Dynamics of Partial Differential Equations

Volume 11 (2014)

Number 1

Partial regularity of solutions to the four-dimensional Navier-Stokes equations

Pages: 53 – 69

DOI: https://dx.doi.org/10.4310/DPDE.2014.v11.n1.a3

Authors

Hongjie Dong (Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Xumin Gu (School of Mathematical Sciences, Fudan University, Shanghai, China)

Abstract

In this paper, we consider suitable weak solutions of incompressible Navier-Stokes equations in four spatial dimensions. We obtain two $\epsilon$ regularity criteria in terms of certain scale-invariant quantities. As a consequence, we show that the two-dimensional space-time Hausdorff measure of the set of singular points is equal to zero.

Keywords

Navier-Stokes equations, partial regularity, Hausdorff’s dimension

2010 Mathematics Subject Classification

35Q30, 76D03, 76D05

Published 8 April 2014