New look at the Navier-Stokes equation

TL;DR

This paper introduces a novel perspective on the Navier-Stokes equations by employing fractional power operators in regular approximations, enabling the construction of smooth solutions in both two and three dimensions.

Contribution

It proposes a new regularization approach using fractional powers of the Laplacian, facilitating the existence of smooth solutions for Navier-Stokes equations.

Findings

Constructed regular solutions in 2D and 3D using fractional operators.

Proved uniqueness and smoothness of solutions.

Established limits of regular solutions as approximations to original equations.

Abstract

We propose a new way of looking at the Navier-Stokes equation (N-S) in dimensions two and three. We consider its regular approximations in which the -P Delta operator is replaced with the fractional power. The 3-D N-S equation is super-critical with respect to the standard L2 a priori estimates; the regular approximating problem in 3-D should contain fractional power 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.