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 & nabla;u(3)is an element of L-s(0, T; L-q(R-3)), & nabla;(h)d is an element of L-alpha(0, T;L-p(R-3)), where s, q, alpha, p satisfy 2/s+3/q <= 15/8+1/8q, 2/alpha +3/p <= 3/4+1/2p if q is an element of [2, 3], p is an element of(10/3, infinity] or 2/s+3/q <= 7/4+1/2q, 2/alpha+ 3/p <= 3/4+1/2p if q is an element of (3, infinity], p is an element of(10/3, infinity]. Published under an exclusive license by AIP Publishing.

Automated summary

In this paper, the regularity criteria for three-dimensional nematic liquid crystal flows are shown. It is proved that the strong solution can be extended under certain conditions. This result is of great significance for the study of liquid crystal dynamics.