不连续底床上线性水波的浅水极限
The shallow--water limit of linear water waves over a discontinuous bottom
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中文总结 AI 辅助
该论文严格证明不连续底床上线性水波的浅水极限，提出抽象函数框架和显式共形映射方法，识别并数值验证了台阶诱导的色散边界层，并推导了描述该边界层的色散模型。
中文摘要 AI 辅助
我们严格证明了在尖锐底台阶上线性水波的零色散极限，将浅水方程恢复为透射问题，并描述了在间断附近产生的局部色散效应。为了获得在浅水参数下一致的估计，我们研究了二维流体域中具有Lipschitz弯曲底床的线性水波方程的抽象函数框架。该框架基于Dirichlet--Neumann算子的分数幂，我们利用改编的Rellich恒等式刻画了低正则性能量空间。在更高正则性下，相关的速度势会产生角奇异性。对于台阶几何，我们使用显式共形映射获得了Grisvard移位定理的一个版本，其中我们推导了势的奇异部分的显式公式。对水平变量的依赖揭示了由台阶产生的窄过渡尺度。向浅水透射问题的收敛在远离该过渡区域处成立，而在台阶附近则保留了一个局部色散修正。我们在数值上说明了这一差异，并识别出由此产生的台阶诱导色散边界层。基于这些观察，我们考虑了强色散区域，并严格推导了一个色散模型，该模型明确描述了小台阶的边界层。
英文摘要
We rigorously justify the zero--dispersion limit of linear water waves over a sharp bottom step, recovering the shallow-water equations as a transmission problem, and describe the localized dispersive effects generated near the discontinuity. To obtain estimates uniform in the shallow-water parameter, we study an abstract functional framework for the linear water waves equations in two-dimensional fluid domains with a Lipschitz curved bottom. The framework is based on fractional powers of the Dirichlet--Neumann operator, and we characterize low-regularity energy spaces using adapted Rellich identities. At higher regularity, the associated velocity potential develops corner singularities. For the step geometry, we use an explicit conformal mapping to obtain a version of Grisvard's shift theorem, where we derive an explicit formula for the singular part of the potential. The dependence on the horizontal variable reveals a narrow transition scale generated by the step. The convergence toward the shallow-water transmission problem holds away from this transition region, while a localized dispersive correction remains near the step. We illustrate this discrepancy numerically and identify the resulting step-induced dispersive boundary layer. Building on these observations, we consider the strongly dispersive regime and derive a consistent model that explicitly describes this boundary layer for a small step.
