On the regularity criteria for liquid crystal flows involving the gradient of one velocity component

In this paper, we show the regularity criteria for three-dimensional nematic liquid crystal flows. More precisely, we prove that the strong solution (u, d) can be extended beyond T, provided ∇u3 ∈ Ls(0, T; Lq(R3)), ∇hd ∈ Lα(0, T; Lp(R3)), where s, q, α, p satisfy 2s+3q≤158+18q,2α+3p≤34+12p if q∈[2,3],p∈(103,∞] or 2s+3q≤74+12q,2α+3p≤34+12p if q∈3,∞,p∈103,∞.