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

Max Hirsch, Franziska Weber, Yukun Yue

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 $σ$ in $L^\infty(0,T;H^1(\dom))$, where $σ$ is the inertial constant. Furthermore, we present an energy stable finite element scheme that is stable and convergent for all $σ$ 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.

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

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 in , where is the inertial constant. Furthermore, we present an energy stable finite element scheme that is stable and convergent for all 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.

Paper Structure

This paper contains 17 sections, 17 theorems, 207 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Definitions of the weak and strong solutions
  5. Global Strong solution
  6. Higher-order Energy Estimates
  7. Galerkin Approximation
  8. Existence and uniqueness results
  9. Finite Element Scheme
  10. Convergence to the strong solution
  11. Convergence as $\sigma\to0$
  12. Numerical Results
  13. Convergence test
  14. Refinement in space
  15. Refinement in time
  16. ...and 2 more sections

Key Result

Theorem 3.1

For fixed $T>0$, the problem eq:Qt has a unique strong solution $Q$ which satisfies Definition def:strong_solution given $Q_0\in H^2(\Omega), Q_{0,t}\in H^1(\Omega)$. As a consequence, $Q\in L^\infty\left([0,T];L^\infty(\Omega)\right)$. In addition, this solutions is unique among all weak solutions

Figures (3)

  • Figure 1: Space refinement experiment $H^1$ errors for $(Q_h)_{11}$ and $(Q_h)_{12}$ and $L^2$ error for $r_h$. An $O(h)$ reference line is given in green without point markers.
  • Figure 2: Time refinement experiment $H^1$ errors for $(Q_h)_{11}$ and $(Q_h)_{12}$ and $L^2$ error for $r_h$. An $O(\Delta t)$ reference line is given in green without point markers.
  • Figure 3: Convergence rates in $\sigma$ for various perturbations rates of the initial value and initial time derivative.

Theorems & Definitions (41)

  • Definition 2.1: Weak solutions of \ref{['eq:qtensorflow']}
  • Remark 2.2
  • Definition 2.3: Weak solutions of \ref{['eq:reformulation']}
  • Definition 2.4: Strong solution of \ref{['eq:qtensorflow']}
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Remark 3.5
  • ...and 31 more