On Inhibition of Rayleigh--Taylor Instability by a Horizontal Magnetic Field in Non-resistive MHD Fluids: the Viscous Case

Abstract

It is still open whether the phenomenon of inhibition of Rayleigh-Taylor (RT) instability by a horizontal magnetic field can be mathematically verified for a non-resistive viscous magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, since it was roughly proved in the linearized case by Wang in [41]. In this paper, we prove such inhibition phenomenon by the (nonlinear) inhomogeneous, incompressible, viscous case with Navier (slip) boundary condition. More precisely, we show that there is a critical number of field strength m C , such that if the strength |m| of a horizontal magnetic field is bigger than m C , then the small perturbation solution around the magnetic RT equilibrium state is algebraically stable in time. In addition, we also provide a nonlinear instability result for the case |m| ∈ [0, m C ). The instability result presents that a horizontal magnetic field can not inhibit the RT instability, if it's strength is too small.