The Oberbeck-Boussinesq approximation in critical spaces

TL;DR

This paper rigorously analyzes the Oberbeck-Boussinesq approximation for compressible viscous gases, proving global existence of solutions and convergence to the Boussinesq system in three dimensions, considering heat conductivity effects.

Contribution

It establishes the global existence and convergence of solutions in critical spaces, using Strichartz estimates and uniform Mach number bounds, for the first time in this context.

Findings

Global existence of strong solutions for small perturbations

Convergence to the Boussinesq system with explicit decay rates

Validity of the approximation in critical regularity spaces

Abstract

In this paper we study the validity of the so-called Oberbeck-Boussinesq approximation for compressible viscous perfect gases in the whole three-dimensional space. Both the cases of uids with positive heat conductivity and zero conductivity are considered. For small perturbations of a constant equilibrium, we establish the global existence of unique strong solutions in a critical regularity functional framework. Next, taking advantage of Strichartz estimates for the associated system of acoustic waves, and of uniform estimates with respect to the Mach number, we obtain all-time convergence to the Boussinesq system with a explicit decay rate.