  We study the dynamics of a gas bubble in a fluid with surface tension,
initially near a spherical equilibrium. While there are many studies and
applications of radial bubble dynamics, the theory of general deformations from
a spherical equilibrium is less developed. We aim to understand how
asymmetrically perturbed equilibrium bubbles evolve toward spherical
equilibrium due to thermal or viscous dissipation in an incompressible liquid.
We focus on the isobaric approximation [Prosperetti, JFM, 1991], under which
the gas pressure within the bubble is spatially uniform and obeys the ideal gas
law. The liquid outside the bubble is incompressible, irrotational, and has
surface tension. We prove that any equilibrium gas bubble must be spherical by
showing that the bubble boundary is a closed surface of constant mean
curvature. We then study the initial value problem (IVP) for the coupled PDEs,
constitutive laws and interface conditions of the isobaric approximation for
general (asymmetric) small initial perturbations of the spherical bubble in the
linearized approximation. Our first result, considering thermal damping without
viscosity, proves that the linearized IVP is globally well-posed. The monopole
(radial) component of the perturbation decays exponentially over time, while
the multipole (non-radial) components undergo undamped oscillations. This
indicates a limitation of the isobaric model for non-spherical dynamics. Our
second result, incorporating viscous dissipation, shows that the IVP is linear
and nonlinearly ill-posed due to an incompatibility of normal stress boundary
conditions, for non-spherical solutions, and the irrotationality assumption.
Our study concludes that to accurately capture the dynamics of general
deformations of a gas bubble, the model must account for either vorticity
generated at the bubble-fluid boundary, spatial non-uniformities in the gas
pressure, or both.
