发表机构
Jilin University; RICAM, Austrian Academy of Sciences（吉林大学; 奥地利科学院计算与应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对界面附近小气泡的米纳特共振，建立统一渐近理论，涵盖远分离、临界和近接触三种间距区间，揭示共振频率从孤立气泡极限到接触极限的连续变化规律。
AI 中文摘要
我们为声学背景中置于材料不连续界面 $\Gamma$ 附近的小气泡 $D_\varepsilon$ 所产生的米纳特共振发展了一套统一的渐近理论。气泡的特征尺寸为 $\varepsilon>0$，其密度和体积模量通过米纳特高对比度标度依赖于同一小参数。分析按尺度化间距 \\[ \Theta_\varepsilon:= \frac{\operatorname{dist}(D_\varepsilon,\Gamma)}{\varepsilon} \\] 进行组织，并涵盖三个自然区间：\\[ \Theta_\varepsilon\to+\infty, \qquad \Theta_\varepsilon\to\Theta_*\in(0,\infty), \qquad \Theta_\varepsilon\to0: \\] 1) 在远分离区间，界面在首阶不可见，经典的全空间米纳特定律得以恢复。2) 在临界区间，重标度几何同时保留气泡和界面且间距有限，牛顿电容被由极限切向透射问题确定的平界面电容所取代。3) 在近接触区间，该族收敛到接触极限电容，对所有满足 $\operatorname{dist}(D_\varepsilon,\Gamma) = o(\varepsilon)$ 的间隙标度产生相同的首阶米纳特共振。这三个区间提供了共振频率随气泡接近界面而变化（从经典孤立气泡极限到接触极限）的统一描述。分析依赖于将共振问题统一约化为标量方程以及对界面相关电容的变分刻画；特别是，相关透射能量的 Mosco 收敛产生了直到奇异接触构型的电容势和最小能量的收敛。
英文摘要
We develop a unified asymptotic theory for Minnaert resonances generated by a small bubble $D_\varepsilon$ placed near an interface $Γ$ of material discontinuity in an acoustic background. The bubble has characteristic size $\varepsilon>0$, with its density and bulk modulus depending on the same small parameter through the Minnaert high-contrast scaling. The analysis is organized by the scaled separation \[ Θ_\varepsilon := \frac{\operatorname{dist}(D_\varepsilon,Γ)}{\varepsilon}, \] and covers the three natural regimes \[ Θ_\varepsilon\to+\infty, \qquad Θ_\varepsilon\toΘ_*\in(0,\infty), \qquad Θ_\varepsilon\to0: \] 1) In the far-separation regime, the interface is invisible at leading order and the classical full-space Minnaert law is recovered. 2) In the critical regime, the rescaled geometry retains both the bubble and the interface at finite separation, and the Newtonian capacitance is replaced by a flat-interface capacitance determined by the limiting tangent transmission problem. 3) In the near-contact regime, this family converges to a contact-limiting capacitance, yielding the same leading-order Minnaert resonance for all gap scalings satisfying $\operatorname{dist}(D_\varepsilon,Γ) = o(\varepsilon)$. These three regimes provide a unified description of how the resonance frequency changes as the bubble approaches the interface, from the classical isolated-bubble limit to the contact limit. The analysis relies on a uniform reduction of the resonance problem to a scalar equation and on a variational characterization of the interface-dependent capacitances; in particular, Mosco convergence of the associated transmission energies yields convergence of the capacitary potentials and minimum energies up to the singular contact configuration.
