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

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.