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

We study the partial regularity problem of the incompressible Navier--Stokes equations. In this paper, we show that a reverse H\"older inequality of velocity gradient with increasing support holds 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 $L^\infty(0,T;L^{3,w}(\mathbb{R}^3))$ where $L^{3,w}(\mathbb{R}^3)$ denotes the standard weak Lebesgue space.