Almost global well-posedness of Ericksen-Leslie's hyperbolic liquid crystal model for small data in two dimensions

This article is devoted to the simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model in two spatial dimensions, which is a nonlinear coupling of incompressible Navier-Stokes equations with wave map to circle $S^1$. The structure of second-order material derivative is exploited in order to close the energy estimates. We established the almost global well-posedness for small and smooth initial data near the constant equilibrium. Our proof relies on the idea of vector-field method and ghost weight method.