Some regularity criteria via the signed curl operators for the 3D Navier-Stokes equations

By making full use of the Fourier localization technique and the Littlewood-Paley decomposition, we establish refined estimates for the product operates under different frameworks. This together with a decomposition of the curl operator enables us to obtain three regularity criteria of the weak solution for the 3D incompressible Navier-Stokes equations involving the signed curl operators. To our knowledge, this is the first instance of exhibiting regularity criteria in terms of 'partial vorticity'. Our results also improve some known regularity criteria for the Navier-Stokes equations.