The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces

Abstract

This work is the continuation of the recent paper \cite{D2} devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type $B_{\infty ,r}^s$ embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of Hölder spaces and of the endpoint Besov space $B_{\infty ,1}^1.$ For such data and under the nonvacuum assumption, we establish the local well-posedness and a continuation criterion in the spirit of that of Beale, Kato and Majda in \cite{BKM}. In the last part of the paper, we give lower bounds for the lifespan of a solution. In dimension two, we point out that the lifespan tends to infinity when the initial density tends to be a constant. This is, to our knowledge, the first result of this kind for the density-dependent incompressible Euler equations.

References in corpus (1)

• On the well-posedness of the incompressible density-dependent Euler equations in the $L^p$ framework