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We consider the dynamics of a gas bubble immersed in an incompressible fluidof fixed temperature, and focus on the relaxation of an expanding andcontracting spherically symmetric bubble due to thermal effects. We study twomodels, both systems of PDEs with an evolving free boundary: the fullmathematical model as well as an approximate model, arising for example in thestudy of sonoluminescence. For fixed physical parameters (surface tension ofthe gas-liquid interface, liquid viscosity, thermal conductivity of the gas,etc.), both models share a family of spherically symmetric equilibria, smoothlyparametrized by the mass of the gas bubble. Our main result concerns theapproximate model. We prove the nonlinear asymptotic stability of the manifoldof equilibria with respect to small spherically symmetric perturbations. Therate of convergence is exponential in time. To prove this result we first provea weak form of nonlinear asymptotic stability -- with no explicit rate oftime-decay -- using the energy dissipation law, and then, via a center manifoldanalysis, bootstrap the weak time-decay to exponential time-decay.  We also study the uniqueness of the family of spherically symmetricequilibria within each model. The family of spherically symmetric equilibriacaptures all regular spherically symmetric equilibria of the approximatesystem. However within the full model, this family is embedded in a largerfamily of spherically symmetric solutions. For the approximate system, we provethat all equilibrium bubbles are spherically symmetric, by an application ofAlexandrov's theorem on closed surfaces of constant mean curvature.

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