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

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 $\pi/2$, 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.