Vortex reconnections in classical and quantum fluids

Abstract

We review recent rigorous results on the phenomenon of vortex reconnection in classical and quantum fluids. In the context of the Navier¿Stokes equations in T3 we show the existence of global smooth solutions that exhibit creation and destruction of vortex lines of arbitrarily complicated topologies. Concerning quantum fluids, we prove that for any initial and final configurations of quantum vortices, and any way of transforming one into the other, there is an initial condition whose associated solution to the Gross¿Pitaevskii equation realizes this specific vortex reconnection scenario. Key to prove these results is an inverse localization principle for Beltrami fields and a global approximation theorem for the linear Schrödinger equation.