An Antisymmetric Solution of the 3D Incompressible Navier–Stokes Equations with “Tornado-Like” Behavior

Abstract

In the framework of the Global Regularity Problem for the incompressible Navier–Stokes (NS) equations in the whole space ${{\mathbb{R}}^{3}}$, Li and Sinai in [13] proved that there are smooth complex solutions that become singular (“blow-up”). We discuss the possible extension of the Li-Sinai approach to real solutions and report results obtained by computer simulations on the behavior of a particular solution related to the complex blow-up. Although a blow-up is excluded by axial symmetry, the solution is a good model of a “tornado-like” behavior, with a sharp increase of speed and vorticity concentrated in an annular region around the symmetry axis. We conclude with a discussion on the search of possible candidates for a real blow-up in the framework of the Li-Sinai approach.