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

In this paper, we investigate the regularity criterion of the tridimensional Navier-Stokes equations via one velocity component. Our strategy is to establish the following version of regularity criterions of Leray-Hopf weak solutions in the framework of anisotropic Lebesgue space. This allows us to obtain regularity criterion of Leray-Hopf weak solutions via only one element Λ_i^γu_j with γ∈[0, 1] and i, j∈{1, 2, 3}, that is, Here {Digamma}₁, {Digamma}₂ and {Digamma} are the sets of indexes (α, β) which appear in our results and the fractional operator Λi:=-∂i2. This extends and improves some known regularity criterions of Leray-Hopf weak solutions in term of one velocity component, including the notable works of C. Cao and E.S. Titi [4]. More importantly, by making full use of the Bony paraproduct decomposition, we show that Leray-Hopf weak solutions are smooth on [0, T] if, which fill the gap of endpoint α=∞.