Local regularity criteria of the 3D Navier–Stokes and related equations

In this paper, we are concerned with the local regularity of suitable weak solutions to the 3D Navier–Stokes equations and its generalized case with fractional power of the Laplacian ( − Δ ) α ( 3 / 4 ≤ α 1 ). In the first part, we derive some ϵ -regularity criteria in terms of the deformation tensor D ( u ) for the classical Navier–Stokes equations. In the second part, for the fractional case, we obtain some regularity conditions for suitable weak solutions including the velocity u , the gradient of the velocity ∇ u , the rotation tensor Ω ( u ) and the deformation tensor D ( u ) . This generalizes the results obtained by Gustafson et al. (2007).