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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]]===
===[[Spring 2018 | Tentative schedule for Spring 2018]]===


= Seminar Schedule Spring 2013 =
== PDE GA Seminar Schedule Fall 2017 ==
{| 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 11
|February 4
|Mihaela Ifrim (UW)
|[http://math.postech.ac.kr/~mjbae/ Myoungjean Bae (POSTECH)]
|[[#Mihaela Ifrim|  Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation]]
|[[#Myoungjean Bae (POSTECH) |
| Kim & Tran
  Transonic shocks for Euler-Poisson system and related problems]]
|-
|Feldman
|September 18
|-
|Longjie Zhang (University of Tokyo)
|-
|[[#Longjie Zhang |  On curvature flow with driving force starting as singular initial curve in the plane]]
|February 18
|  Angenent
|Mike Cullen (Met. Office, UK)
|-
|[[ #Mike Cullen (Met. Office, UK)|
|September 22,
Modelling the uncertainty in predicting large-scale atmospheric circulations.]]
VV 9th floor hall, 4:00pm
|Feldman
|Jaeyoung Byeon (KAIST)  
|-
|[[#Jaeyoung Byeon|  Colloquium: Patterns formation for elliptic systems with large interaction forces]]
|-
|  Rabinowitz
|March 18
|-
|Mohammad Ghomi(Math. Georgia Tech)
|September 25
|[[ #Mohammad Ghomi(Math. Georgia Tech)|
| Tuoc Phan (UTK)
Tangent lines, inflections, and vertices of closed curves.]]
|[[#Tuoc Phan Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application]]
|Angenent
| Tran
|-
|-  
|-
|September 26,
|April 8
VV B139 4:00pm
|[http://www.maths.ox.ac.uk/contact/details/xiang Wei Xiang (Oxford)]
| Hiroyoshi Mitake (Hiroshima University)
|[[ #Wei Xiang (Oxford)|
|[[#Hiroyoshi Mitake | Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application]]
Shock Diffraction Problem to the
| Tran
Two Dimensional Nonlinear Wave System and Potential Flow Equation.]]
|-  
|Feldman
|September 29,
|-
VV901 2:25pm
|-
| Dongnam Ko (CMU & SNU)
|Thursday April 25, Time: 3:30, Room: 901 Van Vleck (Note: Special Day, Room)
|[[#Dongnam Ko | a joint seminar with ACMS: On the emergence of local flocking phenomena in Cucker-Smale ensembles ]]
|[http://math.wvu.edu/~adriant/CV1.html Adrian Tudorascu (West Virginia University)]
| Shi Jin & Kim
|[[ #Adrian Tudorascu (West Virginia University)|One-dimensional pressureless
|-
Euler/Euler-Poisson systems with/without viscosity
|October 2
.]]
| No seminar due to a KI-Net conference
|Feldman
|
|-
|
|-
|-  
|May 6
|October 9
|[http://www.icmat.es/dcordoba Diego Cordoba (Madrid)]
| Sameer Iyer (Brown University)
|[[#Diego Cordoba (Madrid) |
|[[#Sameer Iyer | Global-in-x Steady Prandtl Expansion over a Moving Boundary ]]
Interface dynamics for incompressible fluids. ]]
| Kim
|Kiselev
|-  
|-
|October 16
| Jingrui Cheng (UW)
|[[#Jingrui Cheng |  A 1-D semigeostrophic model with moist convection ]]
| Kim & Tran
|-
|October 23
| Donghyun Lee (UW)
|[[#Donghyun Lee | The Vlasov-Poisson-Boltzmann system in bounded domains ]]
| 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==


===Myoungjean Bae (POSTECH)===
===Mihaela Ifrim===
''Transonic shocks for Euler-Poisson system and related problems''


Abstract: Euler-Poisson system models various physical phenomena
Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation
including the propagation of electrons in submicron semiconductor
devices and plasmas, and the biological transport of ions for channel
proteins. I will explain difference between Euler system and
Euler-Poisson system and mathematical difficulties arising due to this
difference. And, recent results about subsonic flow and transonic flow
for Euler-Poisson system will be presented. This talk is based on
collaboration with Ben Duan and Chunjing Xie.


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.


===Mike Cullen (Met. Office, UK)===
===Longjie Zhang===
''Modelling the uncertainty in predicting large-scale atmospheric circulations''


Abstract: This talk describes work on quantifying the uncertainty in climate models by using data assimilation techniques;
On curvature flow with driving force starting as singular initial curve in the plane
and in estimating the properties of large-scale atmospheric flows by using asymptotic limit solutions.


===Mohammad Ghomi(Math. Georgia Tech)===
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.


>> Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow with surgery, is based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane and the sphere which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's ``tennis ball theorem".
===Jaeyoung Byeon===


===Wei Xiang (Oxford)===
Title: Patterns formation for elliptic systems with large interaction forces


Abstract: The vertical shock which initially separates two
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.
piecewise constant Riemann data, passes the wedge from left to right,
then shock diffraction phenomena will occur and the incident shock
becomes a transonic shock. Here we study this problem on nonlinear
wave system as well as on potential flow equations. The existence and
the optimal regularity across sonic circle of the solutions to this
problem is established. The comparison of these two systems is
discussed, and some related open problems are proposed.


===Adrian Tudorascu (West Virginia University)===


Abstract:
===Tuoc Phan===
This chalk & board presentation will start with an interesting motivation for these systems (the simplest of which was shown to model adhesion dynamics/ballistic aggregation by Zeldovich in 1970 in a paper published in "Astronomy & Astrophysics"). A general existence result for one-dimensional pressureless Euler/Euler-Poisson systems with/without viscosity will be discussed. This is achieved via entropy solutions for some appropriate scalar conservation laws; I will show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. I will also discuss the stability and uniqueness of solutions, which is obtained via a contraction principle in the Wasserstein metric. This is based on joint work with T. Nguyen (U. Akron).
Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.


===Diego Cordoba (Madrid)===
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
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.


Abstract: We consider the evolution of an interface generated between two immiscible,  
===Hiroyoshi Mitake===
incompressible and irrotational fluids. Specifically we study the Muskat equation (the  
Derivation of multi-layered interface system and its application
interface between oil and water in sand) and water wave equation (interface between water
 
and vacuum). For both equations we will study well-posedness and the existence of smooth
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.
initial data for which the smoothness of the interface breaks down in finite time. We
 
will also discuss some open problems.
 
===Dongnam Ko===
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.
 
===Sameer Iyer===
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.
 
===Jingrui Cheng===
 
A 1-D semigeostrophic model with moist convection.
 
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.
 
===Donghyun Lee===
 
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.

Revision as of 12:37, 14 October 2017

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 Spring 2018

PDE GA Seminar Schedule Fall 2017

date speaker title host(s)
September 11 Mihaela Ifrim (UW) Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation Kim & Tran
September 18 Longjie Zhang (University of Tokyo) On curvature flow with driving force starting as singular initial curve in the plane Angenent
September 22,

VV 9th floor hall, 4:00pm

Jaeyoung Byeon (KAIST) Colloquium: Patterns formation for elliptic systems with large interaction forces Rabinowitz
September 25 Tuoc Phan (UTK) Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application Tran
September 26,

VV B139 4:00pm

Hiroyoshi Mitake (Hiroshima University) Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application Tran
September 29,

VV901 2:25pm

Dongnam Ko (CMU & SNU) a joint seminar with ACMS: On the emergence of local flocking phenomena in Cucker-Smale ensembles Shi Jin & Kim
October 2 No seminar due to a KI-Net conference
October 9 Sameer Iyer (Brown University) Global-in-x Steady Prandtl Expansion over a Moving Boundary Kim
October 16 Jingrui Cheng (UW) A 1-D semigeostrophic model with moist convection Kim & Tran
October 23 Donghyun Lee (UW) The Vlasov-Poisson-Boltzmann system in bounded domains Kim & Tran
October 30 Myoungjean Bae (POSTECH) TBD Feldman
November 6 Jingchen Hu (USTC and UW) TBD Kim & Tran
December 4 Norbert Pozar (Kanazawa University) TBD Tran

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.

Jaeyoung Byeon

Title: Patterns formation for elliptic systems with large interaction forces

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.


Tuoc Phan

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 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

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.


Dongnam Ko

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.

Sameer Iyer

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.

Jingrui Cheng

A 1-D semigeostrophic model with moist convection.

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.

Donghyun Lee

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.