Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection

Abstract

This paper is concerned with the initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. We show that the system has a unique classical solution for H 3 initial data, and the non-slip boundary condition for velocity field and the perfectly conducting wall condition for magnetic field. In addition, we show that the kinetic energy is uniformly bounded in time.

Stability of Couette flow for 2D Boussinesq system in a uniform magnetic field

“…Two fundamental problems, the global regularity problem and the stability problem, have been among the main driving forces in advancing the mathematical theory on the Boussinesq-MHD system. Significant progress has been made on the global regularity of the nonlinear Boussinesq-MHD system [7,8,9,10,23,24,26,37,38]. The goal of this paper is the nonlinear stability around the Couette flow (u sh = (y, 0), b sh = (1, 0), p sh = 0, θ sh = 0).…”

Initial-boundary value problem to 2D Boussinesq equations for MHD convection with stratification effects

“…Compared with Cauchy problems, solutions of the initial boundary value problems usually exhibit different behaviors and much richer phenomena. In [7], the author obtained the global well-posedness for Boussinese-MHD system (1.1) in bounded domain with constant viscosity. Nevertheless, for initial-boundary value problem to the system (1.1) with temperaturedependent viscosity, it is still open.…”

On the stability of hydrostatic equilibrium in a uniform magnetic field with partial dissipation

“…For more physics, numerical simulations, and experiments, the interested readers may refer to previous studies [1][2][3] and the references therein. Recently, important progress has been made on the global stability for Boussinesq-MHD system; see other works [4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein. And Bian et al 17 have proved the nonlinear stability of Couette flow in a uniform magnetic field for this system with only vertical dissipation, in which some ideas are shared from the paper of Deng et al 18 for 2D Boussinesq system with vertical dissipation.…”