On the Rayleigh–Taylor Instability for the Incompressible Viscous Magnetohydrodynamic Equations

Abstract

We study the Rayleigh-Taylor instability problem for two incompressible, immiscible, viscous magnetohydrodynamic (MHD) flows with zero resistivity and surface tension (or without surface tension), evolving with a free interface under presence of a uniform gravitational field. First, we reformulate the MHD free boundary problem in an infinite slab as a Navier-Stokes system in Lagrangian coordinates with a force term induced by the fluid flow map. Then, we analyze the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, and both fluids being in (unstable) equilibrium. By studying a family of modified variational problems, we construct smooth (when restricted to each fluid domain) solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces, thus leading to an global instability result for the linearized problem. Finally, using these pathological solutions, we prove the global instability for the corresponding nonlinear problem in an appropriate sense. Moreover, we evaluate that the so-called critical number indeed is equal to , and analyze the effect of viscosity and surface tension on the instability.