Localization of Beltrami fields: global smooth solutions and vortex reconnection for the Navier-Stokes equations

We introduce a class of divergence-free vector fields on $\mathbb{R}^3$ obtained after a suitable localization of Beltrami fields. First, we use them as initial data to construct unique global smooth solutions of the three dimensional Navier-Stokes equations. The relevant fact here is that these initial data can be chosen to be large in any critical space for the Navier-Stokes problem, however they satisfy the nonlinear smallness assumption introduced in [10]. As a further application of the method, we use these vector fields to provide analytical example of vortex-reconnection for the three-dimensional Navier-Stokes equations on $\mathbb{R}^3$. To do so, we exploit the ideas developed in [14] but differently from this latter we cannot rely on the non-trivial homotopy of the three-dimensional torus. To overcome this obstacle we use a different topological invariant, i.e. the number of hyperbolic critical points of the vector field.