Seminars and Colloquia by Series

Incompressible MHD Without Resistivity: structure and regularity

Series
PDE Seminar
Time
Tuesday, August 29, 2023 - 15:03 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ronghua PanGeorgia Tech

We study the global existence of classical solutions to the incompressible viscous MHD system without magnetic diffusion in 2D and 3D. The lack of resistivity or magnetic diffusion poses a major challenge to a global regularity theory even for small smooth initial data. However, the interesting nonlinear structure of the system not only leads to some significant challenges, but some interesting stabilization properties, that leads to the possibility of the theory of global existence of classical and/or strong solutions. This talk is based on joint works with Yi Zhou, Yi Zhu, Shijin Ding, Xiaoying Zeng, and Jingchi Huang.

Optimal blowup stability for wave maps

Series
PDE Seminar
Time
Tuesday, April 25, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Roland DonningerUniversity of Vienna

I discuss some recent results, obtained jointly with David Wallauch, on the stability of self-similar wave maps under minimal regularity assumptions on the perturbation. More precisely, we prove the asymptotic stability of an explicitly known self-similar wave map in corotational symmetry. The key tool are Strichartz estimates for the linearized equation in similarity coordinates.

Global well-posedness for the one-phase Muskat problem

Series
PDE Seminar
Time
Tuesday, April 18, 2023 - 15:00 for
Location
Skiles 006
Speaker
Huy NguyenUniversity of Maryland, College Park

 

We will discuss the one-phase Muskat problem concerning the free boundary of Darcy fluids in porous media. It is known that there exists a class of non-graph initial boundary leading to self-intersection at a single point in finite time (splash singularity). On the other hand, we prove that the problem has a unique global-in-time solution if the initial boundary is a periodic Lipschitz graph of arbitrary size. This is based on joint work with H. Dong and F. Gancedo. 

Nontrivial global solutions to some quasilinear wave equations in three space dimensions

Series
PDE Seminar
Time
Tuesday, April 11, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Dongxiao YuUniversity of Bonn

In this talk, I will present a method to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. Starting from a global solution to the geometric reduced system satisfying several pointwise estimates, we find a matching exact global solution to the original quasilinear wave equations. As an application of this method, we will construct nontrivial global solutions to Fritz John's counterexample $\Box u=u_tu_{tt}$ and the 3D compressible Euler equations without vorticity for $t\geq 0$.

Transport equations and connections with mean field games

Series
PDE Seminar
Time
Tuesday, April 4, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ben SeegerUniversity of Texas at Austin

Transport equations arise in the modelling of several complex systems, including mean field games. Such equations often involve nonlinear dependence of the solution in the drift. These nonlinear transport equations can be understood by developing a theory for transport equations with irregular drifts. In this talk, I will outline the well-posedness theory for certain transport equations in which the drift has a one-sided bound on the divergence, yielding contractive or expansive behavior, depending on the direction in which the equation is posed. The analysis requires studying the relationship between the transport and continuity equations and the associated ODE flow. The theory is then used to discuss certain nonlinear transport equations arising in the study of finite state-space mean field games. This is joint work with P.-L. Lions.

The Scattering Problem of the Intermediate Long Wave Equation

Series
PDE Seminar
Time
Tuesday, March 14, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yilun WuUniversity of Oklahoma

The Intermediate Long Wave equation (ILW) describes long internal gravity waves in stratified fluids. As the depth parameter in the equation approaches zero or infinity, the ILW formally approaches the Kortweg-deVries equation (KdV) or the Benjamin-Ono equation (BO), respectively. Kodama, Ablowitz and Satsuma discovered the formal complete integrability of ILW and formulated inverse scattering transform solutions. If made rigorous, the inverse scattering method will provide powerful tools for asymptotic analysis of ILW. In this talk, I will present some recent results on the ILW direct scattering problem. In particular, a Lax pair formulation is clarified, and the spectral theory of the Lax operators can be studied. Existence and uniqueness of scattering states are established for small interaction potential. The scattering matrix can then be constructed from the scattering states. The solution is related to the theory of analytic functions on a strip. This is joint work with Peter Perry.

The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes

Series
PDE Seminar
Time
Tuesday, March 7, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lili HeJohns Hopkins University

I will discuss the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. The proof uses tools from microlocal analysis and a detailed description of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator at low frequencies.

On the collision of two kinks for the phi^6 model with equal low speed

Series
PDE Seminar
Time
Tuesday, February 28, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Abdon MoutinhoLAGA, Université Sorbonne Paris Nord

We will talk about our results on the elasticity and stability of the 
collision of two kinks with low speed v>0 for the nonlinear wave 
equation of dimension 1+1 known as the phi^6 model. We will show that 
the collision of the two solitons is "almost" elastic and that, after 
the collision, the size of the energy norm of the remainder and the size 
of the defect of the speed of each soliton can be, for any k>0, of the 
order of any monomial v^{k} if v is small enough.

References:
This talk is based on our current works:
On the collision problem of two kinks for the phi^6 model with low speed 
   [https://arxiv.org/abs/2211.09749]
Approximate kink-kink solutions for the phi^6 model in the low-speed 
limit [https://arxiv.org/abs/2211.09714]

On co-dimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

Series
PDE Seminar
Time
Tuesday, February 21, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonas LührmannTexas A&M University

Solitons are particle-like solutions to dispersive evolution equations 
whose shapes persist as time goes by. In some situations, these solitons 
appear due to the balance between nonlinear effects and dispersion, in 
other situations their existence is related to topological properties of 
the model. Broadly speaking, they form the building blocks for the 
long-time dynamics of dispersive equations.

In this talk I will present joint work with W. Schlag on long-time decay 
estimates for co-dimension one type perturbations of the soliton for the 
1D focusing cubic Klein-Gordon equation (up to exponential time scales), 
and I will discuss our previous work on the asymptotic stability of the 
sine-Gordon kink under odd perturbations. While these two problems are 
quite similar at first sight, we will see that they differ by a subtle 
cancellation property, which has significant consequences for the 
long-time dynamics of the perturbations of the respective solitons.

Regularity of Hele-Shaw flow with source and drift: Flat free boundaries are Lipschitz

Series
PDE Seminar
Time
Tuesday, February 14, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yuming Paul ZhangAuburn University

The classical Hele-Shaw flow describes the motion of incompressible viscous fluid, which occupies part of the space between two parallel, nearby plates. With source and drift, the equation is used in models of tumor growth where cells evolve with contact inhibition, and congested population dynamics. We consider the flow with Hölder continuous source and Lipschitz continuous drift. We show that if the free boundary of the solution is locally close to a Lipschitz graph, then it is indeed Lipschitz, given that the Lipschitz constant is small. This is joint work with Inwon Kim.

Pages