Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. (English) Zbl 0837.35113

Constantin, Peter; Fefferman, Charles

Indiana Univ. Math. J. 42, No. 3, 775-789 (1993).

The authors study the problem of global regularity for the three- dimensional incompressible Navier-Stokes equation. As is well-known, the existence of smooth solutions of the Cauchy problem for all time given smooth, localized divergence-free initial data, is still open. The authors argue that, unlike the two-dimensional case, in which the problem of global regularity has an affirmative answer, the developing time singularities may be due to an existing nonlinear stretching mechanism for the vorticity magnitude. The main point in favour of this idea is the proof that if the direction of vorticity is sufficiently well behaved in regions of high vorticity magnitude, then the solution is smooth.

Reviewer: F.Rosso (Firenze)

MSC:

35Q30	Navier-Stokes equations
35B65	Smoothness and regularity of solutions to PDEs
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

global regularity; existence of smooth solutions; Cauchy problem; vorticity