On Inhibition of the Rayleigh-Taylor Instability by a Horizontal Magnetic Field in 2D Non-Resistive MHD Fluids: The Viscous Case

Abstract

We investigate whether the inhibition phenomenon of the 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. This phenomenon was mathematically analyzed by Wang (J. Math. Phys., 53:073701, 2012) for stratified MHD fluids in the linearized case. To our best knowledge, the mathematical verification of this inhibition phenomenon in the nonlinear case still remains open. In this paper, we prove such inhibition phenomenon for the (nonlinear) inhomogeneous, incompressible, viscous case with Navier (slip) boundary condition. More precisely, we show that there is a critical number of the 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. Moreover, we also provide a nonlinear instability result when $|m|∈[0,m_C).$