The viscous damping of three dimensional spherical gas bubble inside unbounded compressible liquid

TL;DR

This paper proves the global existence and stability of a spherical gas bubble in a viscous, compressible liquid, demonstrating how viscous damping leads to decay of perturbations with explicit rates.

Contribution

It establishes the global existence and asymptotic stability of spherical bubbles in viscous compressible liquids using energy methods, providing explicit decay rates.

Findings

Global existence of solutions proved

Asymptotic stability of spherical equilibrium established

Explicit decay rates for perturbations derived

Abstract

The present paper considers a homogeneous bubble inside an unbounded polytropic compressible liquid with viscosity. The system is governed by the Navier-Stokes equation with free boundary which is determined by the kinematic and dynamic boundary conditions on the bubble-liquid interface. The global existence of solution is proved, and the $\dot{H}^1$ asymptotic stability of the spherical equilibrium in terms of viscous damping together with a explicit decay rate is given in bare energy methods.