# American Institute of Mathematical Sciences

October  2016, 21(8): 2767-2784. doi: 10.3934/dcdsb.2016072

## On the Rayleigh-Taylor instability for the compressible non-isentropic inviscid fluids with a free interface

 1 Department of Mathematics, Shanghai Normal University, Shanghai 200234 2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240

Received  November 2014 Revised  June 2016 Published  September 2016

In this paper, we study the Rayleigh-Taylor instability phenomena for two compressible, immiscible, inviscid, ideal polytropic fluids. Such two kind of fluids always evolve together with a free interface due to the uniform gravitation. We construct the steady-state solutions for the denser fluid lying above the light one. With an assumption on the steady-state temperature function, we find some growing solutions to the related linearized problem, which in turn demonstrates the linearized problem is ill-posed in the sense of Hadamard. By such an ill-posedness result, we can finally prove the solutions to the original nonlinear problem does not have the property EE(k). Precisely, the $H^3$ solutions to the original nonlinear problem can not Lipschitz continuously depend on their initial data.
Citation: Jing Wang, Feng Xie. On the Rayleigh-Taylor instability for the compressible non-isentropic inviscid fluids with a free interface. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2767-2784. doi: 10.3934/dcdsb.2016072
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