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

Abstract

In this paper, we consider the Boussinesq equations with magnetohydrodynamics convection in the domain T × R and establishes the nonlinear stability of the Couette flow (u sh = (y, 0), b sh = (1, 0), p sh = 0, θ sh = 0). The novelty in this paper is that we design a new Fourier multiplier operator by using the properties of the enhanced dissipation to overcome the difficult term ∂ xy (−∆) −1 j in the linearized and nonlinear system. Then, we prove the asymptotic stability for the linearized system. Finally, we establish the nonlinear stability for the full system by bootstrap principle.