Compressible, In viscid Rayleigh-Taylor Instability

作者：Guo Yan^*; Tice Ian

来源：Indiana University Mathematics Journal, 2011, 60(2): 677-711.

• compressible Euler equation
• Rayleigh-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 et root vertical bar xi vertical bar, 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.

• 出版日期2011