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

TL;DR

This paper proves local well-posedness for the density-dependent Euler equations in the whole space within Besov spaces, extending previous results beyond the $L^2$ framework and small perturbations, using new a priori estimates.

Contribution

It establishes well-posedness results in Besov spaces for the density-dependent Euler equations without restrictions to $L^2$ or small perturbations, and introduces new a priori estimates for elliptic equations with variable coefficients.

Findings

Local-in-time well-posedness in Besov spaces including borderline cases.

A continuation criterion similar to Beale-Kato-Majda for these equations.

New a priori estimates for elliptic equations with nonconstant coefficients.

Abstract

The present paper is devoted to the study of the well-posedness issue for the density-dependent Euler equations in the whole space. We establish local-in-time results for the Cauchy problem pertaining to data in the Besov spaces embedded in the set of Lipschitz functions, including the borderline case $B^{\frac Np+1}_{p,1}(\R^N).$ A continuation criterion in the spirit of the celebrated one by Beale-Kato-Majda for the classical Euler equations, is also proved. In contrast with the previous work dedicated to this system in the whole space, our approach is not restricted to the $L^2$ framework or to small perturbations of a constant density state: we just need the density to be bounded away from zero. The key to that improvement is a new a priori estimate in Besov spaces for an elliptic equation with nonconstant coefficients.