A variational principle for three-dimensional water waves over Beltrami flows

  title={A variational principle for three-dimensional water waves over Beltrami flows},
  author={Evgeniy Lokharu and Erik Wahl'en},
  journal={Nonlinear Analysis},

An Existence Theory for Small-Amplitude Doubly Periodic Water Waves with Vorticity

We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the

Traveling water waves — the ebb and flow of two centuries

This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of

A variational formulation for steady surface water waves on a Beltrami flow

This paper considers steady surface waves ‘riding’ a Beltrami flow (a three-dimensional flow with parallel velocity and vorticity fields). It is demonstrated that the hydrodynamic problem can be



Variational formulations for steady water waves with vorticity

For free-surface water flows with a vorticity that is monotone with depth, we show that any critical point of a functional representing the total energy of the flow adjusted with a measure of the

A Hamiltonian Formulation of Water Waves with Constant Vorticity

We show that the governing equations for two-dimensional water waves with constant vorticity can be formulated as a canonical Hamiltonian system, in which one of the canonical variables is the

A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves

This article presents a rigorous existence theory for small-amplitude threedimensional travelling water waves using a centre-manifold reduction technique by which the problem is reduced to a locally equivalent Hamiltonian system with a finite number of degrees of freedom.

Non-existence of three-dimensional travelling water waves with constant non-zero vorticity

  • E. Wahlén
  • Mathematics
    Journal of Fluid Mechanics
  • 2014
Abstract We prove that there are no three-dimensional bounded travelling gravity waves with constant non-zero vorticity on water of finite depth. The result also holds for gravity–capillary waves

Three-dimensional overturned traveling water waves

Traveling Two and Three Dimensional Capillary Gravity Water Waves

The main results of this paper are existence theorems for traveling gravity and cap- illary gravity water waves in two dimensions, and capillary gravity water waves in three dimensions, for any

A variational principle for a fluid with a free surface

The full set of equations of motion for the classical water wave problem in Eulerian co-ordinates is obtained from a Lagrangian function which equals the pressure. This Lagrangian is compared with

An exact solution for equatorially trapped waves

[1] We present an exact solution of the nonlinear governing equations for geophysical water waves in the β-plane approximation near the Equator. The solution describes in the Lagrangian framework

On three-dimensional Gerstner-like equatorial water waves

  • D. Henry
  • Geology
    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2017
A number of exact Gerstner-like solutions which have been derived to model various geophysical oceanic waves, and wave–current interactions, in the equatorial region are surveyed.

Existence of solutions for three dimensional stationary incompressible Euler equations with nonvanishing vorticity

In this paper, solutions with nonvanishing vorticity are established for the three dimensional stationary incompressible Euler equations on simply connected bounded three dimensional domains with