On the Nonlinear Rayleigh–Taylor Instability of Nonhomogeneous Compressible Elastic Fluids

Abstract

This paper focuses on the Rayleigh–Taylor problem of two-dimensional nonhomogeneous compressible elastic fluid within a horizontally periodic domain of infinite height. First, we utilize a variational method to establish linear unstable solutions for the elastic RT problem. Subsequently, inspired by Grenier’s approach in [Comm. Pure Appl. Math., 2000, 53(9): 1067–1091], we proceed to construct higher-order growing mode approximate solutions for the elastic RT problem, considering its inviscid nature. We then derive error estimates between these approximate solutions and the nonlinear solutions of the elastic RT problem. Finally, by adapting the bootstrap instability method of Hwang–Guo in [Arch. Ration. Mech. Anal., 2003, 167(3): 235–253], we demonstrate the existence of escape points, leading to the nonlinear RT instability result. This study shows that RT instability can manifest in compressible elastic fluids with a small elasticity coefficient.

Keywords

Compressible elastic fluids / Rayleigh–Taylor instability / approximate solutions / bootstrap method / 35Q35 / 35B35 / 76E09

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