Seminars and Colloquia by Series

Degenerating Einstein spaces

Series
PDE Seminar
Time
Tuesday, October 29, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ruobing Zhang – Stony 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. LeFloch – Sorbonne 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 Lim – ShanghaiTech 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 Taskovic – Emory 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 Yu – University 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 Li – University 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 Mou – UCLA –

 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.

Analysis on Keller-Segel Models in Chemotaxis.

Series
PDE Seminar
Time
Tuesday, April 16, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Li Chen – University of Mannheim –

I this talk I will summerize some of our contributions in the analysis of parabolic elliptic Keller-Segel system, a typical model in chemotaxis. For the case of linear diffusion, after introducing the critical mass in two dimension, I will show our result for blow-up conditions for higher dimension. The second part of the talk is concentrated in the critical exponent for Keller-Segel system with porus media type diffusion. In the end, motivated from the result on nonlocal Fisher-KPP equation, we show that the nonlocal reaction will also help in preventing the blow-up of the solutions.  

On the motion of a rigid body with a cavity filled with a viscous liquid

Series
PDE Seminar
Time
Tuesday, April 9, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Gieri Simonett – Vanderbilt University –
I will consider the motion of a rigid body with an interior cavity that is completely filled with a viscous fluid. The equilibria of the system will be characterized and their stability properties are analyzed. It will be shown that the fluid exerts a stabilizing effect, driving the system towards a state where it is moving as a rigid body with constant angular velocity. In addition, I will characterize the critical spaces for the governing evolution equation, and I will show how parabolic regularization in time-weighted spaces affords great flexibility in establishing regularity and stability properties for the system. The approach is based on the theory of Lp-Lq maximal regularity. (Joint work with G. Mazzone and J. Prüss).

Validity of Steady Prandtl Expansio

Series
PDE Seminar
Time
Tuesday, April 2, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
skiles 006
Speaker
Professor Yan Guo – Brown University –

In a joint work with Sameer Iyer, the validity of steady Prandtl layer expansion is established in a channel. Our result covers the celebrated Blasius boundary layer profile, which is based on uniform quotient estimates for the derivative Navier-Stokes equations, as well as a positivity estimate at the flow entrance.

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