Regularity results for rough solutions of the incompressible Euler equations via interpolation methods

TL;DR

This paper demonstrates that solutions to the incompressible Euler equations with spatial Besov regularity also have enhanced temporal regularity, and the pressure term is twice as regular as the velocity, using interpolation techniques.

Contribution

It extends previous results by showing improved regularity properties for Euler solutions and pressure in Besov spaces through interpolation methods.

Findings

Velocity solutions gain temporal regularity from spatial Besov regularity.

Pressure is shown to be twice as regular as the velocity.

Generalizes previous H"older space results to Besov spaces.

Abstract

Given any solution $u$ of the Euler equations which is assumed to have some regularity in space - in terms of Besov norms, natural in this context - we show by interpolation methods that it enjoys a corresponding regularity in time and that the associated pressure $p$ is twice as regular as $u$. This generalizes a recent result by Isett [16] (see also Colombo and De Rosa [8]), which covers the case of H\"older spaces.