Boussinesq状态下波浪与部分浸没障碍物的相互作用
Waves Interacting With A Partially Immersed Obstacle In The Boussinesq Regime
摘要：
本文将波-结构相互作用问题转化为Boussinesq系统的传输问题，并对其进行了推导和数学分析。d= 1维$2\times2$双曲型方程组的初边值问题和传递问题已得到很好的理解。然而，对于许多应用，特别是对于表面水波的描述，双曲系统的色散扰动必须考虑。我们在这里考虑的配置，其中的波的运动是由Boussinesq系统（双曲型非线性浅水方程的色散扰动），并在存在一个固定的部分浸没的障碍。我们将坚持与标准双曲情形的区别和相似之处，并把我们的注意力集中在一个新的现象上，即弥散边界层的出现。为了获得在相关的时间尺度上的解的存在性和统一的边界，有必要控制这种色散边界层和它产生的振荡。这一分析导致了一个新的概念的兼容性条件，符合标准的双曲型兼容性条件时，色散参数设置为零。据作者所知，这是第一次，这些现象（可能发挥核心作用，在分析初边值问题的色散扰动双曲型系统）展出。
英文摘要：
This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension d= 1 for $2\times2$ hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive perturbations of hyperbolic systems must be considered. We consider here a configuration where the motion of the waves is governed by a Boussinesq system (a dispersive perturbation of the hyperbolic nonlinear shallow water equations), and in the presence of a fixed partially immersed obstacle. We shall insist on the differences and similarities with respect to the standard hyperbolic case, and focus our attention on a new phenomenon, namely, the apparition of a dispersive boundary layer. In order to obtain existence and uniform bounds on the solutions over the relevant time scale, a control of this dispersive boundary layer and of the oscillations in time it generates is necessary. This analysis leads to a new notion of compatibility condition that is shown to coincide with the standard hyperbolic compatibility conditions when the dispersive parameter is set to zero. To the authors' knowledge, this is the first time that these phenomena (likely to play a central role in the analysis of initial boundary value problems for dispersive perturbations of hyperbolic systems) are exhibited.
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