On Well-Posedness of Ericksen--Leslie's Hyperbolic Incompressible Liquid Crystal Model

We study Ericksen--Leslie's hyperbolic incompressible liquid crystal model. Under some constraints on the Leslie coefficients which ensure the basic energy law is dissipative, we prove the local-in-time existence and uniqueness of the classical solution to the system with finite initial energy. Furthermore, with an additional assumption on the coefficients which provides a damping effect, and the smallness of the initial energy, the unique global classical solution can be established.