Stability and exponential decay of the magnetic Bénard system with horizontal dissipation

Abstract

This article is concerned with the global stability problem for the two-dimensional magnetic Bénard system. The focus here involves only horizontal dissipation, horizontal magnetic diffusion, and horizontal thermal diffusivity. The lack of vertical dissipation makes it difficult to study the stability and long-time behavior of the magnetic Bénard system. To handle these, we use a new method to decompose the velocity u, magnetic field b, and temperature Θ into the horizontal average (u‾,b‾,Θ‾) and the oscillation part (u˜,b˜,Θ˜). Then we establish the global stability by using several anisotropic inequalities when the initial data are small in H2(Ω). Moreover, we mathematically show that the H1-norm of the oscillation part eventually decays exponentially to zero. Furthermore, the solution (u,b,Θ) approaches the horizontal average (u‾,b‾,Θ‾) asymptotically.