Partial regularity of Solutions of Navier-Stokes equations

TL;DR

This paper investigates the size of the singular set in 3D Navier-Stokes solutions, showing it has Hausdorff dimension less than 1 under certain integrability conditions using backward uniqueness techniques.

Contribution

It establishes a new partial regularity result for Navier-Stokes solutions by bounding the Hausdorff dimension of the singular set under specific integrability criteria.

Findings

Hausdorff dimension of singular set is less than 1

Under certain integrability conditions, singularities are highly restricted

Backward uniqueness is used to derive regularity results

Abstract

In this paper, we study the singular set of 3-dimensional Navier-Stokes equations. Under the condition$\frac{1}{R^{\frac{3s}{q}+2-s}}\int^{R^{2}}_{0}(\int_{B_{R}}|u|^{q}dx)^{\frac{s}{q}}ds <C,$ for $(q,s)\in\{(2,5),(5,2)\},$ we use the backward uniqueness of parabolic equations to show that the Hausdorff dimension of the singular set is less than 1.