Vol. 14, No. 8, 2021

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

Elia Bruè and Quoc-Hung Nguyen

Vol. 14 (2021), No. 8, 2539–2559
Abstract

The aim of this note is to prove sharp regularity estimates for solutions of the continuity equation, associated to W1,p vector fields for p > 1. The regularity is of “logarithmic order” and is measured by means of suitable seminorms.

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Keywords
ordinary differential equations with nonsmooth vector fields, continuity equation, transport equation, regular Lagrangian flow, BV function, log-Sobolev space, Bressan's mixing conjecture
Mathematical Subject Classification 2010
Primary: 34A12, 35F10, 35F25
Milestones
Received: 27 October 2019
Revised: 24 May 2020
Accepted: 31 July 2020
Published: 19 December 2021
Authors
Elia Bruè
Scuola Normale Superiore
Pisa
Italy
School of Mathematics
Institute for Advanced Study
Princeton, NJ
United States
Quoc-Hung Nguyen
Scuola Normale Superiore
Pisa
Italy
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing
China