On Instability and Stability of Three-Dimensional Gravity Driven Viscous Flows in a Bounded Domain

TL;DR

This paper analyzes the stability and instability of three-dimensional gravity-driven viscous flows in bounded domains, demonstrating linear and nonlinear instability for heavier densities and stability for lighter densities under certain conditions.

Contribution

It provides a rigorous analysis of stability criteria for gravity-driven viscous flows, including new energy functionals and bootstrap methods for nonlinear stability analysis.

Findings

Heavier steady densities are linearly and nonlinearly unstable.

Lighter steady densities are linearly globally stable.

Stability depends on specific density conditions and domain properties.

Abstract

We investigate the instability and stability of some steady-states of a three-dimensional nonhomogeneous incompressible viscous flow driven by gravity in a bounded domain $\Omega$ of class $C^2$. When the steady density is heavier with increasing height (i.e., the Rayleigh-Taylor steady-state), we show that the steady-state is linear unstable (i.e., the linear solution grows in time in $H^2$) by constructing a (standard) energy functional and exploiting the modified variational method. Then, by introducing a new energy functional and using a careful bootstrap argument, we further show that the steady-state is nonlinear unstable in the sense of Hadamard. When the steady density is lighter with increasing height, we show, with the help of a restricted condition imposed on steady density, that the steady-state is linearly globally stable and nonlinearly locally stable in the sense of Hadamard.