On the return to equilibrium problem for axisymmetric floating structures in shallow water

In this paper we address the return to equilibrium problem for an axisymmetric floating structure in shallow water. First we show that the motion of the solid object can be reduced to a delay differential equation involving an extension-trace operator whose role is to describe the influence of the fluid equations on the solid motion. It turns out that the compatibility conditions on the initial data for the return to equilibrium configuration are not satisfied, so we cannot use the results from [3] for the nonlinear problem. Hence we linearize the equations in the exterior domain supposing small amplitude waves and we keep the nonlinear equations in the interior domain. For such configurations, the extension-trace operator can be computed explicitly and the delay term in the delay differential equation can be put in convolution form. The solid motion is governed by a nonlinear second order integro-differential equation, whose linearization is the well-known Cummins equation. By writing it as a functional differential equation with infinite delay, we show the global in time existence and uniqueness of the solution using the conservation of the total energy. Finally the local asymptotic stability of the equilibrium position is shown.