Analysis of an inviscid zero-Mach number system in endpoint Besov spaces with finite-energy initial data

Abstract

The present paper is the continuation of work [14], devoted to the study of an inviscid zero-Mach number system in the framework of \emph{endpoint} Besov spaces of type $B_{\infty ,r}^s({\mathbb{R}}^d)$, $r\in [1,\infty ]$, $d\ge 2$, which can be embedded in the Lipschitz class $C^{0,1}$. In particular, the largest case $B_{\infty ,1}^1$ and the case of Hölder spaces $C^{1,\alpha }$ are permitted. The local in time well-posedness result is proved, under an additional $L^2$ hypothesis on the initial inhomogeneity and velocity field. A new a priori estimate for parabolic equations in endpoint spaces $B_{\infty ,r}^s$ is presented, which is the key to the proof. In dimension two, we are able to give a lower bound for the lifespan, such that the solutions tend to be globally defined when the initial inhomogeneity is small. There we will show a refined a priori estimate in endpoint Besov spaces for transport equations with \emph{non solenoidal} transport velocity field.

This submission supersedes the second part of arXiv:1305.1131

References in corpus (2)

• A global existence result for a zero Mach number system
• The well-posedness issue for an inviscid zero-Mach number system in general Besov spaces