On well-posedness of Ericksen-Leslie's paraboloc-hyperbolic liquid crystal model

We establish the following well-posedness results on Ericksen-Leslie's parabolic-hyperbolic liquid crystal model: 1, if the dissipation coefficients \beta = \mu_4 - 4 \mu_6 > 0, and the size of the initial energy E^{in} is small enough, then the life span of the solution is at least -O(\ln E^{in}); 2, for the special case that the coefficients \mu_1 = \mu_2 = \mu_3 = \mu_5 = \mu_6 = 0, for which the model is the Navier-Stokes equations coupled with the wave map from \mathbb{R}^n to \mathbb{S}^2, the same existence result holds but without the smallness restriction on the size of the initial data; 3, with further constraints on the coefficients, namely \alpha = \mu_4 - 4 \mu_6 - \tfrac{ (|\lambda_1| - 7 \lambda_2)^2 }{\eta} - \tfrac{ 2 ( 7 |\lambda_1| - 2\lambda_2 )^2 }{ |\lambda_1| } > 0 and \mu_2 < \mu_3, the global classical solution with small initial data can be established. A relation between the Lagrangian multiplier and the geometric constraint |d|=1 plays a key role in the proof.