Stability of 2D inviscid MHD equations with only fractional magnetic diffusion in the horizontal direction

This paper focuses on a special 2D magnetohydrodynamic (MHD) system with no viscosity and only fractional magnetic diffusion in the horizontal direction on the domain Q = T x R and T = [0, 1] be a periodic box. Due to the lack of the velocity dissipation, 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 its solution grow rather rapidly. However, when coupled to the magnetic field in such an MHD system, our result in this paper then shows the stabilization effect. Moreover, we will derive the exponentially decay of solutions on horizontal direction.