IUMJ

Title: Compressible, inviscid Rayleigh-Taylor instability

Authors: Yan Guo and Ian Tice

Issue: Volume 60 (2011), Issue 2, 677-712

Abstract:

We consider the Rayleigh-Taylor problem for two compressible, immiscible, inviscid, barotropic fluids evolving with a free interface in the presence of a uniform gravitational field. After constructing Rayleigh-Taylor steady-state solutions with a denser fluid lying above the free interface with the second fluid, we turn to an analysis of the equations obtained from linearizing around such a steady state. By a natural variational approach, we construct normal mode solutions that grow exponentially in time with rate like $e^{t \sqrt{|\xi|}}$, where $\xi$ is the spatial frequency of the normal mode. A Fourier synthesis of these normal mode solutions allows us to construct solutions that grow arbitrarily quickly in the Sobolev space $H^k$, which leads to an ill-posedness result for the linearized problem. Using these pathological solutions, we then demonstrate ill-posedness for the original nonlinear problem in an appropriate sense. More precisely, we use a contradiction argument to show that the nonlinear problem does not admit reasonable estimates of solutions for small time in terms of the initial data.