# Radiative decay of bubble oscillations

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Michael Weinstein, Columbia University
Fine Hall 110

We consider the dynamics of a gas bubble in an unbounded, inviscid and compressible fluid with surface tension. Kinematic and dynamic (Young-Laplace) boundary conditions couple the dynamics of bubble surface deformations to the dynamics of waves in the fluid. We study the linear decay estimates for the fluid and deforming bubble near the spherical equilibrium. The local energy decay is exponential in time, $exp(-\Gamma t)$. $\Gamma$ is determined by a non-selfadjoint scattering resonance spectral problem. The scattering resonances which limit the time-decay rate are of a high order multipole character and are due to surface tension. The decay rate for general solutions ($\Gamma$, exponentially small in the Mach number) is much, much slower than for spherically symmetric solutions ($\Gamma_{radial}$, linear in the Mach number). The analysis makes use of results on the Neumann to Dirchlet map for the wave equation, and results on the location of complex scattering resonances for the wave equation with the above boundary conditions.