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

Jing Wang; Feng Xie

doi:10.3934/dcdsb.2016072

Article Contents

2016, Volume 21, Issue 8: 2767-2784. Doi: 10.3934/dcdsb.2016072

This issue Previous Article On the compressible Navier-Stokes-Korteweg equations Next Article A reaction-convection-diffusion model for cholera spatial dynamics

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

Jing Wang^1, and
Feng Xie^2,

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

Received: October 31, 2014

Revised: May 31, 2016

Published: September 2016

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Abstract

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.

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Mathematics Subject Classification: 35L04, 35L65, 76E17.

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