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∫0T‖‖ui(τ)‖Lip‖Lj,kαβdτ<∞,2β+2α+1p⩽34+12α,(p,α)∈Ϝ1, or∫0T‖‖ui(τ)‖Ljp‖Li,kαβdτ<∞,2β+2α+1p⩽34+14α+14p,(p,α)∈Ϝ2,fori≠j. This allows us to obtain regularity criterion of Leray–Hopf weak solutions via only one element Λiγuj with γ∈[0,1]γ∈[0,1] and i,j∈{1,2,3}i,j∈{1,2,3}, that is∫0T‖Λiγuj(τ)‖Lαβdτ<∞,(α,β)∈Ϝ. Here Ϝ1Ϝ1, Ϝ2Ϝ2 and Ϝ   are the sets of indexes (α,β)(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][0,T] if∫0T‖uj(τ)‖BMO83dτ<∞,j∈{1,2,3}, or∫0T‖Λiγuj(τ)‖B˙∞,2083−2γdτ<∞,γ∈[0,1],fori,j∈{1,2,3}, which fill the gap of endpoint α=∞α=∞.