A regularity upgrade of pressure

TL;DR

This paper improves the understanding of pressure regularity in incompressible Euler equations by establishing optimal estimates using advanced function space techniques and generalizing classical lemmas.

Contribution

It introduces new optimal regularity estimates for pressure in various function spaces and generalizes the Div-Curl lemma as a fractional Leibniz rule in Hardy spaces.

Findings

Optimal pressure regularity estimates in Sobolev, Besov, Hardy spaces

Generalization of the Div-Curl lemma as a fractional Leibniz rule

Counterexamples demonstrating sharpness of results

Abstract

For the incompressible Euler equations the pressure formally scales as a quadratic function of velocity. We provide several optimal regularity estimates on the pressure by using regularity of velocity in various Sobolev, Besov and Hardy spaces. Our proof exploits the incompressibility condition in an essential way and is deeply connected with the classic Div-Curl lemma which we also generalise as a fractional Leibniz rule in Hardy spaces. To showcase the sharpness of results, we construct a class of counterexamples at several end-points.