Global strong solution to the three-dimensional liquid crystal flows of Q-tensor model

Abstract

A complex hydrodynamic system that models the fluid of nematic liquid crystals in a bounded domain in R3 is studied. The system is a forced incompressible Navier–Stokes equation coupled with a parabolic type equation of Q-tensors. We invoke the maximal regularity of the Stokes operators and parabolic operators in Besov spaces to obtain the local strong solution if the initial Q-tensor is not too “wild”. In addition, it is showed that such solution can be extended to a global one if the initial data is a sufficiently small perturbation around the trivial equilibrium state. Finally, it is proved that the global strong solution obtained here is identical to those weak solutions obtained in Paicu and Zarnescu [26].