The Zero Inertia Limit for the Q-Tensor Model of Liquid Crystals: Analysis and Numerics

TL;DR

This paper rigorously analyzes the zero inertia limit in the Q-tensor model of liquid crystals, proving convergence, developing a stable numerical scheme, and providing error estimates for simulations.

Contribution

It establishes well-posedness including inertia, proves convergence to the non-inertial model, and introduces a stable finite element scheme with error analysis.

Findings

Solutions with inertia converge to non-inertial solutions at rate σ

Developed an energy stable finite element scheme for all σ

Provided error estimates for the fully discrete numerical scheme

Abstract

The goal of this work is to rigorously study the zero inertia limit for the Q-tensor model of liquid crystals. Though present in the original derivation of the Ericksen-Leslie equations for nematic liquid crystals, the inertia term of the model is often neglected in analysis and applications. We show wellposedness of the model including inertia and then show using the relative entropy method that solutions of the model with inertia converge to solutions of the model without inertia at a rate ${\sigma}$ in $L^\infty(0,T;H^1(\dom))$, where $\sigma$ is the inertial constant. Furthermore, we present an energy stable finite element scheme that is stable and convergent for all $\sigma$ and study the zero inertia limit numerically. We also present error estimates for the fully discrete scheme with respect to the discretization parameters in time and space.