Well-posedness and the zero activity limit for the active nematic liquid crystal model

In this paper, we establish the existence of global-in-time smooth solution for the hydrodynamics of incompressible active nematic liquid crystal model in the torus Td with d = 2, 3. Furthermore, for the dimensionless system obtained by choosing an appropriate time scale, we prove rigorously the limit of the active liquid crystal model is the corresponding passive model as the activity parameter epsilon -> 0, which is exactly consistent with the relevant physical conclusion.