Stability of perturbations near a background magnetic field of the 2D incompressible MHD equations with mixed partial dissipation

Abstract

The stability and large-time behavior problem on some partially dissipated systems is not well-understood. The vorticity gradient of the 2D incompressible Euler equation can grow double exponentially in time while the same quantity to the 2D Navier-Stokes equation decays algebraically in time. However, the stability and large-time behavior of the vorticity gradients of the 2D Navier-Stokes equation with only vertical or horizontal dissipation appears to be unknown. This paper presents a global stability result on perturbations near a background magnetic field to the 2D incompressible magnetohydrodynamic (MHD) equations with vertical dissipation and horizontal magnetic diffusion. This stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit decay rate for the solution of this nonlinear system.