3-D incompressible Navier-Stokes equations: Complex blow-up and related real flows

In the framework of the Global Regularity Problem for the Navier-Stokes (NS) Equations in R3 Li and Sinai proved the existence of singular complex solutions (“blow-up”). We give an outline of their approach and discuss the perspectives of its extension to real solutions. We also illustrate, with the help of computer simulations, the behaviour of a real solution related to the complex blow-up. It does not blow up, due to its approximate axial symmetry, but it shows a remarkable tornado-like behaviour, with a rapid concentration and increase of speed and vorticity.