On the Rayleigh-Taylor instability for the two coupled fluids

In this paper, we considered the Rayleigh-Taylor (RT) instability for two incompressible, immisicible, invisid coupled fluids, which were Euler and magnetohydrodynamic with zero resistivity. Under the action of the uniform gravitational field, the two fluids interacted at a free interface. We utilized the flow map to denote the Lorentz force under the Lagrangian coordinates. We first showed the ill-posedness to the linear problem around the RT steady state solution. By virtue of such an ill-posed result, we showed that the nonlinear system is also ill-posed.