Seminars and Colloquia by Series

Non-uniqueness and vanishing viscosity

Series
PDE Seminar
Time
Tuesday, April 8, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dallas Albritton – University of Wisconsin-Madison –

The forced 2D Euler equations exhibit non-unique solutions with vorticity in $L^p$, $p > 1$, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit $\nu \to 0^+$ from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of ``resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity $\nu$ and consider $O(\varepsilon)$-size perturbations of his initial datum. We discover a uniqueness threshold $\varepsilon \sim \nu^{\kappa_{\rm c}}$, below which the vanishing viscosity solution is unique and radial, and at which certain vanishing viscosity solutions converge to non-unique, non-radial solutions. Joint work with Maria Colombo and Giulia Mescolini (EPFL).

Recovery of Schrödinger nonlinearities from the scattering map

Series
PDE Seminar
Time
Tuesday, April 1, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jason Murphy – University of Oregon –

We will discuss time-dependent, nonlinear “inverse scattering” in the setting of nonlinear Schrödinger equations.  In particular, we will show that it is possible to recover an unknown nonlinearity from the small-data scattering behavior of solutions.  Time permitting, we will also discuss stability estimates for reconstruction, as well as recovery from modified scattering behavior.  This talk will include some joint work with R. Killip and M. Visan, as well as with G. Chen.

Domain branching in ferromagnets: elliptic regularity in action

Series
PDE Seminar
Time
Tuesday, March 25, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tobias Reid – Georgia Tech –

The Landau-Lifshitz model of micromagnetics is a powerful continuum theory that describes the occurrence of magnetization patterns in a ferromagnetic body. In this talk I will discuss  domain branching in strongly uniaxial materials resulting from the competition between a short-range attractive interaction (surface energy), a long-range repulsive interaction (stray field energy), and a non-convex constraint coming from the strong uniaxiality. 

 

On a mathematical level, we use modern tools from elliptic regularity theory, convex duality, ideas from statistical physics, and fine geometric constructions to describe the occurrence of domain branching through local energy estimates at the boundary of the sample (where the branching is infinitely fine). Our approach provides a robust framework for other domain branching problems and is the first step to prove self-similarity in a statistical sense.

 

(Joint work with Carlos Román)

Construction of multi-soliton solutions for semilinear equations in dimension 3

Series
PDE Seminar
Time
Tuesday, March 4, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Istvan Kadar – Princeton University –

The existence of multi black hole solutions in General Relativity is one of the expectations from the final state conjecture, the analogue of soliton resolution. In this talk, I will present preliminary works in this direction via a semilinear model, the energy critical wave equation, in dimension 3. In particular, I show 1) an algorithm to construct approximate solutions to the energy critical wave equation that converge to a sum of solitons at an arbitrary polynomial rate in (t-r); 2) a robust method to solve the remaining error terms for the nonlinear equation. The methods apply to energy supercritical problems.

Around the convergence problem in mean field control theory and the associated Hamilton-Jacobi equations

Series
PDE Seminar
Time
Tuesday, February 25, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE: https://gatech.zoom.us/j/95641893035?pwd=rZeIGeDdpL0abXWa4t94JDuRKV9wPa.1
Speaker
Samuel Daudin – Université Paris Cité –

The aim of this talk is to discuss recent advances around the convergence problem in mean field control theory and the study of associated nonlinear PDEs.

We are interested in optimal control problems involving a large number of interacting particles and subject to independent Brownian noises. As the number of particles tends to infinity, the problem simplifies into a McKean-Vlasov type optimal control problem for a typical particle. I will present recent results concerning the quantitative analysis of this convergence. More precisely, I will discuss an approach based on the analysis of associated value functions. These functions are solutions of Hamilton-Jacobi equations in high dimension and the convergence problem translates into a stability problem for the limit equation which is posed on a space of probability measures.

I will also discuss the well-posedness of this limiting equation, the study of which seems to escape the usual techniques for Hamilton-Jacobi equations in infinite dimension.

Hamilton-Jacobi equations on Wasserstein spaces

Series
PDE Seminar
Time
Wednesday, February 19, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ben Seeger – University of North Carolina at Chapel Hill –

Please Note: Please note the unusual time and place.

The study of differential equations on infinite spaces of probability measures has become an active field of research in recent years. Such equations arise when describing nonlinear effects in systems with a large number of interacting particles or agents. A rigorous well-posedness theory for such equations then leads to results about large deviations for interacting particle systems, limiting statements about free energies in mean field spin glasses, and mean field descriptions in high-dimensional optimization and game theory, among many other examples.
 
This talk will give an overview of some recent well-posedness results for Hamilton-Jacobi equations on probablity spaces. The nonlinear and infinite-dimensional nature of the underlying space necessitates a mix of techniques from functional analysis and probability theory. We also discuss how these PDE techniques can be used to deduce qualitative and quantitative convergence results for stochastic control and differential games for a large system of interacting agents. This talk is based on joint work with Joe Jackson (University of Chicago) and Samuel Daudin (Université Paris Cité, Laboratoire Jacques Louis Lions).

Blow-up of the Modified Benjamin-Ono Equation

Series
PDE Seminar
Time
Tuesday, February 11, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Justin Holmer – Brown University –

We prove that negative energy solutions to the modified Benjamin-Ono (mBO) equation, which is L^2 critical, with mass slightly above the ground state mass, blow-up in finite or infinite time.   These blow-up solutions lie adjacent to those constructed by Martel & Pilod (2017) that have mass exactly equal to the ground state mass.  The solutions that we construct, with mass slightly above the ground state mass, are numerically observable and expected to be stable.  This is joint work with Svetlana Roudenko and Kai Yang.

On the hydrostatic Euler equations: singularity formation, effect of rotation, and regularization by noise

Series
PDE Seminar
Time
Tuesday, February 4, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Quyuan Lin – Clemson University –

The hydrostatic Euler equations, also known as the inviscid primitive equations, are derived from the Euler equations by taking the hydrostatic limit. They are commonly used when the aspect ratio of the domain is small, such as the ocean and atmosphere in the planetary scale. In this talk, I will first present the stability of finite-time blowup of smooth solutions to this model, then discuss the effect of fast rotation (from Coriolis force) in prolonging the lifespan of solutions. Finally, I will talk about the regularization effects that arise when the model is driven by certain random noise.

Nonlinear Scattering Theory for Asymptotically de Sitter Vacuum Solutions

Series
PDE Seminar
Time
Tuesday, November 19, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Serban Cicortas – Princeton University

We will talk about recent work establishing a quantitative nonlinear scattering theory for asymptotically de Sitter solutions of the Einstein vacuum equations in (n+1) dimensions with n ≥ 4 even, which are determined by small scattering data at future infinity and past infinity. We will also explain why the case of even spatial dimension n poses significant challenges compared to its odd counterpart and was left open by the previous works in the literature.

Mean viability and 2nd-order Hamilton-Jacobi-Bellman equations

Series
PDE Seminar
Time
Tuesday, November 12, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian Keller – University of Central Florida –
I present a new approach of proving uniqueness for viscosity solutions of fully nonlinear 2nd-order Hamilton-Jacobi-Bellman equations. 
This approach is purely probabilistic. It uses the concept of mean viability and the closely related notion of quasi-contingent solution. 
Unlike all existing methods in the literature, my approach does not rely on finite-dimensional results. 
This is of relevance for genuinely infinite-dimensional open problems.

 

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