Local existence and blow‐up criterion for the Euler equations in the Besov spaces

Abstract

We prove the local in time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler equations of inviscid incompressible fluid flows in Rⁿ, n≥2. As a corollary we obtain the persistence of Besov space regularity for the solutions of the 2‐D Euler equations with initial velocity belonging to the Besov spaces. For the proof of the results we establish a logarithmic inequality of the Beale–Kato–Majda type and a Moser type of inequality in the Besov spaces.