From water waves to fusion - mathematical analysis of steady ideal flows

The Euler equations are a system of nonlinear partial differential equations derived already in the 18th century, but whose properties are not yet fully understood. We will consider the stationary equations, which model the steady flow of ideal fluids such as water.

Remarkably, they also model a plasma equilibrium, with applications in fusion reactors where a plasma is confined by a magnetic field. Here one is particularly interested in toroidal domains. In the axisymmetric case the equations reduce to an elliptic PDE with a rich theory. In general the equations are elliptic-hyperbolic and rigorous mathematical theory is lacking.

In the project we will develop such a theory. In addition, we will consider domains with free boundaries, modelling steady water waves with vorticity. Here there is a well-developed theory in 2D where again the problem is elliptic. The 3D theory has until recently been limited to irrotational flow.

In order to model wave-current interactions it is important to allow for vorticity. Such a theory has only recently started to emerge and the goal of the project is to develop it to a mature stage. We will also consider unresolved problems in 2D. Tools from nonlinear analysis will be used, but new methods are needed since the problem is nonstandard.

Considering water waves and plasma equilibria in parallel will lead to synergy effects.

The research will be carried out by the PI, one PhD student and four postdocs together with international collaborators.