On the Rayleigh-Taylor Instability for Incompressible, Inviscid Magnetohydrodynamic Flows

We study the Rayleigh-Taylor instabolity for two incompressible, immiscible, inviscid magnetohydrodynamic (MHD) fluids with zero resistivity, evolving with a free interface in the presence of a uniform gravitational field. We first construct the Rayleigh-Taylor steady-state solution with a denser fluid lying above the light one. Then, we turn to an analysis of the equations obtained from linearizing around such a steady state. By solving a system of ordinary differential equations, we construct the normal mode solutions to the linearized problem that grow exponentially in time. A Fourier synthesis of these normal mode solutions allows us to construct solutions that grow arbitrarily quickly in the Sobolev space H, thus leading to an ill-posedness result for the linearized problem in the sense of Hadamard. Using these pathological solutions, we can then demonstrate the ill-posedness of the original non-linear problem in some sense.