Instability of the Abstract Rayleigh--Taylor Problem and Applications

TL;DR

This paper proves the existence of a unique unstable solution for an abstract Rayleigh--Taylor problem in stratified viscous fluids, extending to various fluid models and coordinate systems using a bootstrap instability method.

Contribution

It introduces a novel bootstrap instability method to construct unstable solutions for the abstract RT problem, applicable to multiple fluid models and coordinate frameworks.

Findings

Existence of a unique unstable solution in L^1-norm for the abstract RT problem.

Extension of unstable solutions to viscoelastic, MHD, and viscous fluids.

Method for modifying initial data to satisfy compatibility conditions.

Abstract

We prove the existence of a unique unstable strong solution in the sense of $L^1$-norm for an abstract Rayleigh--Taylor (RT) problem arising from stratified viscous fluids in Lagrangian coordinates based on a bootstrap instability method. In the proof, we develop a method to modify the initial data of the linearized abstract RT problem based on an existence theory of unique solution of stratified (steady) Stokes problem and an iterative technique, so that the obtained modified initial data satisfy necessary compatibility conditions of the (original) abstract RT problem. Applying an inverse transformation of Lagrangian coordinates to the obtained unstable solution, and then taking proper values of parameters, we can further get unstable solutions for the RT problems in viscoelastic fluids, magnetohydrodynamics (MHD) fluids with zero resistivity and pure viscous fluids (with or without interface intension) in Eulerian coordinates. Our results can be also extended to the corresponding inhomogeneous case (without interface).