Analysis Seminar

On the (in)stability of the identity map in optimal transportation

Yash Jhaveri
Member, School of Mathematics
October 14, 2019

In the optimal transport problem, it is well-known that the geometry of the target domain plays a crucial role in the regularity of the optimal transport. In the quadratic cost case, for instance, Caffarelli showed that having a convex target domain is essential in guaranteeing the optimal transport’s continuity. In this talk, we shall explore how, quantitatively, important convexity is in producing continuous optimal transports.

Weak solutions of the Navier-Stokes equations may be smooth for a.e. time

Maria Colombo
École Polytechnique Fédérale de Lausanne; von Neumann Fellow, School of Mathematics
October 7, 2019

In a recent result, Buckmaster and Vicol proved non-uniqueness of weak solutions to the Navier-Stokes equations which have bounded kinetic energy and integrable vorticity. 

 

We discuss the existence of such solutions, which in addition are regular outside a set of times of dimension less than 1. 

The inviscid limit for the Navier-Stokes equations with data analytic only near the boundary

Fei Wang
University of Maryland
May 30, 2019

We address the inviscid limit for the Navier-Stokes equations in a half space, with initial datum that is analytic only close to the boundary of the domain, and has finite Sobolev regularity in the complement. We prove that for such data the solution of the Navier-Stokes equations converges in the vanishing viscosity limit to the solution of the Euler equation, on a constant time interval.

Singularity formation for some incompressible Euler flows

Tarek Elgindi
University of California, San Diego
May 6, 2019

We describe a recent construction of self-similar blow-up solutions of the incompressible Euler equation. A consequence of the construction is that there exist finite-energy $C^{1,a}$ solutions to the Euler equation which develop a singularity in finite time for some range of $a>0$. The approach we follow is to isolate a simple non-linear equation which encodes the leading order dynamics of solutions to the Euler equation in some regimes and then prove that the simple equation has stable self-similar blow-up solutions.

Loops in hydrodynamic turbulence

Katepalli Sreenivasan
New York University; Member, School of Mathematics
April 17, 2019
An important question in hydrodynamic turbulence concerns the scaling proprties in the inertial range. Many years of experimental and computational work suggests---some would say, convincingly shows---that anomalous scaling prevails. If so, this rules out the standard paradigm proposed by Kolmogorov. The situation is not so obvious if one considers circulation around loops as the scaling objects instead of the traditional velocity increments.

Two-dimensional random field Ising model at zero temperature

Jian Ding
The Wharton School, The University of Pennsylvania
April 5, 2019
I will discuss random field Ising model on $Z^2$ where the external field is given by i.i.d. Gaussian
variables with mean zero and positive variance. I will present a recent result that at zero temperature the effect of boundary conditions on the magnetization in a finite box decays exponentially in the distance to the boundary. This is based on joint work with Jiaming Xia.

Localization and delocalization for interacting 1D quasiperiodic particles.

Ilya Kachkovskiy
Michigan State University
March 15, 2019
We consider a system of two interacting one-dimensional quasiperiodic particles as an operator on $\ell^2(\mathbb Z^2)$. The fact that particle frequencies are identical, implies a new effect compared to generic 2D potentials: the presence of large coupling localization depends on symmetries of the single-particle potential.