Regularity Of Flow Solutions In Three-dimensional Nematic Liquid Crystals

In this paper,we study the regularity criteria of the solutions for threedimensional incompressible nematic liquid crystal flows in anisotropic Lebesgue space.We discuss it in two parts:in the first part,we give the regularity criteria for the conditions of velocity field u and directional field gradient ?d,in the second part,we give the regularity criteria for the conditions of velocity field gradient ?u and directional field gradient ?d.Specifically speaking:1.Suppose(u0,d0)?H1(R3)× H2(R3).Let(u,d)? R3 ×(0,T)be a weak solution of three-dimensional incompressible nematic liquid crystal flows,with T<?.If(u,d)satifies(?) or(u,d)satisfies(?) then(u,d)can extend to T smoothly.Here,M is a positive constant,p=(p1,p2,p3),q=(q1,q,q3)?(2,?)3,1/p=1/p1+1/p2+1/p3,1/q=1/q1+1/q2+1/q3.2.Suppose(u0,d0)? H1(R3)× H2(R3).Let(u,d)? R3 ×(0,T)be a weak solution of three-dimensional incompressible nematic liquid crystal flows,with T<?.If(u,d)satisfies(?) then(u,d)can extend to T smoothly.Here,p,q ?(2,?)3,M is a positive constant.