Stability of the 3D MHD equations without vertical dissipation near an equilibrium

Abstract

Important progress has been made on the standard Laplacian case with mixed partial dissipation and diffusion. The stability problem of the 3D incompressible magnetohydrodynamic (MHD) equations without vertical dissipation but with the fractional velocity dissipation $ (-\Delta)^\alpha u $ and magnetic diffusion $ (-\Delta)^\beta b $ is unfortunately not often well understood for many ranges of fractional powers. This paper discovers that there are new phenomena with the case $ \alpha, \beta \leq 1 $. We establish that, if an initial datum ($ u_0, b_0 $) in the Sobolev space $ H^3(\mathbb{R}^3) $ is close enough to the equilibrium state, and we replace the terms $ (-\Delta)^\alpha u $ and $ (-\Delta)^\beta b $ by $ (-\Delta_h)^\alpha u $ and $ (-\Delta_h)^\beta b $, respectively, the resulting equations with $ \alpha, \beta \in(\frac{1}{2}, 1] $ then always lead to a steady solution, where $ \Delta_h = \partial_{x_1}^2+\partial_{x_2}^2 $.

Keywords

• magnetohydrodynamic equations
• background magnetic field
• partial dissipation
• stability