A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component (Q2349920)

In the paper, the author investigates the regularity criterion of the three-dimensional Navier-Stokes equations via one velocity component. The strategy consists in establishing regularity criteria of Leray-Hopf weak solutions in the framework of anisotropic Lebesgue space. This extends and improves some known regularity criteria of Leray-Hopf weak solutions in term of one velocity component. By making full use of the Bony paraproduct decomposition, the author shows the conditions for Leray-Hopf weak solutions to be smooth.