Almost global well-posedness of 2-D Ericksen-Leslie's hyperbolic liquid crystal model for small data

This article is devoted to the two dimensional simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model, where the direction $d$ of liquid crystal molecules satisfies a wave map equation with an acoustical metric. 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. There are two key ingredients in our proof: (i) Inspired by the gauge theory in Tataru \cite{Tataru,Tataru05}, we reformulate the wave map equation into a free wave equation with acoustical metric, where the nonlinearity is annihilated due to the geometry of $\mathbb S^1$; (ii) Motivated by the ghost weight method in Alinhac \cite{A01}, we introduce a new and important ``good unknown", the velocity $u$, which provides the additional dissipation $u/\langle t-r\rangle\in L^2_tL^2_x$. These new observations turn out to be extremely crucial in resolving the system in low dimensions.