Compressible, inviscid Rayleigh-Taylor instability Yan GuoIan Tice 35Q3576E3076E19compressible Euler equationRayleigh-Taylor instability 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. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4193 10.1512/iumj.2011.60.4193 en Indiana Univ. Math. J. 60 (2011) 677 - 712 state-of-the-art mathematics http://iumj.org/access/