Some explicit solutions of the three-dimensional Euler equations with a free surface (Q2089724)

There are two fundamental ways of describing the motion of an incompressible, inviscid fluid, viz. Lagrangian \& Eulerian methods. While the first one describes the evolution of the fluid through particle trajectories in the space of measure-preserving diffeomorphisms, the latter deals with the velocity field, pressure function and the shape of the free surface at any time and space. The author have produced a family of radial solutions in the Eulerian coordinates to the three dimensional Euler equations in a fluid domain with a free surface and having finite depth. The flows described by these solutions display a density that depends on depth, which in turn gives the velocity field a non-trivial vertical structure. Some of the solutions presented in this article are not only explicit in terms of the velocity field and pressure function, but also with regard to the free surface. This happens only when the density function has a particular form. From the point of view of application in oceanography, these solutions accommodate a variable pressure on the free surface which makes it relevant to model turbulence in the ocean. To obtain other (quasi) solutions to the Euler equation, the short-wavelength perturbation method for the basic flow solutions has been used. More precisely, perturbations of the velocity field and pressure function along the streamlines of the basic flow have been introduced in the form of Wentzel-Kramers-Brillouin (WKB) ansatz. The components of the amplitude vector of the perturbation satisfy a system of ODE, which is equivalent to Hill's equation. A quite involved analysis is used to show boundedness (in time) of solutions to Hill's equation. This proves that the amplitude of the perturbation is bounded, which is equivalent to the fact that the basic flow is stable.