# Mathematical aspects of three-dimensional water waves with vorticity (3DWATERWAVES)

## Abstract

The goal of this project is to develop a mathematical theory for steady three-dimensional water waves with vorticity. The mathematical model consists of the incompressible Euler equations with a free surface, and vorticity is important for modelling the interaction of surface waves with non-uniform currents. In the two-dimensional case, there has been a lot of progress on water waves with vorticity in the last decade. This progress has mainly been based on the stream function formulation, in which the problem is reformulated as a nonlinear elliptic free boundary problem. An analogue of this formulation is not available in three dimensions, and the theory has therefore so far been restricted to irrotational flow. In this project we seek to go beyond this restriction using two different approaches. In the first approach we will adapt methods which have been used to construct three-dimensional ideal flows with vorticity in domains with a fixed boundary to the free boundary context (for example Beltrami flows). In the second approach we will develop methods which are new even in the case of a fixed boundary, by performing a detailed study of the structure of the equations close to a given shear flow using ideas from infinite-dimensional bifurcation theory. This involves handling infinitely many resonances.

## Team

## News

#### April 28, 2022:

A preprint by D. Svensson Seth, K. Varholm and E. Wahlén on the bifurcation of small-amplitude symmetric doubly periodic water waves with small vorticity is now available (see below).

#### April 22, 2022:

A preprint by E. Wahlén and J. Weber on global bifurcation for pure gravity waves with possibly overhanging profiles and critical layers for flows with general vorticity is now available (see below).

#### February 4, 2022:

A preprint on global bifurcation for axisymmetric capillary waves with general vorticity and swirl by A. Erhardt, E. Wahlén and J. Weber is now available (see below).

#### September 20, 2021:

An overview paper on water waves based on the ONEPAS seminar has now been published on arXiv. One of the section gives an overview of three-dimensional steady waves with and without vorticity (see below).

#### September 14, 2021:

A preprint by E. Wahlén and J. Weber on global bifurcation of capillary-gravity waves with possibly overhanging profiles and critical layers for flows with general vorticity is noow available (see below).

#### June 22, 2020:

A preprint by Douglas Svensson Seth concerning steady three-dimensional ideal flows with vorticity in fixed geometries with edges is now available (see below).

#### June 8, 2020:

The paper "An existence theory for small-amplitude doubly periodic water waves with vorticity" has been accepted for publication in Archive for Rational Mechanics and Analysis. The preprint is available below.

#### August 7, 2019:

A preprint concerning doubly-periodic steady water waves over Beltrami flows is now available (see below). Comments are welcome!

#### November 27, 2018:

There is an open call for a postdoctoral position at Lund University within the project. Deadline December 31. Follow this link for more information.

#### April 11, 2018:

The **Lund Workshop on Fluid Dynamics and Dispersive Equations** will take place June 25-29, 2018. See this link for more information.

A preprint by E. Lokharu and E. Wahlén on a variational principle for three-dimensional steady water waves over Beltrami flows is now available (see below). Comments are welcome!

#### September 18, 2017:

A preprint concerning steady three-dimensional ideal flows with vorticity in fixed geometries is now available (see below). This is a joint work with Boris Buffoni. Our idea is to use a formulation of the steady incompressible Euler equations in terms of two "stream functions". There is a nice associated variational formulation, but the variational functional is not coercive and we are therefore forced to use a Nash-Moser approach. Comments are welcome!

## Publications

#### Symmetric doubly periodic gravity-capillary waves with small vorticity

#### Large-amplitude steady gravity water waves with general vorticity and critical layers

#### Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl

#### Traveling water waves -- the ebb and flow of two centuries

journal version (Quarterly of Applied Mathematics)

#### Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticity

#### On the two and one-half dimensional Vlasov-Poisson system with an external magnetic field: global well-posedness and stability of confined steady states

journal version (Nonlinear Analysis: Real World Applications, open access)

#### Steady three-dimensional ideal flows with nonvanishing vorticity in domains with edges

journal version (Journal of Differential Equations, open access)

#### An existence theory for small-amplitude doubly periodic water waves with vorticity

journal version (Archive for Rational Mechanics and Analysis, open access)

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

journal version (Nonlinear Analysis, open acces)

#### Steady three-dimensional rotational flows: an approach via two stream functions and Nash-Moser iteration

journal version (Analysis & PDE)

## Funding source

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 678698