Global Weak Solutions for Boussinesq System with Temperature dependent Viscosity and bounded Temperature

TL;DR

This paper proves the global existence of weak solutions for a Boussinesq system with temperature-dependent viscosity, assuming bounded initial temperature and specific conditions on initial velocity and viscosity close to a constant.

Contribution

It establishes the global weak solutions for the Boussinesq system with variable viscosity under new conditions on initial data and viscosity proximity to a constant.

Findings

Global weak solutions exist under specified conditions.

Viscosity close to a positive constant is sufficient.

Bounded initial temperature with critical velocity space suffices.

Abstract

In this paper we obtain a result about the global existence of weak solutions for the $d$-dimensional Bussinesq system, with viscosity dependent on temperature. The initial temperature is just supposed to be bounded, while the initial velocity belongs to some critical Besov Space, invariant to the scaling of this system. We suppose the viscosity close enough to a positive constant, and the $L^\infty$ norm of their difference plus the Besov norm of the horizontal component of the initial velocity is supposed to be exponentially small with respect to the vertical component of the initial velocity. On Preliminaries and in the appendix we consider some $L^p L^q$ regularity Theorems for the heat kernel, which play an important role in the main proof of this article.