Nonlinear Rayleigh-Taylor instability for nonhomogeneous incompressible viscous magnetohydrodynamic flows

We investigate the nonlinear instability of a smoothRayleigh-Taylor steady-state solution (including the case of heavier density with increasingheight) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field.We first analyze the linearized equations around the steady-state solution.Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$,thus leading to the linear instability. With the help of the constructedunstable solutions of the linearized problem and a local well-posedness result of smooth solutionsto the original nonlinear problem, we establish the instability of the density, the horizontal and vertical velocities in the nonlinear problem.Moreover, when the steady magnetic field is vertical and small, we prove the instability of the magnetic field. This verifiesthe physical phenomenon: instability of the velocity leads to the instability of the magnetic field through the induction equation.