The well-posedness issue in endpoint spaces for an inviscid low-Mach number limit system

TL;DR

This paper investigates the well-posedness of a low-Mach number limit system with heat conduction in Besov spaces, providing new estimates and conditions for existence, uniqueness, and lifespan of solutions in various function space settings.

Contribution

It extends well-posedness results for the low-Mach system in Besov spaces, introducing new a priori estimates and handling cases with different integrability and smallness conditions.

Findings

Established well-posedness in Besov spaces for various p values.

Derived a new a priori estimate for parabolic equations.

Provided a lower bound for solution lifespan, especially in 2D.

Abstract

The present paper is devoted to the well-posedness issue for a low-Mach number limit system with heat conduction but no viscosity. We will work in the framework of general Besov spaces $B^s_{p,r}(\R^d)$, $d\geq 2$, which can be embedded into the class of Lipschitz functions. Firstly, we consider the case of $p\in[2,4]$, with no further restrictions on the initial data. Then we tackle the case of any $p\in\,]1,\infty]$, but requiring also a finite energy assumption. The extreme value $p=\infty$ can be treated due to a new a priori estimate for parabolic equations. At last we also briefly consider the case of any $p\in ]1,\infty[$ but with smallness condition on initial inhomogeneity. A continuation criterion and a lower bound for the lifespan of the solution are proved as well. In particular in dimension 2, the lower bound goes to infinity as the initial density tends to a constant.