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

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.