Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields

TL;DR

This paper establishes sharp logarithmic regularity estimates for solutions to the continuity equation driven by Sobolev vector fields of class W^{1,p}, enhancing understanding of solution regularity under minimal smoothness assumptions.

Contribution

It provides the first sharp regularity estimates of logarithmic order for solutions of the continuity equation with Sobolev vector fields of class W^{1,p}.

Findings

Sharp logarithmic regularity estimates are proved.

Regularity is measured using Gargliardo's seminorms.

Results apply to vector fields with p>1 in Sobolev space.

Abstract

The aim of this note is to prove sharp regularity estimates for solutions of the continuity equation associated to vector fields of class $W^{1,p}$ with $p>1$. The regularity is of "logarithmic order" and is measured by means of suitable versions of Gargliardo's seminorms.