On the regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations

In this paper, we consider the regularity criterion of axisymmetric weak solutions to the Navier-Stokes equations in R-3. Let u be an axisymmetric weak solution in R-3 x (0, T), w = curl u, and w(theta) be the azimuthal component of w in the cylindrical coordinates. It is proved that u becomes a regular solution if w(theta) is an element of L2/2-s (0, T; (M) over dot(2,3/s)), where (M) over dot(2,3/s) is the critical Morrey-Campanato space. (C) 2010 Elsevier Ltd. All rights reserved.