An Improved Result on Rayleigh--Taylor Instability of Nonhomogeneous Incompressible Viscous Flows

TL;DR

This paper improves the understanding of Rayleigh--Taylor instability in nonhomogeneous incompressible viscous flows by removing previous restrictive conditions, demonstrating nonlinear instability through refined energy estimates.

Contribution

It establishes nonlinear instability of Rayleigh--Taylor steady-states without the earlier restrictive assumption on the density derivative.

Findings

Nonlinear instability proven without restrictive density conditions

Refined energy estimates enhance stability analysis

Bootstrap argument confirms instability results

Abstract

In [F. Jiang, S. Jiang, On instability and stability of three-dimensional gravity driven viscous flows in a bounded domain, Adv. Math., 264 (2014) 831--863], Jiang et.al. investigated the instability of Rayleigh--Taylor steady-state of a three-dimensional nonhomogeneous incompressible viscous flow driven by gravity in a bounded domain $\Omega$ of class $C^2$. In particular, they proved the steady-state is nonlinearly unstable under a restrictive condition of that the derivative function of steady density possesses a positive lower bound. In this article, by exploiting a standard energy functional and more-refined analysis of error estimates in the bootstrap argument, we can show the nonlinear instability result without the restrictive condition.