论文标题
界面重力波的Kakinuma模型的数学分析。第二部分：作为浅水近似的理由
A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part II: Justification as a shallow water approximation
论文作者
论文摘要
我们认为Kakinuma模型是界面重力波的运动。 Kakinuma模型是一个近似Lagrangian的Euler-Lagrange方程的系统，该系统是通过近似完整模型Lagrangian中的速度势来获得的。 Kakinuma模型的结构及其初始价值问题的适当性在同伴论文[Arxiv：2103.12392]中分析。在本文中，我们表明，Kakinuma模型是界面重力波的较高浅水近似值，并在$ O（δ_1^{4n+2}+2}+δ_2^{4n+2^{4n+2} $ $的情况下，在一致性的意义上，其中$δ_1$和$Δ_2$Δ_2$ $Δ大致说明Kakinuma模型的大小，分别是典型的水平波长和$ n $的下层。此外，在对完整模型的解决方案存在的假设下，通过在Kakinuma模型的解决方案和完整模型的解决方案之间给出错误估计，证明了Kakinuma模型的严格理由。还提供了Kakinuma模型的哈密顿量与完整模型的错误估计。
We consider the Kakinuma model for the motion of interfacial gravity waves. The Kakinuma model is a system of Euler-Lagrange equations for an approximate Lagrangian, which is obtained by approximating the velocity potentials in the Lagrangian of the full model. Structures of the Kakinuma model and the well-posedness of its initial value problem were analyzed in the companion paper [arXiv:2103.12392]. In this present paper, we show that the Kakinuma model is a higher order shallow water approximation to the full model for interfacial gravity waves with an error of order $O(δ_1^{4N+2}+δ_2^{4N+2})$ in the sense of consistency, where $δ_1$ and $δ_2$ are shallowness parameters, which are the ratios of the mean depths of the upper and the lower layers to the typical horizontal wavelength, respectively, and $N$ is, roughly speaking, the size of the Kakinuma model and can be taken an arbitrarily large number. Moreover, under a hypothesis of the existence of the solution to the full model with a uniform bound, a rigorous justification of the Kakinuma model is proved by giving an error estimate between the solution to the Kakinuma model and that of the full model. An error estimate between the Hamiltonian of the Kakinuma model and that of the full model is also provided.
