New look at the Navier-Stokes equation

Abstract

We propose a new way of looking at the Navier-Stokes equation (N-S) in dimensions two and three. In 2-D that problem is critical with respect to the standard L 2 a priori estimates. We consider its regular approximations in which the −P ∆ operator is replaced with the fractional power (−P ∆) 1+α , α > 0 small, where P is the projector on the space of divergence-free functions. The 3-D N-S equation is super-critical with respect to the standard L 2 a priori estimates; the regular approximating problem in 3-D should contain fractional power (−P ∆) s with s > 5 4 . Using Dan Henry's semigroup approach we construct regular solutions to such approximations. The solutions are unique, smooth and regularized through the equation in time. Solution to 2-D and 3-D N-S equations are obtained next as a limit of the regular solutions of the above approximations. Moreover, since the nonlinearity of the N-S equation is of quadratic type, the solutions corresponding to small initial data and small f are shown to be global in time and regular.