Bienvenue - Laboratoire Jacques-Louis Lions

David Lannes (Université de Bordeaux 1)
Waves interacting with a partially immersed obstacle in the Boussinesq regime

Cette séance du séminaire s’inscrit dans le cadre des Journées Tarantola : Défis en géosciences (6 et 7 juin 2019)
https://ange-geophysics.sciencesconf.org

Résumé
In this talk we shall present 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) 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 conditions that is shown to coincide with the standard hyperbolic compatibility conditions when the dispersive parameter is set to zero. These phenomena are likely to play a central role in the analysis of initial boundary value problems for dispersive perturbations of hyperbolic systems.
This is joint work with D. Bresch and G. Métivier, see arXiv:1902.04837