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

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 (2003 arXiv:1307.056517) (see also Colombo and De Rosa (2020 SIAM J. Math. Anal. 52 221–238)), which covers the case of Hölder spaces.