Report
Teaser, summary, work performed and final results
Periodic Reporting for period 2 - 3DWATERWAVES (Mathematical aspects of three-dimensional water waves with vorticity)
Teaser
The goal of this project is to develop a mathematical theory for steady three-dimensional water waves with vorticity. This is important for modelling the interaction of surface waves with non-uniform currents. Such currents can for example be caused by wind or bottom friction...
Summary
The goal of this project is to develop a mathematical theory for steady three-dimensional water waves with vorticity. This is important for modelling the interaction of surface waves with non-uniform currents. Such currents can for example be caused by wind or bottom friction and can have a profound effect on the surface waves.
The research is based on the Euler equations - a set of nonlinear partial differential equations which go back to the 18th century but are notorious for their complexity. Three-dimensional waves are waves whose surfaces vary in all horizontal directions. They may for example be doubly periodic, that is, periodic in two distinct directions - a situation which frequently occurs when waves travelling in different directions meet. In the two-dimensional case (that is, when the surface and fluid flow is independent of one horizontal direction), 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, which is a nice simplification of the Euler equations only available in the two-dimensional setting. An analogue of this formulation is not available in three dimensions, and the theory has therefore so far been restricted to irrotational flow. Irrotational flow can be said to describe what happens when surface waves travel on still water or currents that are uniform in the vertical direction. In this project we seek to go beyond this restriction. In order to do so, we will both investigate particular situations when the Euler equations simplify (such as Beltrami flows) and try to work directly with the Euler equations. In the latter approach we will need to overcome difficulties involving infinitely many resonances. We will also consider new methods for studying flows with vorticity in fixed geometries as a natural part of the project.
Work performed
In a joint work with B. Buffoni (EPFL), we have constructed steady three-dimensional rotational ideal flows near a given parallel flow by using a formulation based on two stream functions. Instead of prescribing the relationship between the vorticity and the stream function (as in the two-dimensional setting), we prescribe the relationship between the Bernoulli function and the stream functions. In addition, the values of the stream functions are given on the inflow and outflow sets of the domain. Although this formulation has been known before, this is the first time it is used in a general existence theory. The paper has been published in the journal Analysis & PDE, but is also available in postprint form: https://arxiv.org/abs/1709.05957.
In a paper by postdoc E. Lokharu and the PI we have derived a variational principle for doubly periodic three-dimensional travelling water waves over Beltrami flows. Variational principles have previously played a major role in various existence theories for irrotational water waves in two and three dimensions as well as for rotational water waves in two dimensions, and it can therefore be expected that our new variational principle will be useful for constructing three-dimensional water waves with vorticity in the future. In particular, we hope that an extension of the principle will be useful for constructing fully localised solitary waves, that is waves which are asymptotically flat in all horizontal directions. The paper has been published in the journal Nonlinear Analysis (open access), but is also available in preprint form: https://arxiv.org/abs/1804.04172.
In addition, progress has been made on an existence theory for small-amplitude doubly periodic three-dimensional travelling waters over Beltrami flows. These waves bifurcate from flows with a flat surface in which the velocity is constant at each depth but the direction of the velocity field is depth dependent. The theory is based on a multi-parameter bifurcation approach. A preprint by postdoc E. Lokharu, PhD student D. Svensson Seth and the PI is soon to appear.
A Workshop on Fluid Dynamics and Dispersive Equations was organised as a part of the project at Lund University, June 25-29, 2018. It gathered many renowned experts on mathematical aspects of fluid dynamics and waves, including several ERC grantees. The 25 talks delivered during the workshop included topics ranging from the shoreline problem for the nonlinear shallow water equations to a mathematical description of the phenomenon of vortex reconnection. In addition, thanks to the support of the ERC, a number of junior researchers participated and contributed with poster presentations.
Final results
Our results on water waves over Beltrami flows are the first existence results for three-dimensional travelling water waves with non-zero vorticity of any kind. This opens up a new avenue of research.
Moreover, our construction of steady rotational flows using two stream functions is the first time this approach has been used in a general, rigorous existence theory. An important challenge will be to generalise this approach to different geometries, including domains with free boundaries (so as to describe waves).