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===[[Previous PDE/GA seminars]]===
===[[Previous PDE/GA seminars]]===
===[[Fall 2015 | Tentative schedule for Fall 2015]]===
===[[Fall 2018 | Tentative schedule for Fall 2018]]===
 
 
 
== PDE GA Seminar Schedule Spring 2018 ==
 


= Seminar Schedule Spring 2015 =
{| 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)
|-
 
|January 21 (Departmental Colloquium: 4PM, B239)
|-
|Jun Kitagawa (Toronto) 
|January 29, '''3-3:50PM,  B341 VV.'''
|[[#Jun Kitagawa (Toronto)  | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics  ]]
| Dan Knopf (UT Austin)
|Feldman
|[[#Dan Knopf | Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons]]
|-
| Angenent
|'''February 9'''  
|-  
|Jessica Lin (Madison)
|February 5,  '''3-3:50PM,  B341 VV.'''
|[[#Jessica Lin (Madison) | Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations ]]
| Andreas Seeger (UW)
|Kim
|[[#Andreas Seeger |  Singular integrals and a problem on mixing flows ]]
|-
| Kim & Tran
|February 17 ('''Tuesday''') (joint with Analysis Seminar: 4PM, B139)
|-  
|Chanwoo Kim (Madison)  
|February 12
|[[#Chanwoo Kim (Madison) | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier ]]
| Sam Krupa (UT-Austin)
|Seeger
|[[#Sam Krupa | Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]
|-
| Lee
|February 23 (special time*, 3PM, B119)
|-  
| Yaguang Wang (Shanghai Jiao Tong)
|February 19
|[[ #Yaguang Wang | Stability of Three-dimensional Prandtl Boundary Layers ]]
| Maja Taskovic (UPenn)
|Jin
|[[#Maja Taskovic | Exponential tails for the non-cutoff Boltzmann equation ]]
|-
| Kim
|March 2
|-  
|Benoit Pausader (Princeton)
|February 26
|[[#Benoit Pausader (Princeton) | Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions]]
| Ashish Kumar Pandey (UIUC)
|Kim
|[[#  |  Instabilities in shallow water wave models ]]
|-
| Kim & Lee
|March 9
|-  
|Haozhao Li (University of Science and Technology of China)  
|March 5
|[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]]
| Khai Nguyen (NCSU)
|Wang
|[[#Khai Nguyen | Burgers Equation with Some Nonlocal Sources ]]
|-
| Tran
|March 16
|-  
| Jennifer Beichman (Madison)  
|March 12
|[[#Jennifer Beichman (Madison) | ]]
| Hongwei Gao (UCLA)
| Kim
|[[#Hongwei Gao Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations ]]
|-
| Tran
|March 23
|-  
| Ben Fehrman (University of Chicago)
|March 19
|[[#Ben Fehrman (University of Chicago | TBA ]]
| Huy Nguyen (Princeton)
| Lin
|[[#Huy Nguyen Compressible fluids and active potentials ]]
|-
| Lee
|March 30
| Spring recess Mar 28-Apr 5 (S-N)
|[[# |  ]]
|  
|-
|April 6
| Vera Hur (UIUC)
|[[# |  ]]
| Yao
|-
|April 13
|
|[[#  |  ]]
|
|-
|April 20
|Yuan Lou (Ohio State)
|[[#Yuan Lou (Ohio State) | TBA]]
|Zlatos
|-
|-
|April 27
|March 26
|
|[[#  |  ]]
|
|-
|May 4
|
|[[#  |  ]]
|  
|  
|[[#  |  Spring recess (Mar 24-Apr 1, 2018) ]]
|-
|-
|April 2
| In-Jee Jeong (Princeton)
|[[#In-Jee Jeong |  Singularity formation for the 3D axisymmetric Euler equations ]]
| Kim
|-
|April 9
| Jeff Calder (Minnesota)
|[[#Jeff Calder |  TBD ]]
| Tran
|-
|April 21-22 (Saturday-Sunday)
| Midwest PDE seminar
|[[#Midwest PDE seminar |  ]]
| Angenent, Feldman, Kim, Tran.
|-
|April 25 (Wednesday)
| Hitoshi Ishii (Wasow lecture)
|[[#Hitoshi Ishii |  TBD]]
| Tran.
|}
|}


== Abstracts ==
===Dan Knopf===


== Abstracts ==
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.


===Jun Kitagawa (Toronto)===
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.


Regularity theory for generated Jacobian equations: from optimal transport to geometric optics


Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.
===Ashish Kumar Pandey===


===Jessica Lin (Madison)===
Title: Instabilities in shallow water wave models


Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations
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.


We establish error estimates for the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. Based on the approach of Armstrong and Smart in the elliptic setting, we construct a quantity which captures the geometric behavior of solutions to parabolic equations. The error estimates are shown to be of algebraic order. This talk is based on joint work with Charles Smart.


===Khai Nguyen===


===Yaguang Wang (Shanghai Jiao Tong)===
Title: Burgers Equation with Some Nonlocal Sources


Stability of Three-dimensional Prandtl Boundary Layers
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.


In this talk, we shall study the stability of the Prandtl boundary layer
===Hongwei Gao=== 
equations in three space variables. First, we obtain a well-posedness
result of the three-dimensional Prandtl equations under some constraint on
its flow structure. It reveals that the classical Burgers equation plays an
important role in determining this type of flow with special structure,
that avoids the appearance of the complicated secondary flow in the
three-dimensional Prandtl boundary layers. Second, we give an instability
criterion for the Prandtl equations in three space variables. Both of
linear and nonlinear stability are considered. This criterion shows that
the monotonic shear flow is linearly stable for the three dimensional
Prandtl equations if and only if the tangential velocity field direction is
invariant with respect to the normal variable, which is an exact complement
to the above well-posedness result for a special flow. This is a joint work
with Chengjie Liu and Tong Yang.


Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations


===Benoit Pausader (Princeton)===
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.


Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions
===Huy Nguyen===


It is well known that pure compressible fluids tend to develop shocks, even from small perturbation. We study how self consistent electromagnetic fields can stabilize these fluids. In a joint work with A. Ionescu and Y. Deng, we consider a compressible fluid of electrons in 2D, subject to its own electromagnetic field and to a field created by a uniform background of positively charged ions. We show that small smooth and irrotational perturbations of a uniform background at rest lead to solutions that remain globally smooth, in contrast with neutral fluids. This amounts to proving small data global existence for a system of quasilinear Klein-Gordon equations with different speeds.
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.


===Haozhao Li (University of Science and Technology of China)===
===In-Jee Jeong===


Regularity scales and convergence of the Calabi flow
Title: Singularity formation for the 3D axisymmetric Euler equations


We define regularity scales to study the behavior of the Calabi flow.
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.
Based on estimates of the regularity scales, we obtain convergence theorems
of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence
of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for
the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra
assumption that the scalar curvature is uniformly bounded.

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.