A regularity criterion for the solution of nematic liquid crystal flows in terms of the -norm

In this paper, we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions ( n = 3 ) and in two dimensions ( n = 2 ). We prove that the solution ( u , d ) is smooth up to time T provided that there exists a positive constant ε 0 > 0 such that (i) for n = 3 , ‖ ( u , ∇ d ) ‖ L ∞ ( 0 , T ; B ̇ ∞ , ∞ − 1 ) ≤ ε 0 , and (ii) for n = 2 , ‖ ∇ d ‖ L ∞ ( 0 , T ; B ̇ ∞ , ∞ − 1 ) ≤ ε 0 .