# Waves Interacting With A Partially Immersed Obstacle In The Boussinesq Regime

## TL;DR

This paper derives and analyzes a wave-structure interaction model involving a Boussinesq system with a partially immersed obstacle, highlighting dispersive boundary layers and establishing conditions for solution existence and bounds.

## Contribution

It introduces a new analysis of dispersive boundary layers in Boussinesq systems with obstacles, extending hyperbolic theory to dispersive wave-structure interactions.

## Key findings

- Identification of dispersive boundary layers near obstacles
- Development of compatibility conditions for dispersive systems
- Establishment of existence and uniform bounds for solutions

## Abstract

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.