Existence of knotted vortex structures in stationary solutions of the Euler equations

Abstract

In this paper, we review recent research on certain geometric aspects of the vortex lines of stationary ideal fluids. We mainly focus on the study of knotted and linked vortex lines and vortex tubes, which is a topic that can be traced back to Lord Kelvin and was popularized by the works of Arnold and Moffatt on topological hydrodynamics in the 1960s. In this context, we provide a leisurely introduction to some recent results concerning the existence of stationary solutions to the Euler equations in Euclidean space with a prescribed set of vortex lines and vortex tubes of arbitrarily complicated topology. The content of this paper overlaps substantially with the one the authors published in the Newsletter of the European Mathematical Society in June 2015.