Calculus of Variations and Geometric Measure Theory

E. Bruè - Q. H. Nguyen

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

created by bruè on 08 May 2019
modified on 25 Dec 2022


Published Paper

Inserted: 8 may 2019
Last Updated: 25 dec 2022

Journal: Mathematische Annalen
Year: 2019


It is known, after J16 and ACM18, 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 BC94.