Global weak solutions for Boussinesq system with temperature dependent viscosity and bounded temperature. (English) Zbl 1375.35312

De Anna, Francesco

Adv. Differ. Equ. 21, No. 11-12, 1001-1048 (2016).

Summary: 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.

MSC:

35Q30	Navier-Stokes equations
35Q35	PDEs in connection with fluid mechanics
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05	Navier-Stokes equations for incompressible viscous fluids