Global solution of 2D hyperbolic liquid crystal system for small initial data

We prove the global stability of small perturbation near the constant equilibrium for the two dimensional simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model, where the direction function of liquid crystal molecules satisfies a wave map equation with an acoustical metric. This improves the almost global existence result by Huang-Jiang-Zhao. As byproducts, we obtain the sharp (same as the linear solution) decay estimates for both the heat part and the wave part. Moreover the nonlinear wave part scatters to a linear solution as time goes to infinity. This paper's main contribution is the discovery of a novel null structure within the velocity equation's wave-type quadratic self-interaction. This structure compensates the insufficient decay rate in 2D, which previously hindered the establishment of global regularity for small data.