浅水中轴对称浮式结构物的恢复平衡问题
On the return to equilibrium problem for axisymmetric floating structures in shallow water
摘要：
在本文中，我们解决返回到平衡问题的轴对称浮式结构在浅水中。首先，我们表明，固体运动的方程可以减少到一个延迟微分方程涉及的扩展跟踪运营商的作用是描述固体运动的流体方程的影响。结果表明，返回到平衡位形的初始数据的相容性条件不满足，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，我们线性化的方程在外部域中，我们保持非线性方程在内部域中。对于这样的配置，可以显式地计算扩展跟踪算子，并且可以将微分方程中的延迟项置于卷积形式中。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Cummins方程。利用流体-结构总能量守恒定理证明了该问题解的时间上的全局存在性和唯一性。
英文摘要：
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.
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