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

## TL;DR

This paper investigates the return to equilibrium problem for axisymmetric floating structures in shallow water, deriving a nonlinear integro-differential equation governing the solid motion and establishing global existence and uniqueness of solutions.

## Contribution

It introduces a reduced delay differential equation involving an extension-trace operator and derives explicit forms under small amplitude wave assumptions.

## Key findings

- The solid motion follows a nonlinear second order integro-differential equation.
- The linearized model corresponds to the Cummins equation.
- Global existence and uniqueness of solutions are proven using energy conservation.

## Abstract

In this paper we address the return to equilibrium problem for an axisymmetric floating structure in shallow water. First we show that the equation for the solid motion 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 result from [3] for the nonlinear problem. Hence, assuming small amplitude waves, we linearize the equations in the exterior domain 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 differential equation can be put in convolution form. The solid motion is therefore governed by a nonlinear second order integro-differential equation, whose linearization is the well-known Cummins equation. We show global in time existence and uniqueness of the solution using the conservation of the total fluid-structure energy.