On Classical Solutions of Rayleigh--Taylor Instability in Inhomogeneous Incompressible Viscous Fluids in Bounded Domains

TL;DR

This paper establishes the existence of classical solutions demonstrating Rayleigh--Taylor instability in inhomogeneous viscous fluids within bounded domains, using an iterative approach and bootstrap method.

Contribution

It introduces a new method to construct classical solutions of the nonlinear RT problem from linear solutions, advancing understanding of instability in viscous fluids.

Findings

Existence of local-in-time classical solutions for the RT problem.

Construction of initial data leading to instability.

Application of bootstrap instability method.

Abstract

We study the existence of unstable classical solutions of the Rayleigh--Taylor instability problem (abbr. RT problem) of an inhomogeneous incompressible viscous fluid in a bounded domain. We find that, by using an existence theory of (steady) Stokes problem and an iterative technique, the initial data of classical solutions of the linearized RT problem can be modified to new initial data, which can generate local-in-time classical solutions of the RT problem, and are close to the original initial data. Thus, we can use a classical bootstrap instability method to further obtain classical solutions of (nonlinear) RT instability based on the ones of linear RT instability.