Seminar Analysis: Elia Bruè (Scuola Normale Superiore di Pisa)

Propagation of regularity and mixing estimates for solutions of the incompressible continuity equation with Sobolev vector field

Since the work by DiPerna and Lions (1989) the continuity and transport equation under mild regularity assumptions on the vector field have been extensively studied, becoming a florid research field. The applicability of this theory is very wide, especially in the study of partial differential equations and very recently also in the field of non-smooth geometry.

The aim of this talk is to give an overview of the quantitative side of the theory initiated by Crippa and De Lellis. We address the problem of mixing and propagation of regularity for solutions to the continuity equation drifted by Sobolev fields. The problem is well understood when the vector field enjoys a Sobolev regularity with integrability exponent p>1 and basically nothing is known (at the quantitative level) in the case p=1.

We present sharp regularity estimates for the case p>1 and new attempts to attack the challenging question in the case p=1. This is a join work with Quoc-Hung Nguyen.

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