Mathematics – Analysis of PDEs
Scientific paper
2010-04-06
Mathematics
Analysis of PDEs
to appear in SIAM Journal on Mathematical Analysis
Scientific paper
Consider the dynamics of a gas bubble in an inviscid, compressible liquid with surface tension. Kinematic and dynamic boundary conditions couple the bubble surface deformation dynamics with the dynamics of waves in the fluid. This system has a spherical equilibrium state, resulting from the balance of the pressure at infinity and the gas pressure within the bubble. We study the linearized dynamics about this equilibrium state in a center of mass frame: 1) We prove that the velocity potential and bubble surface perturbation satisfy point-wise in space exponential time-decay estimates. 2) The time-decay rate is governed by scattering resonances, eigenvalues of a non-selfadjoint spectral problem. These are pole singularities in the lower half plane of the analytic continuation of a resolvent operator from the upper half plane, across the real axis into the lower half plane. 3) The time-decay estimates are a consequence of resonance mode expansions for the velocity potential and bubble surface perturbations. 4) For small compressibility (Mach number, a ratio of bubble wall velocity to sound speed, \epsilon), this is a singular perturbation of the incompressible limit. The scattering resonances which govern the anomalously slow time-decay, are {\it Rayleigh resonances}. Asymptotics, supported by high-precision numerical studies, indicate that the Rayleigh resonances which are closest to the real axis satisfy | \frac{\Im \lambda_\star(\epsilon)}{\Re \lambda_\star(\epsilon)} | = {\cal O} (\exp(-\kappa\ \We\ \epsilon^{-2})), \kappa>0. Here, \We denotes the Weber number, a dimensionless ratio comparing inertia and surface tension. 5) To obtain the above results we prove a general result, of independent interest, estimating the Neumann to Dirichlet map for the wave equation, exterior to a sphere.
Shapiro Allen M.
Weinstein Michael I.
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