Free boundary problems of PDEs arising from bubble dynamics

Abstract: Bubble dynamics plays a crucial role not only in fundamental and applied physics but also in various engineering and industrial applications. In this talk, we will discuss mathematical models describing the deformation of a gas bubble in a liquid. These models fall under the category of fluid interface problems, a subclass of free boundary problems.

To provide a comprehensive understanding, we will begin with a brief overview of fundamental PDEs governing fluid dynamics, the associated boundary conditions, and the broader context of bubble dynamics. Subsequently, attention will be directed towards the thermal decay of bubble oscillation, particularly examining the approximate model proposed by A. Prosperetti in [J. Fluid Mech. 1991]. This model exhibits a one-parameter manifold of spherical equilibria, parametrized by the bubble mass. We prove that the manifold of spherical equilibria is an attracting centre manifold against small spherically symmetric perturbations and that solutions approach this manifold at an exponential rate as time advances. Moreover, we show that all equilibrium bubbles are spherically symmetric through an application of Alexandrov’s theorem on closed constant-mean-curvature surfaces. Furthermore, the manifold of spherically symmetric equilibria captures all regular spherically symmetric equilibrium.

We also examine the dynamics of the bubble-fluid system subject to a small-amplitude, time-periodic external sound field. We prove that this periodically forced system admits a unique time-periodic solution that is nonlinearly and exponentially asymptotically stable against small spherically symmetric perturbations.

If time permits, I will discuss a work in progress on asymmetric dynamics of these models and future directions. This talk is based on joint work with Michael I. Weinstein ([Arch. Ration. Mech. Anal. 2023], [Nonlinear Anal. 2024], and work in progress).