Appears in collection : 2026 - T2 - WS3 - Idealised mathematical models for geophysical flows

The description of waves, through the water waves equations or simpler asymptotic models (such as the nonlinear shallow water equations or the Boussinesq system) is well understood in a domain without boundaries. In the case of wave-structure interactions, such as the dynamics of the shoreline or of floating objects, the free surface has a boundary formed by the contact line between the surface of the fluid and the surface of the solid. The presence of this boundary induces new difficulties such as the derivation and analysis of boundary conditions but also the analysis of the motion of the boundary itself. In this talk we review some known results on the treatment of boundary conditions for hyperbolic systems (such as the nonlinear shallow water equations), and propose some extensions motivated by wave-structure interactions. We will comment also on the treatment of boundary conditions for dispersive perturbations of hyperbolic systems (such as the Boussinesq equations) and introduce the notion of dispersive boundary layers. Finally, we will comment on the dynamics of the contact line, which is not always a kinematic boundary condition in the sense that a particle located on the contact line can detach from it.