Global weak solutions for Boussinesq system with temperature dependent viscosity and bounded temperature

Abstract

In this paper, we obtain a result about the global existence of weak solutions to the d-dimensional Boussinesq-Navier-Stokes system, with viscosity dependent on temperature. The initial temperature is only 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. In the preliminaries, and in the appendix, we consider some $L^p_t L^q_x$ regularity Theorems for the heat kernel, which play an important role in the main proof of this article.