An antisymmetric solution of the 3D incompressible Navier-Stokes equations

Abstract

We report results on the behavior of a particular incompressible Navier-Stokes (NS) flow in the whole space $\R^{3}$, related to the complex singular solutions introduced by Li and Sinai in \cite{LiSi08} that blow up at a finite time. The flow exhibits for some time a tornado-like behavior: a sharp increase of vorticity and of the maximal velocity, with concentration in an annular region around an axis. There is however no rotation around the axis and the approximate axial symmetry with no swirl excludes a real blow-up. We conclude with a discussion on the possible candidates for a real blow-up.