{"title":"Free Boundary Problem for a Gas Bubble in a Liquid, and Exponential Stability of the Manifold of Spherically Symmetric Equilibria","authors":"Chen-Chih Lai, Michael I. Weinstein","doi":"10.1007/s00205-023-01927-z","DOIUrl":null,"url":null,"abstract":"<div><p>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 and 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 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.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6000,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-023-01927-z","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
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 and 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 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.
期刊介绍:
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.