Compressible, inviscid Rayleigh-Taylor instability (Q2898370)

The authors study the Rayleigh-Taylor problem for two compressible, immiscible, inviscid, barotropic fluids evolving with a free interface in the presence of a uniform gravitational field. The fluids are moving nonstationary and within an infinite 3D-slab. A steady-state solution is constructed with a denser fluid located above the free interface with the second fluid. Then the authors analyze linearized equations obtained by linearization of the original problem over the steady-state solution. Constructing normal mode solutions that grow exponentially in time and applying a Fourier synthesis of these solutions lead to an ill-posedness result for the linearized problem. Finally, it can be shown the ill-posedness for the original nonlinear problem (i.e. the compressible Euler equations). More precisely, the authors show by a contradiction argument that the nonlinear problem does not admit reasonable estimates of solutions for small time in terms of the initial data.