A Time-Domain Pressure-Interface Model for Gas Bubble Dynamics with Surface Tension: Well-Posedness, Classical Limits, and Resonance Branches

We derive and analyze a time-domain pressure--interface evolution model for a compressible gas bubble in a compressible liquid with surface tension. Starting from the nonlinear two-phase Euler free-boundary problem and linearizing about a spherical Young--Laplace equilibrium, we obtain a coupled bulk--surface hyperbolic system for the liquid pressure, the gas pressure, and the normal displacement of the interface. The time-domain analysis is complicated by the fact that the surface-tension quadratic form is indefinite: the $Y_0^0$ component is the volume-changing breathing mode, the $Y_1^m (m = \{-1,0,1\})$ components are neutral translations, and only the higher spherical harmonics give a coercive shape-mode energy. We exploit this decomposition. The coercive sector is treated by a $C^0$-semigroup argument, while the breathing mode is analyzed separately by Fourier--Laplace methods. This yields well-posedness for admissible finite-energy data and classical solutions under the natural compatibility conditions. We then justify two limiting descriptions with quantitative error estimates. In an acoustic quasi-static regime, the model reduces to the linearized Rayleigh--Plesset equation for the breathing mode and to the linearized Rayleigh--Lamb equations for the shape modes. In a different regime, it reduces to the frozen-interface acoustic transmission model. Finally, a frequency-domain analysis identifies the corresponding Minnaert, Rayleigh--Lamb, and Fabry--P\'erot-type resonance mechanisms as different components and limits of the same pressure--interface formulation.