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 $2\times 2$ 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.

本文致力于波结构相互作用问题的推导和数学分析，该问题可以简化为 Boussinesq 系统的传输问题。维数初边值问题与传递问题$d=1$ 为了 $2\times 2$双曲线系统很好理解。然而，对于许多应用，尤其是对地表水波的描述，必须考虑双曲线系统的色散扰动。我们在这里考虑一种配置，其中波浪的运动由 Boussinesq 系统（双曲非线性浅水方程的色散扰动）控制，并且存在固定的部分浸没的障碍物。我们将坚持标准双曲线情况的异同，将注意力集中在一个新现象上，即色散边界层的出现。为了在相关时间尺度上获得解的存在性和统一边界，必须控制该色散边界层及其产生的时间振荡。这种分析导致了兼容性条件的新概念，当色散参数设置为零时，该概念与标准双曲兼容性条件一致。据作者所知，这是第一次展示这些现象（可能在分析双曲系统色散扰动的初始边界值问题中起核心作用）。