Global existence and incompressible limit in critical spaces for compressible flow of liquid crystals

TL;DR

This paper proves global existence and uniqueness of solutions for compressible liquid crystal flow in critical spaces, and analyzes the incompressible limit as Mach number approaches zero, with convergence rates.

Contribution

It improves previous results by relaxing initial data regularity requirements and establishes the incompressible limit with precise convergence rates.

Findings

Global solutions exist and are unique near equilibrium states.

Solutions converge to incompressible model as Mach number tends to zero.

Convergence rates are explicitly obtained.

Abstract

The Cauchy problem for the compressible flow of nematic liquid crystals in the framework of critical spaces is considered. We first establish the existence and uniqueness of global solutions provided that the initial data are close to some equilibrium states. This result improves the work by Hu and Wu [SIAM J. Math. Anal., 45 (2013), pp. 2678-2699] through relaxing the regularity requirement of the initial data in terms of the director field. We then consider the incompressible limit problem for ill prepared initial data. We prove that as the Mach number tends to zero, the global solution to the compressible flow of liquid crystals converges to the solution to the corresponding incompressible model in some function spaces. Moreover, the accurate converge rates are obtained.