WELL-POSEDNESS FOR THE 3D INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL SYSTEM IN THE CRITICAL LP FRAMEWORK

In this paper, we consider the well-posedness of the Cauchy problem of the 3D incompressible nematic liquid crystal system with initial data in the critical Besov space <(B)over dot>(3/p-1)(p,1) (R-3) x <(B)over dot>(3/q)(q,1) (R-3) with 1 < p < infinity, 1 <= q < infinity and -min{1/3, 1/2p} <= 1/q - 1/p <= 1/3. In particular, if we impose the restrictive condition 1 < p < 6, we prove that there exist two positive constants C-0 and 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 [Graphics] where (d) over bar (0) is a constant vector with vertical bar(d) over bar (0)vertical bar = 1. Here nu and mu are two positive viscosity constants.