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

TL;DR

This paper establishes local well-posedness for an inviscid zero-Mach number system in endpoint Besov spaces, introduces a new a priori estimate, and discusses lifespan bounds and global existence in two dimensions.

Contribution

It extends analysis of the zero-Mach number system to endpoint Besov spaces, providing new estimates and lifespan bounds, especially in two dimensions.

Findings

Local well-posedness under $L^2$ initial data

New a priori estimate for parabolic equations in endpoint spaces

Lower bound for solution lifespan in 2D with small initial inhomogeneity

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 $B^s_{\infty,r}(\mathbb{R}^d)$, $r\in [1,\infty]$, $d\geq 2$, which can be embedded in the Lipschitz class $C^{0,1}$. In particular, the largest case $B^1_{\infty,1}$ and the case of H\"older 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 $B^s_{\infty,r}$ 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.