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 R^3 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.