Global solutions to the 3D incompressible nematic liquid crystal system

In this paper, we consider the global well-posedness of the Cauchy problem of the 3D incompressible 3 nematic liquid crystal system with initial data in the critical Besov space B-1/2 (2,1)(R-3) x B-3/2 (2,1) (R-3). In particular, we prove that there exists a positive constant C-0 such that the nematic liquid crystal system has a unique global solution with initial data (u(0), d(0)) = (u(0)(h), u(0)(3) d(0)) which satisfies ((1 + 1/v mu)parallel to d(o) - (d) over bar (0) parallel to B-3/2 (2,1) + 1/v parallel to u(0)(h)parallel to B-1/2 (2,1) ) exp{c(0)/v(2) (parallel to u(0)(3)parallel to B-1/2 (2,1) + 1/mu)(2)}<= c(0) for some c(0) sufficiently small, where (d) over bar (0) is a constant vector with vertical bar(d) over bar (0)vertical bar = 1. Here v and mu are two positive viscosity constants. (C) 2014 Elsevier Inc. All rights reserved.