#### Vol. 14, No. 4, 2021

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Waves interacting with a partially immersed obstacle in the Boussinesq regime

### Didier Bresch, David Lannes and Guy Métivier

Vol. 14 (2021), No. 4, 1085–1124
##### 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 $2×2$ 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.

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##### Keywords
wave-structure interaction, Boussinesq system, free surface, transmission problem, local well-posedness, dispersive boundary layer, oscillations in time, compatibility conditions
##### Mathematical Subject Classification 2010
Primary: 35B30, 35G61, 35Q35, 76B15
##### Milestones
Received: 12 February 2019
Revised: 14 November 2019
Accepted: 20 December 2019
Published: 6 July 2021
##### Authors
 Didier Bresch Université Grenoble Alpes Université Savoie Mont Blanc et CNRS UMR 5127 Le Bourget du Lac France David Lannes Institut de Mathématiques de Bordeaux Université de Bordeaux et CNRS UMR 5251 Talence France Guy Métivier Institut de Mathématiques de Bordeaux Université de Bordeaux et CNRS UMR 5251 Talence France