Navier–Stokes Equation and its Fractional Approximations

Abstract

We consider the Navier–Stokes equation (N-S) in dimensions two and three as limits of the fractional approximations. In 2-D the N-S problem is critical with respect to the standard L2 a priori estimates and we consider its regular approximations with the fractional power operator (- PΔ) 1+α, α> 0 small, where P is the projector on the space of divergence-free functions. In 3-D different properties of the N-S problem with respect to the standard L2 a priori estimate are obtained and the 3-D regular approximating problem involves fractional power operator (- PΔ) s with s>54. Using Dan Henry’s semigroup approach and the Giga-Miyakawa estimates we construct regular solutions to such approximations. The solutions are global in time, 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 such regular solutions of the 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.