Wave-structure interaction for long wave models in the presence of a freely moving body on the bottom

In this paper we address a particular fluid-solid interaction problem in which the solid object is lying at the bottom of a layer of fluid and moves under the forces created by waves travelling on the surface of this layer. More precisely, we consider the water waves problem in a fluid of fixed depth with a flat bottom topography and with an object lying on the bottom, allowed to move horizontally under the pressure forces created by the waves. After establishing the physical setting of the problem, namely the dynamics of the fluid and the mechanics of the solid motion, as well as analyzing the nature of the coupling, we examine in detail two particular shallow water regimes: the case of the (nonlinear) Saint-Venant system, and the (weakly nonlinear) Boussinesq system. We prove an existence and uniqueness theorem for the coupled system in both cases. Using the particular structure of the coupling terms we are able to go beyond the standard scale for the existence time of solutions to the Boussinesq system with a moving bottom.