可压缩流体中气泡振动的辐射衰减
Radiative Decay of Bubble Oscillations in a Compressible Fluid
摘要：
考虑气泡在无粘性、可压缩液体中的动力学，液体具有表面张力。运动学和动力学边界条件将气泡表面变形动力学与流体中的波的动力学耦合。该系统具有球形平衡态，这是由无限远处的压力和气泡内的气体压力的平衡引起的。我们在质心系中研究了这种平衡态的线性动力学：1）证明了速度势和气泡表面扰动满足空间指数时间衰减的逐点估计。2)时间衰减率取决于散射共振，非自伴谱问题的本征值。这些是极点奇点在下半平面的解析延拓的预解算子从上半平面，通过实轴到下半平面。3)的时间衰减估计的共振模式扩展的速度潜力和气泡表面扰动的后果。4)对于小的可压缩性（马赫数，气泡壁速度与声速的比率，ε），这是不可压缩极限的奇异扰动。控制极其缓慢的时间衰减的散射共振是{ \it瑞利共振}。由高精度数值研究支持的渐近性表明，最接近实轴的瑞利共振满足|\frac{\Im λ_\star(ε)}{\Re λ_\star(ε)}| = { \cal 0}（\exp（-κ \We \ ε^{-2}）），κ>0。这里，\We表示韦伯数，即比较惯性和表面张力的无量纲比率。5)为了获得上述结果，我们证明了一个一般的结果，独立的利益，估计诺依曼狄利克雷映射的波动方程，外部的一个球。
英文摘要：
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, ε), 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 λ_\star(ε)}{\Re λ_\star(ε)} | = {\cal O} (\exp(-κ \We\ ε^{-2})), κ>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.
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