Seminars and Colloquia by Series

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 He – Johns 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 Moutinho – LAGA, 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 Lührmann – Texas 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 Zhang – Auburn 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.

Global Existence and Long Time Behavior in the 1+1 dimensional Principal Chiral Model with Applications to Solitons

Series
PDE Seminar
Time
Tuesday, February 7, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jessica Trespalacios Julio – Universidad de Chile –

We consider the 1+1 dimensional vector valued Principal Chiral Field model (PCF) obtained as a simplification of the Vacuum Einstein Field equations under the Belinski-Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step towards the study of its nonlinear stability. 

Optimal control of stochastic delay differential equations

Series
PDE Seminar
Time
Tuesday, January 31, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Filippo de Feo – Politecnico di Milano –

In this talk we will discuss an optimal control problem for stochastic differential delay equations. We will only consider the case with delays in the state. We will show how to rewrite the problem in a suitable infinite-dimensional Hilbert space. Then using the dynamic programming approach we will characterize the value function of the problem as the unique viscosity solution of an infinite dimensional Hamilton-Jacobi-Bellman equation.  We will discuss partial C^{1,α}-regularity of the value function. This regularity result is particularly interesting since it permits to construct a candidate optimal feedback map which may allow to find an optimal feedback control. Finally we will discuss some ideas about the case in which delays also appear in the controls.

This is a joint work with S. Federico and A. Święch.

Smooth ergodic theory for evolutionary PDE

Series
PDE Seminar
Time
Tuesday, January 24, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex Bluementh – Georgia Tech –

Smooth ergodic theory provides a framework for studying systems exhibiting dynamical chaos, features of which include sensitive dependence with respect to initial conditions, correlation decay (even for deterministic systems!) and complicated fractal-like attractor geometry. This talk will be an overview of some of these ideas as they apply to evolutionary PDE, with an emphasis on dissipative semilinear parabolic problems, and a discussion of some of my work in this direction, joint with: Lai-Sang Young and Sam Punshon-Smith. 

Non-uniqueness and convex integration for the forced Euler equations

Series
PDE Seminar
Time
Tuesday, January 17, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stan Palasek – UCLA –

This talk is concerned with α-Hölder-continuous weak solutions of the incompressible Euler equations. A great deal of recent effort has led to the conclusion that the space of Euler flows is flexible when α is below 1/3, the famous Onsager regularity. We show how convex integration techniques can be extended above the Onsager regularity to all α<1/2 in the case of the forced Euler equations. This leads to a new non-uniqueness theorem for any initial data. This work is joint with Aynur Bulut and Manh Khang Huynh.

On the Optimal Control of McKean Vlasov SDE and Mean Field Games in Infinite Dimension

Series
PDE Seminar
Time
Tuesday, January 10, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Fausto Gozzi – Luiss University –

In this talk we report on recent works (with A. Cosso, I. Kharroubi, H. Pham, M. Rosestolato) on the optimal control of (possibly path dependent) McKean-Vlasov equations valued in Hilbert spaces. On the other side we present the first ideas of a work with S. Federico, D. Ghilli and M. Rosestolato, on Mean Field Games in infinite dimension.

We will begin by presenting some examples for both topics and their relations. Then most of the time will be devoted to the first topic and the main results (the dynamic programming principle, the law invariance property of the value function, the Ito formula and the fact that the value function is a viscosity solution of the HJB equation, a first comparison result).

We conclude, if time allows, with the first ideas on the solution of the HJB-FKP system arising in mean Field Games in infinite dimension.

Global-in-space stability of self-similar blowup for supercritical wave maps

Series
PDE Seminar
Time
Tuesday, November 15, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Irfan Glogić – University of Vienna –

A distinctive feature of nonlinear evolution equations is the possibility of breakdown of solutions in finite time. This phenomenon, which is also called singularity formation or blowup, has both physical and mathematical significance, and, as a consequence, predicting blowup and understanding its nature is a central problem of the modern analysis of nonlinear PDEs.

In this talk we concentrate on wave maps – a geometric nonlinear wave equation – and we discuss the existence and stability of self-similar solutions, as in all higher dimensions they appear to drive the generic blowup behavior. We outline a novel framework for studying global-in-space stability of such solutions; we then men-tion some long-awaited results that we thereby obtained, and, finally, we discuss the new mathematical challenges that our approach generates.

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