The F. John model and Cummins' equations for freely floating objects

TL;DR

This work provides the first rigorous well-posedness theory for Fritz John’s linear floating-body problem in 2D and justifies Cummins’ memory-inclusive equations in general geometries with topography. The authors recast the problem in a Hamiltonian, semi-Hilbert framework using the Dirichlet-Neumann operator and Kirchhoff potentials to expose added-mass effects and nonlocal couplings, proving energy conservation and well-posedness in an energy space ${\mathbb X}$. They then develop an abstract higher-regularity theory via a hierarchy ${\mathbb X}^n$, linking regularity to the elliptic properties of the corner domain and the regularity of Kirchhoff potentials, with sharp results for small and right-angle corners. In particular, higher regularity holds when the contact angles are small (or equal to $\pi/2$ under compatibility) and certain components of the Kirchhoff potentials are sufficiently regular; conversely, horizontal motions can introduce intrinsic singularities that cap regularity. The paper also derives and analyzes the Cummins equations from John’s model, proving their well-posedness and detailing how corner geometry and added-mass effects influence the dynamics of the three degrees of freedom.

Abstract

In this paper, we address the well-posedness theory of F. John's problem for freely floating objects in a two-dimensional framework. This problem is a linear description of the interactions between an incompressible, irrotational free-surface fluid and a partially immersed solid object. It is related to the Cummins equations, which are a set of coupled integro-differential equations widely used by naval engineers. Our results provide a rigorous justification of this model and the first proof of its well-posedness. To this end, we show that F. John's problem has a Hamiltonian structure, albeit with a non-definite Hamiltonian which compels us to work in a semi-Hilbertian framework. The presence of corners in the fluid domain also induces a lack of elliptic regularity for the boundary value problem satisfied by the velocity potential. This is why the solution to F. John's problem has in general a limited regularity, even with very smooth initial data. Correspondingly, the solution to Cummins' equation is in general no better than $C^3$. We demonstrate higher regularity when the contact angles are small or equal to $π/2$, provided that some compatibility conditions, propagated by the flow, are satisfied. This allows us to exhibit qualitative differences between the three degrees of freedom of the object. For instance, the motion of an object allowed to move only vertically can be $C^\infty$, while it cannot be better than $C^3$ if it is only allowed to translate horizontally.

Figures (2)

• Figure 1: The fluid domain $\Omega$ is the shaded light blue color with lower boundary $\Gamma^{\rm b}$ in dark brown. Its upper boundary is comprised of; $\Gamma^{\rm w}$ the wetted part of the object in solid black and $\Gamma^{\rm D}$ which the union of the solid blue lines $\mathcal{E}_-$ and $\mathcal{E}_+$.
• Figure 2: Admissible configurations

Theorems & Definitions (93)

• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 1.6
• Proposition 1.7
• Theorem 1.8
• Theorem 1.9
• Definition 2.1
• Remark 2.2
• Definition 2.3