


===Donghyun Lee===
===Khai Nguyen===


FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT.
Title: Burgers Equation with Some Nonlocal Sources


Abstract : Free-boundary problems of incompressible fluids have been studied for several decades. In the viscous case, it is basically solved by Stokes regularity. However, the inviscid case problem is generally much harder, because the problem is purely hyperbolic. In this talk, we approach the problem via vanishing viscosity limit, which is a central problem of fluid mechanics. To correct boundary layer behavior, conormal Sobolev space will be introduced. In the spirit of the recent work by N.Masmoudi and F.Rousset (2012, non-surface tension), we will see how to get local regularity of incompressible free-boundary Euler, taking surface tension into account. This is joint work with Tarek Elgindi.
Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency. This talk  will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations. Some open questions will be discussed.
If possible, we also talk about applying the similar technique to the free-boundary MHD(Magnetohydrodynamics). Especially, we will see that strong zero initial boundary condition is still valid for this coupled PDE. For the general boundary condition (for perfect conductor), however, the problem is still open.


=== Hyung-Ju Hwang===
===Hongwei Gao===


The Fokker-Planck equation in bounded domains
Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations


abstract: In this talk, we consider the initial-boundary value problem for the Fokker-Planck equation in an interval or in a bounded domain with absorbing boundary conditions. We discuss a theory of well-posedness of classical solutions for the problem as well as the exponential decay in time, hypoellipticity away from the singular set, and the Holder continuity of the solutions up to the singular set. This is a joint work with J. Jang, J. Jung, and J. Velazquez.
Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.


=== Minh-Binh Tran ===
===Huy Nguyen===


Nonlinear approximation theory for kinetic equations
Title : Compressible fluids and active potentials


Abstract: Numerical resolution methods for the Boltzmann equation plays a very important role in the practical a theoretical study of the theory of rarefied gas. The main difficulty in the approximation of the Boltzmann equation is due to the multidimensional structure of the Boltzmann collision operator. The major problem with deterministic numerical methods using to solve Boltzmann equation is that we have to truncate the domain or to impose nonphysical conditions to keep the supports of the solutions in the velocity space uniformly compact. I
Abstract: We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.
n this talk, we will introduce our new way to make the connection between nonlinear approximation theory and kinetic theory. Our nonlinear wavelet approximation is nontruncated and based on an adaptive spectral method associated with a new wavelet filtering technique. The approximation is proved to converge and preserve many properties of the homogeneous Boltzmann equation. The nonlinear approximation solves the equation without having to impose non-physics conditions on the equation.


=== Bob Jensen ===
===In-Jee Jeong===


Crandall-Lions Viscosity Solutions of Uniformly Elliptic PDEs
Title: Singularity formation for the 3D axisymmetric Euler equations


Abstract: I will discuss C-L viscosity solutions of uniformly elliptic partial differential equations for operators with only measurable spatial regularity. E.g., $L[u] = \sum a_{i\,j}(x)\,D_{i\,j}u(x)$ where $a_{i\,j}(x)$ is bounded, uniformly elliptic, and measurable in $x$. In general there isn't a meaningful extension of the C-L viscosity solution definition to operators with measurable spatial dependence.  But under uniform ellipticity there is a natural extension. Though there isn't a general comparison principle in this context, we will see that the extended definition is robust and uniquely characterizes the ``right" solutions for such problems.
Abstract: We consider the 3D axisymmetric Euler equations on exterior domains $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2  \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.

Revision as of 18:55, 13 March 2018

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 2018

PDE GA Seminar Schedule Spring 2018

date speaker title host(s)
January 29, 3-3:50PM, B341 VV. Dan Knopf (UT Austin) Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons Angenent
February 5, 3-3:50PM, B341 VV. Andreas Seeger (UW) Singular integrals and a problem on mixing flows Kim & Tran
February 12 Sam Krupa (UT-Austin) Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case Lee
February 19 Maja Taskovic (UPenn) Exponential tails for the non-cutoff Boltzmann equation Kim
February 26 Ashish Kumar Pandey (UIUC) Instabilities in shallow water wave models Kim & Lee
March 5 Khai Nguyen (NCSU) Burgers Equation with Some Nonlocal Sources Tran
March 12 Hongwei Gao (UCLA) Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations Tran
March 19 Huy Nguyen (Princeton) Compressible fluids and active potentials Lee
March 26 Spring recess (Mar 24-Apr 1, 2018)
April 2 In-Jee Jeong (Princeton) Singularity formation for the 3D axisymmetric Euler equations Kim
April 9 Jeff Calder (Minnesota) TBD Tran
April 21-22 (Saturday-Sunday) Midwest PDE seminar Angenent, Feldman, Kim, Tran.
April 25 (Wednesday) Hitoshi Ishii (Wasow lecture) TBD Tran.

Abstracts

Dan Knopf

Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons

Abstract: We describe Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kähler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kähler solutions of Ricci flow that become asymptotically Kähler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kähler metrics under Ricci flow.

Andreas Seeger

Title: Singular integrals and a problem on mixing flows

Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.

Sam Krupa

Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case

Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.


Maja Taskovic

Title: Exponential tails for the non-cutoff Boltzmann equation

Abstract: The Boltzmann equation models the motion of a rarefied gas, in which particles interact through binary collisions, by describing the evolution of the particle density function. The effect of collisions on the density function is modeled by a bilinear integral operator (collision operator) which in many cases has a non-integrable angular kernel. For a long time the equation was simplified by assuming that this kernel is integrable (the so called Grad's cutoff) with a belief that such an assumption does not affect the equation significantly. However, in the last 20 years it has been observed that a non-integrable singularity carries regularizing properties which motivates further analysis of the equation in this setting.

We study behavior in time of tails of solutions to the Boltzmann equation in the non-cutoff regime by examining the generation and propagation of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose we introduce Mittag-Leffler moments which can be understood as a generalization of exponential moments. An interesting aspect of this result is that the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time. This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.


Ashish Kumar Pandey

Title: Instabilities in shallow water wave models

Abstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM, and regularized Boussinesq type equations.


Khai Nguyen

Title: Burgers Equation with Some Nonlocal Sources

Abstract: Consider the Burgers equation with some nonlocal sources, which were derived from models of nonlinear wave with constant frequency. This talk will present some recent results on the global existence of entropy weak solutions, priori estimates, and a uniqueness result for both Burgers-Poisson and Burgers-Hilbert equations. Some open questions will be discussed.

Hongwei Gao

Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations

Abstract: In this talk, we discuss the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of (level-set) convex Hamiltonians and a sequence of (level-set) concave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization. If time permits, we will talk about some homogenization results when the monotonicity assumption breaks down.

Huy Nguyen

Title : Compressible fluids and active potentials

Abstract: We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.

In-Jee Jeong

Title: Singularity formation for the 3D axisymmetric Euler equations

Abstract: We consider the 3D axisymmetric Euler equations on exterior domains $\{ (x,y,z) : (1 + \epsilon|z|)^2 \le x^2 + y^2 \} $ for any $\epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi.