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 nematic liquid crystal system with initial data in the critical Besov space B2,112(R3)×B2,132(R3). In particular, we prove that there exists a positive constant C0C0 such that the nematic liquid crystal system has a unique global solution with initial data (u0,d0)=(u0h,u03,d0) which satisfies((1+1νμ)‖d0−d¯0‖B2,132+1ν‖u0h‖B2,112)exp⁡{C0ν2(‖u03‖B2,112+1μ)2}≤c0, for some c0c0 sufficiently small, where d¯0 is a constant vector with |d¯0|=1. Here ν and μ are two positive viscosity constants.