On the blow-up of some complex solutions of the 3D Navier–Stokes equations: theoretical predictions and computer simulations

We consider some complex-valued solutions of the Navier–Stokes equations in |${\mathbb R}^{3}$| for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of computer simulations, which give a detailed picture of the blow-up for both types. They reveal in particular important features not, as yet, predicted by the theory, such as a concentration of the energy and the enstrophy around a few singular points, while elsewhere the 'fluid' remains quiet.