The well-posedness issue for an inviscid zero-Mach number system in general Besov spaces

TL;DR

This paper investigates the well-posedness of a zero-Mach number system with heat conduction in general Besov spaces, establishing local existence, continuation criteria, and lifespan bounds using advanced harmonic analysis techniques.

Contribution

It extends well-posedness results for zero-Mach number systems to non-homogeneous Besov spaces with new analytical tools and refined estimates.

Findings

Proved local in time well-posedness in Besov spaces

Established continuation criteria for solutions

Derived lower bounds for solution lifespan

Abstract

The present paper is devoted to the study of a zero-Mach number system with heat conduction but no viscosity. We work in the framework of general non-homogeneous Besov spaces $B^s_{p,r}(\mathbb{R}^d)$, with $p\in[2,4]$ and for any $d\geq 2$, which can be embedded into the class of globally Lipschitz functions. We prove a local in time well-posedness result in these classes for general initial densities and velocity fields. Moreover, we are able to show a continuation criterion and a lower bound for the lifespan of the solutions. The proof of the results relies on Littlewood-Paley decomposition and paradifferential calculus, and on refined commutator estimates in Chemin-Lerner spaces.