Rigorous justification of the uniaxial limit from Qian-Sheng's inertial $Q$-tensor theory to the Ericksen-Leslie theory

TL;DR

This paper rigorously demonstrates that as elastic coefficients approach zero, solutions of Qian-Sheng's inertial Q-tensor model converge to those of the full inertial Ericksen-Leslie liquid crystal model, establishing a mathematical connection.

Contribution

It provides a rigorous mathematical proof of the uniaxial limit from the Qian-Sheng inertial Q-tensor model to the Ericksen-Leslie system using Hilbert expansion.

Findings

Convergence of solutions in the uniaxial limit

Mathematical validation of the model connection

Use of Hilbert expansion method

Abstract

In this paper, we rigorously justify the connection between Qian-Sheng's inertial $Q$-tensor model and the full Ericksen-Leslie model for the liquid crystal flow. By using the Hilbert expansion method, we prove that, when the elastic coefficients tend to zero(also called the uniaxial limit), the solution to the Qian-Sheng's inertial model will converge to the solution to the full inertial Ericksen-Leslie system.