# Sobolev estimates for solutions of the transport equation and ODE flows associated to non-Lipschitz drifts

## TL;DR

This paper demonstrates that Sobolev regularity can propagate for solutions of the transport equation and ODE flows with non-Lipschitz drifts when the gradient is exponentially integrable, providing sharp estimates and applications.

## Contribution

It establishes new conditions under which Sobolev regularity propagates for transport equations with non-Lipschitz drifts, extending previous results.

## Key findings

- Sobolev regularity propagates under exponential integrability of the gradient.
- Provides sharp Sobolev estimates for solutions.
- Generalizes a regularity result for the 2D Euler equation.

## Abstract

It is known, after \cite{Jabin16} and \cite{AlbertiCrippaMazzucato18}, that ODE flows and solutions of the transport equation associated to Sobolev vector fields do not propagate Sobolev regularity, even of fractional order. In this paper, we show that some propagation of Sobolev regularity happens as soon as the gradient of the drift is exponentially integrable. We provide sharp Sobolev estimates and new examples. As an application of our main theorem, we generalize a regularity result for the 2D Euler equation obtained by Bahouri and Chemin in \cite{BahouriChemin94}.