Beltrami fields and knotted vortex structures in incompressible fluid flows

Abstract

AbstractThis paper gives a survey on recent results about the existence of knotted vortex structures in incompressible fluids. This includes the proof of Lord Kelvin's conjecture on the existence of knotted vortex tubes in steady Euler flows and a new probabilistic approach to address Arnold's speculation that typical Beltrami fields should exhibit vortex lines of arbitrary topological complexity. We review the key tools to establish these results: the global approximation properties of Beltrami fields, the inverse localization principle in spectral theory, and the theory of Gaussian random Beltrami fields, which permits the introduction of probabilistic considerations in the picture. Finally, we include some applications of these techniques to other problems, such as the existence of vortex reconnections for smooth solutions of the 3D Navier–Stokes equation and the evolution of local hot spots under the heat equation.