Incompressible limit of the Ericksen-Leslie hyperbolic liquid crystal model in compressible flow

TL;DR

This paper rigorously analyzes the incompressible limit of the compressible Ericksen-Leslie liquid crystal model, establishing convergence and energy estimates, and providing the first such result for this coupled hyperbolic-parabolic system.

Contribution

It introduces a framework for deriving uniform energy estimates and proves the global classical solution convergence from compressible to incompressible liquid crystal models.

Findings

Established uniform energy estimates in Mach number limit

Proved convergence of solutions as Mach number approaches zero

Derived convergence rates for well-prepared initial data

Abstract

Ericksen and Leslie proposed a hydrodynamic model for liquid crystals in the format of conservation laws in the 1960s. Their original model includes inertial and compressibility effects, which makes the model a coupled parabolic-hyperbolic system. In this paper we build up the connection between the compressible and incompressible parabolic-hyperbolic liquid crystal model in the framework of classical solutions. We first derive the scaled Ericksen-Leslie system with dimensionless numbers, including Mach, Reynolds, and Ericksen numbers. In particular, we introduce the so-called inertial constant $\chi$ which characterizes the inertial effect of the liquid crystal molecular. Next, we establish the energy estimates uniform in the Mach number $\epsilon$ for both the compressible system and its time-derivative system with small data. Then, we pass to the limit $\epsilon \rightarrow 0$ in the compressible system, so that we establish the global classical solution of the incompressible system by the compactness arguments. Moreover, we also obtain the convergence rates associated with $L^2$-norm in the case of well-prepared initial data. This is the first result on the incompressible limit of the compressible parabolic-hyperbolic liquid crystal model and confirms the relations of different parabolic-hyperbolic liquid crystal model rigorously.