The Foias Lectures

Date: March 5, 2024

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Professor Camillo De Lellis, Institute of Advanced Studies

Title: DiPerna-Lions theory and convex integration

Abstract: After reviewing the fundamental theorems of DiPerna-Lions and Ambrosio on flows of Sobolev vector fields, we will explore a number of sharpness questions related to them. Many of these questions have been answered in the last few years using "convex integration" methods, exported by Modena and Sz\'ekelyhidi to the context of transport equations from that of incompressible fluid dynamics (where they were first introduced 17 years ago by Sz\'ekelyhidi and myself). I will in particular touch upon a striking application: there are divergence-free Sobolev vector fields for which uniqueness of the trajectories fails for a positive measure set of initial conditions, while there is a unique sensible choice of one ``good'' trajectory for almost all initial condition, given by the DiPerna-Lions flow. This theorem was first proved in a joint work of Bru\'e, Colombo, and myself with convex integration techniques and later improved by Kumar using different techniques.