Difference between revisions of "PDE Geometric Analysis seminar"

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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Spring 2018 | Tentative schedule for Spring 2018]]===
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===[[Fall 2018 | Tentative schedule for Fall 2018]]===
 +
 
 +
 
 +
 
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== PDE GA Seminar Schedule Spring 2018 ==
 +
 
  
== PDE GA Seminar Schedule Fall 2017 ==
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!style="width:20%" align="left" | date   
 
!style="width:20%" align="left" | date   
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!align="left" | title
 
!align="left" | title
 
!style="width:20%" align="left" | host(s)
 
!style="width:20%" align="left" | host(s)
 +
 +
|- 
 +
|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
 
|-  
 
|-  
|September 11
+
|February 5,  '''3-3:50PM,  B341 VV.'''
|Mihaela Ifrim (UW)
+
| Andreas Seeger (UW)
|[[#Mihaela IfrimWell-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation]]
+
|[[#Andreas Seeger Singular integrals and a problem on mixing flows ]]
 
| Kim & Tran
 
| Kim & Tran
 
|-  
 
|-  
|September 18
+
|February 12
|Longjie Zhang (University of Tokyo)  
+
| Sam Krupa (UT-Austin)
|[[#Longjie Zhang On curvature flow with driving force starting as singular initial curve in the plane]]
+
|[[#Sam Krupa Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]
| Angenent
+
| Lee
 
|-  
 
|-  
|September 22,
+
|February 19
VV B239 4:00pm
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| Maja Taskovic (UPenn)
|Jaeyoung Byeon (KAIST)  
+
|[[#Maja Taskovic TBD ]]
|[[#Jaeyoung ByeonColloquium: Patterns formation for elliptic systems with large interaction forces]]
+
| Kim
Rabinowitz
+
|-
 +
|February 26
 +
Ashish Kumar Pandey (UIUC)
 +
|[[#  |  Instabilities in shallow water wave models  ]]
 +
| Kim & Lee
 
|-  
 
|-  
|September 25
+
|March 5
| Tuoc Phan (UTK)
+
| Khai Nguyen (NCSU)
|[[#Tuoc Phan Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application]]
+
|[[#Khai Nguyen TBD ]]
 
| Tran
 
| Tran
 
|-  
 
|-  
|September 26,
+
|March 12
VV B139 4:00pm
+
| Hongwei Gao (UCLA)
| Hiroyoshi Mitake (Hiroshima University)
+
|[[#Hongwei Gao TBD ]]
|[[#Hiroyoshi Mitake Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application]]
+
 
| Tran
 
| Tran
 
|-  
 
|-  
|September 29,
+
|March 19
VV901 2:25pm
+
| Huy Nguyen (Princeton)
| Dongnam Ko (CMU & SNU)
+
|[[#Huy Nguyen |  TBD ]]
|[[#Dongnam Ko a joint seminar with ACMS: TBD ]]
+
| Lee
| Shi Jin & Kim
+
|-
 +
|March 26
 +
|
 +
|[[#  |  Spring recess (Mar 24-Apr 1, 2018) ]]
 +
 +
|-
 +
|April 2
 +
 +
|[[# |  TBD ]]
 +
|
 
|-  
 
|-  
|October 2
+
|April 9
| No seminar due to a KI-Net conference
+
| Jeff Calder (Minnesota)
|
+
|[[#Jeff Calder |  TBD ]]
|
+
| Tran
 
|-  
 
|-  
|October 9
+
|April 21-22 (Saturday-Sunday)
| Sameer Iyer (Brown University)
+
| Midwest PDE seminar
|[[#Sameer Iyer Global-in-x Steady Prandtl Expansion over a Moving Boundary ]]
+
|[[#Midwest PDE seminar |  ]]
| Kim
+
| Angenent, Feldman, Kim, Tran.
 
|-  
 
|-  
|October 16
+
|April 25 (Wednesday)
| Jingrui Cheng (UW)
+
| Hitoshi Ishii (Wasow lecture)
|[[#Jingrui Cheng |  TBD ]]
+
|[[#Hitoshi Ishii |  TBD]]
| Kim & Tran
+
| Tran.
|-
+
|October 23
+
| Donghyun Lee (UW)
+
|[[#Donghyun Lee |  TBD ]]
+
| Kim & Tran
+
|-
+
|October 30
+
| Myoungjean Bae (POSTECH)
+
|[[#Myoungjean Bae |  TBD ]]
+
|  Feldman
+
|-
+
|November 6
+
| Jingchen Hu (USTC and UW)
+
|[[#Jingchen Hu |  TBD ]]
+
| Kim & Tran
+
|-
+
|December 4
+
| Norbert Pozar (Kanazawa University)
+
|[[#Norbert Pozar |  TBD ]]
+
| Tran
+
 
|}
 
|}
  
==Abstracts==
+
== Abstracts ==
 
+
===Mihaela Ifrim===
+
 
+
Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation
+
 
+
Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru.
+
 
+
===Longjie Zhang===
+
  
On curvature flow with driving force starting as singular initial curve in the plane
+
===Dan Knopf===
  
We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.
+
Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons
  
===Jaeyoung Byeon===
+
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.
  
Title: Patterns formation for elliptic systems with large interaction forces
+
===Andreas Seeger===
  
Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions.  The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.
+
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.
  
===Tuoc Phan===
+
===Sam Krupa===
Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.
+
  
Abstract: In this talk, we first introduce a problem on the existence of global time smooth solutions for a system of cross-diffusion equations. We then recall some classical results on regularity theories, and show that to solve our problem, new results on regularity theory estimates of Calderon-Zygmund type for gradients of solutions to a class of parabolic equations in Lebesgue spaces are required. We then discuss a result on Calderon-Zygmnud type estimate in the concrete setting to solve our
+
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case
mentioned problem regarding the system of cross-diffusion equations. The remaining part of the talk will be focused on some new generalized results on regularity gradient estimates for some general class of quasi-linear parabolic equations. Regularity estimates for gradients of solutions in Lorentz spaces will be presented. Ideas of the proofs for the results are given.
+
  
===Hiroyoshi Mitake===
+
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.
Derivation of multi-layered interface system and its application
+
  
Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of  the Allen-Cahn equation.  By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.
+
===Ashish Kumar Pandey===
  
===Sameer Iyer===
+
Title: Instabilities in shallow water wave models
Title: Global-in-x Steady Prandtl Expansion over a Moving Boundary.
+
  
Abstract: I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.
+
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.

Revision as of 13:57, 7 February 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) TBD Kim
February 26 Ashish Kumar Pandey (UIUC) Instabilities in shallow water wave models Kim & Lee
March 5 Khai Nguyen (NCSU) TBD Tran
March 12 Hongwei Gao (UCLA) TBD Tran
March 19 Huy Nguyen (Princeton) TBD Lee
March 26 Spring recess (Mar 24-Apr 1, 2018)
April 2 TBD
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.

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.