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

Abstract

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.