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]]===
+
===[[Fall 2018 | Tentative schedule for Fall 2018]]===
 +
 
 +
 
 +
 
 +
== 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   
Line 10: Line 14:
 
!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 9th floor hall, 4:00pm
+
| Maja Taskovic (UPenn)
|Jaeyoung Byeon (KAIST)  
+
|[[#Maja Taskovic Exponential tails for the non-cutoff Boltzmann equation ]]
|[[#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 Burgers Equation with Some Nonlocal Sources ]]
 
| Tran
 
| Tran
 
|-  
 
|-  
|September 26,
+
|March 12
VV B139 4:00pm
+
| Hongwei Gao (UCLA)
| Hiroyoshi Mitake (Hiroshima University)
+
|[[#Hongwei Gao Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations ]]
|[[#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: On the emergence of local flocking phenomena in Cucker-Smale ensembles ]]
+
| Lee
| Shi Jin & Kim
+
|-
|-  
+
|March 26
|October 2
+
|  
| No seminar due to a KI-Net conference
+
|[[#  |  Spring recess (Mar 24-Apr 1, 2018) ]]
|
+
|
|
+
|-
|-  
+
|April 2
|October 9
+
| In-Jee Jeong (Princeton)
| Sameer Iyer (Brown University)
+
|[[#In-Jee Jeong TBD ]]
|[[#Sameer Iyer Global-in-x Steady Prandtl Expansion over a Moving Boundary ]]
+
 
| Kim
 
| Kim
 
|-  
 
|-  
|October 16
+
|April 9
| Jingrui Cheng (UW)
+
| Jeff Calder (Minnesota)
|[[#Jingrui Cheng A 1-D semigeostrophic model with moist convection ]]
+
|[[#Jeff Calder TBD ]]
| Kim & Tran
+
| Tran
 
|-  
 
|-  
|October 23
+
|April 21-22 (Saturday-Sunday)
| Donghyun Lee (UW)
+
| Midwest PDE seminar
|[[#Donghyun Lee The Vlasov-Poisson-Boltzmann system in bounded domains ]]
+
|[[#Midwest PDE seminar |  ]]
| Kim & Tran
+
| Angenent, Feldman, Kim, Tran.
 
|-  
 
|-  
|October 30
+
|April 25 (Wednesday)
| Myoungjean Bae (POSTECH)
+
| Hitoshi Ishii (Wasow lecture)
|[[#Myoungjean Bae |  3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system ]]
+
|[[#Hitoshi Ishii |  TBD]]
|  Feldman
+
| Tran.
|-
+
|November 6
+
| Jingchen Hu (USTC and UW)
+
|[[#Jingchen Hu |  Shock Reflection and Diffraction Problem with Potential Flow Equation ]]
+
| Kim & Tran
+
|-
+
|November 27
+
| Ru-Yu Lai (Minnesota)
+
|[[#Ru-Yu Lai |  TBD ]]
+
| Li
+
|-
+
|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
+
  
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.
+
===Dan Knopf===
  
===Jaeyoung Byeon===
+
Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons
  
Title: Patterns formation for elliptic systems with large interaction forces
+
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.
  
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.
+
===Andreas Seeger===
  
 +
Title: Singular integrals and a problem on mixing flows
  
===Tuoc Phan===
+
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.
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
+
===Sam Krupa===
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===
+
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case
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.
+
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.
  
  
===Dongnam Ko===
+
===Maja Taskovic===
On the emergence of local flocking phenomena in Cucker-Smale ensembles
+
  
Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.
+
Title: Exponential tails for the non-cutoff Boltzmann equation
  
===Sameer Iyer===
+
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.
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.
+
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.
  
===Jingrui Cheng===
 
  
A 1-D semigeostrophic model with moist convection.
+
===Ashish Kumar Pandey===
  
We consider a simplified 1-D model of semigeostrophic system with moisture, which describes moist convection in a single column in the atmosphere. In general, the solution is non-continuous and it is nontrivial part of the problem to find a suitable definition of weak solutions. We propose a plausible definition of such weak solutions which describes the evolution of the probability distribution of the physical quantities, so that the equations hold in the sense of almost everywhere. Such solutions are constructed from a discrete scheme which obeys the physical principles. This is joint work with Mike Cullen, together with Bin Cheng, John Norbury and Matthew Turner.
+
Title: Instabilities in shallow water wave models
  
===Donghyun Lee===
+
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 construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. Moreover we prove an exponential convergence of distribution function toward the global Maxwellian.
 
  
===Myoungjean Bae===
+
===Khai Nguyen===
  
3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system.
+
Title: Burgers Equation with Some Nonlocal Sources
  
I will present a recent result on the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=angular momentum density) componentThis talk is based on a joint work with S. Weng (Wuhan University, China).
+
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 equationsSome open questions will be discussed.
  
===Jingchen Hu===
+
===Hongwei Gao===
  
Shock Reflection and Diffraction Problem with Potential Flow Equation
+
Title: Stochastic homogenization of certain nonconvex Hamilton-Jacobi equations
  
In this talk, we will present our work on nonsymmetric shock reflection and diffraction problem, the equation concerned is potential flow equation, which is a simplification of Euler System, mainly based on the assumption that flow has no vortex. We showed in both nonsymmetric reflection case and diffraction case, that physically admissible solution does not exist. This implies that the formation of vortex is essential to maintain the structural stability of shock reflection and diffraction.
+
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.

Revision as of 23:58, 23 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) 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) TBD Lee
March 26 Spring recess (Mar 24-Apr 1, 2018)
April 2 In-Jee Jeong (Princeton) TBD 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.