On the Global Well-Posedness of 3-D Boussinesq System with Variable Viscosity

Abstract

In this paper, the authors first consider the global well-posedness of 3-D Boussinesq system, which has variable kinematic viscosity yet without thermal conductivity and buoyancy force, provided that the viscosity coefficient is sufficiently close to some positive constant in L ^∞ and the initial velocity is small enough in $\dot{B}_{3,1}^0(\mathbb{R}^3)$. With some thermal conductivity in the temperature equation and with linear buoyancy force θe ₃ on the velocity equation in the Boussinesq system, the authors also prove the global well-posedness of such system with initial temperature and initial velocity being sufficiently small in L ¹(ℝ³) and $\dot{B}_{3,1}^0(\mathbb{R}^3)$ respectively.

Keywords

Boussinesq systems / Littlewood-Paley theory / Variable viscosity / Maximal regularity of heat equation