Free boundary problem for a gas bubble in a liquid, and exponential stability of the manifold of spherically symmetric equilibria

TL;DR

This paper analyzes the stability of spherical gas bubbles in a liquid, proving exponential convergence to equilibrium for an approximate model and exploring the uniqueness of symmetric solutions.

Contribution

It establishes the nonlinear exponential stability of the manifold of equilibria for a free boundary PDE model of gas bubbles, using energy methods and center manifold analysis.

Findings

Exponential decay rate of perturbations towards equilibrium.

Spherical symmetry of all equilibria in the approximate model.

Embedding of the approximate model's equilibria within a larger family in the full model.

Abstract

We consider the dynamics of a gas bubble immersed in an incompressible fluid of fixed temperature, and focus on the relaxation of an expanding and contracting spherically symmetric bubble due to thermal effects. We study two models, both systems of PDEs with an evolving free boundary: the full mathematical model as well as an approximate model, arising for example in the study of sonoluminescence. For fixed physical parameters (surface tension of the gas-liquid interface, liquid viscosity, thermal conductivity of the gas, etc.), both models share a family of spherically symmetric equilibria, smoothly parametrized by the mass of the gas bubble. Our main result concerns the approximate model. We prove the nonlinear asymptotic stability of the manifold of equilibria with respect to small spherically symmetric perturbations. The rate of convergence is exponential in time. To prove this result we first prove a weak form of nonlinear asymptotic stability -- with no explicit rate of time-decay -- using the energy dissipation law, and then, via a center manifold analysis, bootstrap the weak time-decay to exponential time-decay. We also study the uniqueness of the family of spherically symmetric equilibria within each model. The family of spherically symmetric equilibria captures all regular spherically symmetric equilibria of the approximate system. However within the full model, this family is embedded in a larger family of spherically symmetric solutions. For the approximate system, we prove that all equilibrium bubbles are spherically symmetric, by an application of Alexandrov's theorem on closed surfaces of constant mean curvature.