Stability for the two-dimensional anisotropic magnetohydrodynamic equations

The stabilization effect of a background magnetic field on electrically conducting fluids has attracted attention very recently, since a remarkable phenomenon was observed through physical experiments and numerical simulations. To shed light on this phenomenon, we focus on a special two-dimensional anisotropic magnetohydrodynamic equation near a background magnetic field with mixed dissipation. We study the stability problem and large time behavior of solutions. Precisely, we obtain the global stability of smooth solution for the two-dimensional nonlinear magnetohydrodynamic system in the sense of H2. In addition, we also establish the large time behavior with explicit decay rates to the linearized system.