Floating structures in shallow water: local well-posedness in the axisymmetric case

TL;DR

This paper proves the local well-posedness of the axisymmetric shallow water fluid-structure interaction problem involving floating structures, using a coupled hyperbolic PDE and ODE framework under certain initial conditions.

Contribution

It establishes the first well-posedness result for the axisymmetric shallow water floating structure problem with a rigorous mathematical framework.

Findings

Proves local in time well-posedness of the coupled system.

Formulates the problem as a hyperbolic PDE and ODE system.

Requires compatibility conditions on initial data.

Abstract

The floating structure problem describes the interaction between surface water waves and a floating body, generally a boat or a wave energy converter. As shown by Lannes in [18] the equations for the fluid motion can be reduced to a set of two evolution equations on the surface elevation and the horizontal discharge. The presence of the object is accounted for by a constraint on the discharge under the object; the pressure exerted by the fluid on this object is then the Lagrange multiplier associated to this constraint. Our goal in this paper is to prove the well-posedness of this fluid-structure interaction problem in the shallow water approximation under the assumption that the flow is axisymmetric without swirl. We write the fluid equations as a quasilinear hyperbolic mixed initial boundary value problem and the solid equation as a second order ODE coupled to the fluid equations. Finally we prove the local in time well-posedness for this coupled problem, provided some compatibility conditions on the initial data are satisfied.