Local kinetic energy and singularities of the incompressible Navier-Stokes equations

We study the partial regularity problem of the incompressible Navier-Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to L∞(0,T;L3,w(R3)) where L3,w(R3) denotes the standard weak Lebesgue space.