带奇异记忆的粘弹性波动方程的高频波传播
High frequency wave propagation for the viscoelastic wave equation with singular memory
- Purdue University（普渡大学）
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AI总结：
本文研究带空间依赖奇异记忆的粘弹性波动方程高频传播，构造双尺度几何光学精确解，推导局部阻尼逼近方程，证明外部观测可唯一恢复核参数，并通过压缩半群证明适定性与正则性，为几何光学构造提供依据。
AI中文摘要：
我们研究了具有空间依赖的相对历史形式遗传记忆的粘弹性波动方程的高频传播特性。记忆核可具有可积奇异性$\boldsymbol{\frak m}(s,x)=s^{p-1}m(s,x)$，其中$0<p<1$，同时涵盖$p=1$的正则情形。以半振幅传播距离和对应的走时作为单位，当波长$h\to0$时，记忆因子为$\boldsymbol{\frak \tau}=h^{1-p}$。我们构造了具有双尺度几何光学展开的精确解，展开形式为$h^{k+(1-p)\tau}$的幂级数。记忆通过瞬时模量$\boldsymbol{\frak \rho}+\boldsymbol{\frak \tau}\boldsymbol{\frak \rho}_0^\tau\boldsymbol{\frak m}(s,\boldsymbol{\frak \rho})\boldsymbol{\frak \rho}s$修正传播几何结构，而核的奇异性会为主输运方程贡献$C_p=\boldsymbol{\frak \rho}(p)e^{i\boldsymbol{\frak \rho}p/2}$项。当$0<p<1$时，这会产生分数阶尺度、频率相关的衰减以及色散相位校正；当$p=1$时，分数阶层级结构消失，衰减与频率无关，且输运相位校正为零。我们还推导了一个局部阻尼波动方程，其入射高频解在半经典$C^k$范数下以$O(h)$的误差逼近遗传记忆方程的解。对于所有入射方向且$0<h\to0$的情形，外部观测可唯一恢复$\boldsymbol{\frak \rho}_{\boldsymbol{\frak m}}$以及$m$在$s=0$处的完整时间射流，这些量决定了展开式的$O(h^\tau)$余项。最后，通过压缩半群方法，我们证明了空间依赖弱奇异核且给定完整预历史的问题的适定性和任意有限阶Sobolev正则性，给出了显式相容性条件以及关于$\boldsymbol{\frak \tau}$一致的估计，这些估计为几何光学构造提供了理论依据。
英文摘要：
We study high-frequency propagation for a viscoelastic wave equation with spatially dependent hereditary memory in relative-history form. The kernel may have the integrable singularity $\mathfrak m(s,x)=s^{p-1}m(s,x)$, $0<p<1$; the regular case $p=1$ is included. Using the half-amplitude propagation distance and corresponding travel time as units, the wavelength $h\ll1$ yields the memory factor $\varepsilon=h^{1-p}$. We construct exact solutions with full two-scale geometric-optics expansions in powers $h^{k+(1-p)\ell}$. Memory modifies the propagation geometry through the instantaneous modulus $σ+\varepsilon\int_0^\infty\mathfrak m(s,\cdot)\,d s$, while the kernel singularity contributes $C_p=Γ(p)e^{iπp/2}$ to the leading transport equation. For $0<p<1$, this produces fractional scales, frequency-dependent attenuation, and a dispersive phase correction; for $p=1$, the fractional hierarchy disappears, attenuation is frequency independent, and the transport phase correction vanishes. We also derive a local damped wave equation whose incoming high-frequency solutions approximate the hereditary solutions with $O(h)$ error in semiclassical $C^k$ norms. Exterior observations for all incident directions and $0<h\ll1$ uniquely recover $σ_{\mathfrak m}$ and the full temporal jet of $m$ at $s=0$, which determine the expansion modulo $O(h^\infty)$. Finally, a contraction-semigroup argument gives well-posedness and arbitrary finite-order Sobolev regularity for spatially dependent weakly singular kernels and prescribed full prehistory, with explicit compatibility conditions and estimates uniform in $\varepsilon$. These estimates justify the geometric-optics construction.
