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.

中文翻译：

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浅水轴对称浮体回平衡问题

在本文中，我们解决了浅水中轴对称浮动结构的平衡恢复问题。首先，我们证明了固体运动方程可以简化为一个延迟微分方程，其中包含一个扩展迹算子，其作用是描述流体方程对固体运动的影响。事实证明，返回平衡配置的初始数据的兼容性条件不满足，因此我们不能将[3]的结果用于非线性问题。因此，假设小振幅波，我们将外部域中的方程线性化，而将非线性方程保留在内部域中。对于这种配置，可以显式计算扩展迹算子，并且可以将微分方程中的延迟项放入卷积形式。因此，固体运动由非线性二阶积分微分方程控制，其线性化是众所周知的康明斯方程。我们使用总流固能量守恒来显示解的全局时间存在性和唯一性。