Seminars and Colloquia by Series

TBA

Series
PDE Seminar
Time
Tuesday, January 21, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Xiaomin WangSouthern University of Science and Technology

TBA

Invariant Gibbs measures and global strong solutions for 2D nonlinear Schrödinger equations

Series
PDE Seminar
Time
Tuesday, November 19, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrea R. NahmodUniversity of Massachusetts Amherst

In this talk I'll first give an background overview of Bourgain's approach to prove the invariance of the Gibbs measure for the periodic cubic nonlinear Schrodinger equation in 2D and of the para-controlled calculus of Gubinelli-Imkeller and Perkowski in the context of parabolic stochastic equations. I will then present our resolution of the long-standing problem of proving almost sure global well-posedness (i.e. existence /with uniqueness/) for the periodic nonlinear Schrödinger equation (NLS) in 2D on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is achieved by a new method we call /random averaging operators /which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is work with Yu Deng (USC) and Haitian Yue (USC).

Quantitative estimates of propagation of chaos for stochastic systems

Series
PDE Seminar
Time
Tuesday, November 5, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pierre-Emmanuel JabinUniversity of Maryland

We study the mean field limit of large stochastic systems of interacting particles. To treat more general and singular kernels, we propose a modulated free energy combination of the method that we had previously developed and of the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the most singular terms involving the divergence of the flow. Our modulated free energy allows to treat singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller-Segel system in the subcritical regimes, is obtained. This is a joint work with D. Bresch and Z. Wang.

Degenerating Einstein spaces

Series
PDE Seminar
Time
Tuesday, October 29, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ruobing ZhangStony Brook University
In the talk we discuss singularity formation of Einstein metrics as the underlying spaces degenerate or collapse. The usual analytic tools such as uniform Sobolev inequalities and nonlinear a priori estimates are unavailable in this context. We will describe an entirely new way to handle these difficulties, and construct degenerating Ricci-flat metrics with quantitative singularity behaviors.

The seed-to-solution method for the Einstein equations and the asymptotic localization problem

Series
PDE Seminar
Time
Tuesday, October 22, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Philippe G. LeFlochSorbonne University and CNRS

I will present a new method of analysis for Einstein’s
constraint equations, referred to as the Seed-to-Solution Method, which
leads to the existence of asymptotically Euclidean manifolds with
prescribed asymptotic behavior. This method generates a (Riemannian)
Einstein manifold from any seed data set consisting of (1): a Riemannian
metric and a symmetric two-tensor prescribed on a topological manifold
with finitely many asymptotically Euclidean ends, and (2): a density
field and a momentum vector field representing the matter content. By
distinguishing between several classes of seed data referred to as tame
or strongly tame, the method encompasses metrics with the weakest
possible decay (infinite ADM mass) or the strongest possible decay
(Schwarzschild behavior). This analysis is based on a linearization of
the Einstein equations (involving several curvature operators from
Riemannian geometry) around a tame seed data set. It is motivated by
Carlotto and Schoen’s pioneering work on the so-called localization
problem for the Einstein equations. Dealing with manifolds with possibly
very low decay and establishing estimates beyond the critical level of
decay requires significantly new ideas to be presented in this talk. As
an application of our method, we introduce and solve a new problem,
referred to as the asymptotic localization problem, at the critical
level of decay. Collaboration with T. Nguyen. Blog: philippelefloch.org

Isodiametry, variance, and regular simplices from particle interactions

Series
PDE Seminar
Time
Tuesday, October 1, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tongseok LimShanghaiTech University

We study the geometry of minimizers of the interaction energy functional. When the interaction potential is mildly repulsive, it is known to be hard to characterize those minimizers due to the fact that they break the rotational symmetry, suggesting that the problem is unlikely to be resolved by the usual convexity or symmetrization techniques from the calculus of variations. We prove that, if the repulsion is mild and the attraction is sufficiently strong, the minimizer is unique up to rotation and exhibits a remarkable simplex-shape rigid structure. As the first crucial step we consider the maximum variance problem of probability measures under the constraint of bounded diameter, whose answer in one dimension was given by Popoviciu in 1935.

On the relativistic Landau equation

Series
PDE Seminar
Time
Tuesday, September 24, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Maja TaskovicEmory University
In kinetic theory, a large system of particles is described by the particle density function. The Landau equation, derived by Landau in 1936, is one such example. It models a dilute hot plasma with fast moving particles that interact via Coulomb interactions. This model does not include the effects of Einstein’s theory of special relativity. However, when particle velocities are close to the speed of light, which happens frequently in a hot plasma, then relativistic effects become important. These effects are captured by the relativistic Landau equation, which was derived by Budker and Beliaev in 1956. 
 
We study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. The difficulty of the problem lies in the extreme complexity of the kernel in the relativistic collision operator. We present a new decomposition of such kernel. This is then used to prove the global Entropy dissipation estimate, the propagation of any polynomial moment for a weak solution, and the existence of a true weak solution for a large class of initial data. This is joint work with Robert M. Strain.

The energy conservation of inhomogeneous Euler equations

Series
PDE Seminar
Time
Tuesday, September 17, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cheng YuUniversity of Florida

In this talk, I will discuss from a mathematical viewpoint some sufficient conditions that guarantee the energy equality for weak solutions. I will mainly focus on a fluid equation example, namely the inhomogeneous Euler equations. The main tools are the commutator Lemmas.  This is a joint work with Ming Chen.

Compactness and singularity related to harmonic maps

Series
PDE Seminar
Time
Friday, July 26, 2019 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jiayu LiUniversity of Science and Technology of China

In this talk we will review compactness results and singularity theorems related to harmonic maps. We first talk about maps from Riemann surfaces with tension fields bounded in a local Hardy space, then talk about stationary harmonic maps from higher dimensional manifolds, finally talk about heat flow of harmonic maps.

Weak Solutions of Mean Field Game Master Equations

Series
PDE Seminar
Time
Tuesday, April 30, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
skiles 006
Speaker
Chenchen MouUCLA

 In this talk we study master equations arising from mean field game 
problems, under the crucial monotonicity conditions.
Classical solutions of such equations require very strong technical 
conditions. Moreover, unlike the master equations arising from mean 
field control problems, the mean field game master equations are 
non-local and even classical solutions typically do not satisfy the 
comparison principle, so the standard viscosity solution approach seems 
infeasible. We shall propose a notion of weak solution for such 
equations and establish its wellposedness. Our approach relies on a new 
smooth mollifier for functions of measures, which unfortunately does not 
keep the monotonicity property, and the stability result of master 
equations. The talk is based on a joint work with Jianfeng Zhang.

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