The Compressible Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability

Juhi Jang,Ian Tice,Yanjin Wang

doi:10.1007/s00205-015-0960-0

Abstract

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We are concerned with the Rayleigh-Taylor instability when the upper fluid is heavier than the lower fluid along the equilibrium interface. When the surface tension at the free internal interface is below the critical value, we prove that the problem is nonlinear unstable.

Highlights

The Rayleigh-Taylor instability, one of the classic examples of hydrodynamic instability, is an interfacial instability between two fluids of different densities that occurs when a heavy fluid initially lies above a lighter one in a gravitational field
In our companion paper [11], we have proved the global existence of solutions decaying to the equilibrium state (0, 0, 0) in the problem (1.32) when σ− > σc
The goal of this paper is to show that when σ− < σc, the equilibrium state (0, 0, 0) is unstable in the compressible viscous surface-internal wave problem (1.32)

Summary

Introduction

The Rayleigh-Taylor instability, one of the classic examples of hydrodynamic instability, is an interfacial instability between two fluids of different densities that occurs when a heavy fluid initially lies above a lighter one in a gravitational field. Theorem 1.2 together with our results in [11] establish sharp nonlinear stability criteria for the equilibrium state in the compressible viscous surface-internal wave problem. We summarize these and the rates of decay to equilibrium in the following table. Since linear instability can be established in the same way as that for the compressible viscous internal wave problem in [6], the heart of the proof of Theorem 1.2 is the passage from linear instability to nonlinear instability This is in general a delicate issue for PDEs since the spectrum of the linear part is fairly complicated and the unboundedness of the nonlinear part usually yields a loss in derivatives. We allow for composition of derivatives in this counting scheme in a natural way; for example, we write

Growing mode solution to the linearized equations

Growth of solutions to the linear inhomogeneous equations

Nonlinear energy estimates

Nonlinear instability