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On an Initial Value Problem Describing the Small Oscillations of a Floating Cylinder

Vicente Ocqueteau, Marius Tucsnak

TL;DR

The paper addresses the time-domain well-posedness of a coupled PDE-ODE model for small vertical oscillations of a floating infinite circular cylinder interacting with a one-dimensional free surface in infinite depth. It develops an explicit Dirichlet-to-Neumann map on the non-smooth, unbounded exterior domain and proves that this map defines a positive self-adjoint operator, enabling a $C^0$-group formulation of the dynamics. By embedding the problem in a semigroup framework and applying a bounded perturbation argument, the authors establish existence, uniqueness, and regularity results for the evolution, with enhanced smoothness under smoother data. This work provides a rigorous time-domain foundation for wave–structure interactions in a simplified geometric setting, bridging Dirichlet-to-Neumann map theory and semigroup methods in unbounded domains with corners.

Abstract

We study a coupled PDE-ODE system modeling the small oscillations of a floating cylinder interacting with small water waves. We consider the case when the floating is supposed to be an infinite circular cylinder, so that the equations of the free surface of the fluid can be written in one space dimension. The governing equations are formulated as an abstract evolution equation in a suitable Hilbert space, and we establish the well-posedness of the associated initial value problem. A key element of the proof is the analysis of a partial Dirichlet-to-Neumann map on an unbounded domain with a non-smooth boundary.

On an Initial Value Problem Describing the Small Oscillations of a Floating Cylinder

TL;DR

The paper addresses the time-domain well-posedness of a coupled PDE-ODE model for small vertical oscillations of a floating infinite circular cylinder interacting with a one-dimensional free surface in infinite depth. It develops an explicit Dirichlet-to-Neumann map on the non-smooth, unbounded exterior domain and proves that this map defines a positive self-adjoint operator, enabling a -group formulation of the dynamics. By embedding the problem in a semigroup framework and applying a bounded perturbation argument, the authors establish existence, uniqueness, and regularity results for the evolution, with enhanced smoothness under smoother data. This work provides a rigorous time-domain foundation for wave–structure interactions in a simplified geometric setting, bridging Dirichlet-to-Neumann map theory and semigroup methods in unbounded domains with corners.

Abstract

We study a coupled PDE-ODE system modeling the small oscillations of a floating cylinder interacting with small water waves. We consider the case when the floating is supposed to be an infinite circular cylinder, so that the equations of the free surface of the fluid can be written in one space dimension. The governing equations are formulated as an abstract evolution equation in a suitable Hilbert space, and we establish the well-posedness of the associated initial value problem. A key element of the proof is the analysis of a partial Dirichlet-to-Neumann map on an unbounded domain with a non-smooth boundary.
Paper Structure (8 sections, 13 theorems, 118 equations, 1 figure)

This paper contains 8 sections, 13 theorems, 118 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Some background on the Dirichlet problem and on the Dirichlet-to-Neumann map in the half plane
  4. A boundary value problem for the Laplacian
  5. A Dirichlet-to-Neumann map on $\Omega$
  6. Proof of the main result
  7. Data availability statement
  8. Conflict of interest

Key Result

Theorem 2.1

Let $f\in L_{{\rm loc}}^1((0,\infty))$, $h_0, h_1\in \mathbb{R}$, $v_0\in W^{\frac{1}{2},2}(\mathcal{E})$ and $v_1\in L^2(\mathcal{E})$. Then, the system reformulated_system with initial data $h(0)=h_0, \dot h(0)=h_1, v(0,x)=v_0(x)$ and $\frac{\partial v}{\partial t}(0,x)=v_1(x)$ has a unique soluti Moreover, if $v_0\in W^{1,2}(\mathcal{E})$ and $v_1\in W^{\frac{1}{2},2}(\mathcal{E})$ and $f$ is o

Figures (1)

  • Figure 1: Floating cylinder

Theorems & Definitions (30)

  • Theorem 2.1
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Proposition 4.1
  • proof
  • ...and 20 more