Flows for non-smooth vector fields with subexponentially integrable divergence

TL;DR

This paper investigates the behavior of flows generated by Sobolev vector fields with divergence that is subexponentially integrable, using transport equations to analyze regularity and existence.

Contribution

It introduces a new approach based on quantitative estimates of transport equations to study flows with subexponentially integrable divergence.

Findings

Established regularity results for flows with subexponentially integrable divergence.

Extended DiPerna-Lions theory to a broader class of vector fields.

Provided new estimates for solutions to the transport equation.

Abstract

In this paper, we study flows associated to Sobolev vector fields with subexponentially integrable divergence. Our approach is based on the transport equation following DiPerna-Lions [DPL89]. A key ingredient is to use a quantitative estimate of solutions to the Cauchy problem of transport equation to obtain the regularity of density functions.