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

TL;DR

This paper investigates the well-posedness of density-dependent incompressible Euler equations in endpoint Besov spaces, establishing local existence, continuation criteria, and lifespan estimates, especially highlighting the case where initial density approaches a constant.

Contribution

It extends well-posedness results to endpoint Besov spaces for density-dependent Euler equations and provides new lifespan bounds in two dimensions.

Findings

Established local well-posedness in endpoint Besov spaces.

Derived continuation criteria similar to Beale-Kato-Majda.

Showed lifespan tends to infinity as initial density approaches a constant in 2D.

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 $B^s_{\infty,r}$ embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of H\"older spaces and of the endpoint Besov space $B^1_{\infty,1}.$ 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.