Student Analysis

High regularity local wellposedness for Navier Stokes and small data global wellposedness in critical spaces.

Following the contents of last weeks talk, we'll discuss H^3 wellposedness for Euler and various blowup criterion that allow us to detect the breakdown of solutions. Time permitting, we'll discuss how these improved criterion allow us to show that all solutions are global in 2D, and extensions…

An introduction to the Euler and Navier Stokes equations; basic derivation and local well-posedness.

We will give a brief exposition of how the Monge-Ampère equation is related to optimal transport, and how it's regularity theory can be applied to understand the smoothness of transport maps.

On the isoperimetric inequality: a proof using Brenier's theorem.

I'll be giving an informal recap of what we've learned so far about optimal transport, mostly by drawing pictures. Then, I'll discuss some of the ideas in the work of Lott and Villani, which gives a way using optimal transport to make sense of having lower bounds on Ricci curvature for…

We will cover the optimal transportation theorem for quadratic cost functions.

We will complete the proof of the characterization theorem for displacement interpolation. We will derive equations for the minimizers of the primal and dual optimization problems for dynamic coupling.

We will start our study of displacement interpolation, which is the continuous analogue of optimal transport. We'll cover coercive Lagrangian actions, examples of smooth/discrete systems and the notion of dynamical coupling.

Proof of the existence of optimal couplings and basic properties of Wasserstein spaces.