Partial regularity of Solutions of Navier-Stokes equations

Abstract

In this paper, we study the singular set of 3-dimensional Navier-Stokes equations. Under the condition$\frac{1}{R^{\frac{3s}{q}+2-s}}{\int }_0^{R^2}({\int }_{B_R}|u{|}^{q}dx{)}^{\frac{s}{q}}ds<C,$ 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.