Stability of 2D inviscid MHD equations with only vertical magnetic diffusion on T2

Physical experiments and numerical simulations have observed a remarkable phenomenon that a background magnetic field can smooth and stabilize the electrically conducting turbulent fluids. To understand the mechanism of this phenomenon, we will focus on a special 2D magnetohydrodynamic (MHD) system with no viscosity and partial magnetic resistive and examine the stability near a background magnetic field. Due to the lack of dissipation for velocity field, this stability problem is not trivial. Without the presence of a magnetic field, the fluid velocity is governed by the 2D incompressible Euler equation. And it is well known that solutions to the 2D incompressible Euler equation can grow rather rapidly. Our result in this paper then shows the stabilization effect of magnetic filed for conductive fluids. By coupling a suitable magnetic filed, we can obtain the uniform upper bound of solutions. Moreover, we will derive the exponentially decay of solutions on one direction. The approach is based on delicate energy estimate together with some observations such as Poincáre’s inequality in one direction and cancellation structure.