Computing a Class of Blow-up Solutions for the Navier-Stokes Equations

The three-dimensional incompressible Navier-Stokes equations play a fundamental role in a large number of applications to fluid motions, and a large amount of theoretical and experimental studies were devoted to it. Our work is in the context of the Global Regularity Problem, i.e., whether smooth solutions in the whole space R3 can become singular (“blow-up”) in a finite time. The problem is still open and also has practical importance, as the singular solutions would describe new phenomena. Our work is mainly inspired by a paper of Li and Sinai, who proved the existence of a blow-up for a class of smooth complex initial data. We present a study by computer simulations of a larger class of complex solutions and also of a related class of real solutions, which is a natural candidate for evidence of a blow-up. The numerical results show interesting features of the solutions near the blow-up time. They also show some remarkable properties for the real flows, such as a sharp increase of the total enstrophy and a concentration of high values of velocities and vorticity in small regions.