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Rigorous uniaxial limit of the Qian--Sheng inertial Q-tensor hydrodynamics for liquid crystals

Sirui Li, Wei Wang, Qi Zeng

TL;DR

This work provides a rigorous bridge between inertial QS Q-tensor hydrodynamics and Ericksen–Leslie theory for liquid crystals by deploying a Hilbert expansion in the small elastic parameter ε and an inertial parameter η=ε^m. It constructs explicit leading-order uniaxial states Q_0 = s(nn − I/3) and higher-order corrections, then derives a remainder system controlled by specially designed energy functionals that handle the singular ε^{-1} terms. The analysis treats two regimes: m=0, linking QS to the full inertial EL model, and m≥1, linking QS to the noninertial EL model, with stability ensured by projection onto Ker H_n and dissipative estimates. The results extend prior work by relaxing strong damping-inertia assumptions and by handling simultaneous vanishing of elastic and inertial effects, thereby validating the formal limit procedures used to derive EL-type dynamics from Q-tensor hydrodynamics. The methodology provides a robust template for singular-limits in hyperbolic–parabolic Q-tensor systems and clarifies the role of uniaxial minimizers in governing limiting behavior.

Abstract

This article is concerned with the rigorous connections between the inertial Qian--Sheng model and the Ericksen--Leslie model for the liquid crystal flow, under a more general condition of coefficients. More specifically, in the framework of Hilbert expansions, we show that: (i) when the elastic coefficients tend to zero (also called the uniaxial limit), the smooth solution to the inertial Qian--Sheng model converges to that to the full inertial Ericksen--Leslie model; (ii) when the elastic coefficients and the inertial coefficient tend to zero simultaneously, the smooth solution to the inertial Qian--Sheng model converges to that to the noninertial Ericksen--Leslie model.

Rigorous uniaxial limit of the Qian--Sheng inertial Q-tensor hydrodynamics for liquid crystals

TL;DR

This work provides a rigorous bridge between inertial QS Q-tensor hydrodynamics and Ericksen–Leslie theory for liquid crystals by deploying a Hilbert expansion in the small elastic parameter ε and an inertial parameter η=ε^m. It constructs explicit leading-order uniaxial states Q_0 = s(nn − I/3) and higher-order corrections, then derives a remainder system controlled by specially designed energy functionals that handle the singular ε^{-1} terms. The analysis treats two regimes: m=0, linking QS to the full inertial EL model, and m≥1, linking QS to the noninertial EL model, with stability ensured by projection onto Ker H_n and dissipative estimates. The results extend prior work by relaxing strong damping-inertia assumptions and by handling simultaneous vanishing of elastic and inertial effects, thereby validating the formal limit procedures used to derive EL-type dynamics from Q-tensor hydrodynamics. The methodology provides a robust template for singular-limits in hyperbolic–parabolic Q-tensor systems and clarifies the role of uniaxial minimizers in governing limiting behavior.

Abstract

This article is concerned with the rigorous connections between the inertial Qian--Sheng model and the Ericksen--Leslie model for the liquid crystal flow, under a more general condition of coefficients. More specifically, in the framework of Hilbert expansions, we show that: (i) when the elastic coefficients tend to zero (also called the uniaxial limit), the smooth solution to the inertial Qian--Sheng model converges to that to the full inertial Ericksen--Leslie model; (ii) when the elastic coefficients and the inertial coefficient tend to zero simultaneously, the smooth solution to the inertial Qian--Sheng model converges to that to the noninertial Ericksen--Leslie model.
Paper Structure (18 sections, 20 theorems, 223 equations)

This paper contains 18 sections, 20 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Ericksen--Leslie theory
  3. Landau--de Gennes theory
  4. Main results
  5. Critical points and the linearized operator
  6. The Hilbert expansion and the remainder system
  7. The Hilbert expansion
  8. The remainder system
  9. From the Qian--Sheng model to the full inertial Ericksen--Leslie model
  10. Existence of the Hilbert expansion
  11. Uniform estimates for the remainder system
  12. From the Qian--Sheng model to the noninertial Ericksen--Leslie model
  13. Existence of the Hilbert expansion
  14. Uniform estimates for the remainder system
  15. Proof of Theorem \ref{['main theorem']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $m$ be a nonnegative integer, assume that $(\mathbf{n}(t,\mathbf{x}), \mathbf{v}(t,\mathbf{x}))$ is a smooth solution of the system (EL-vv)--(EL-div) on $[0, T]$ with coefficients defined by (coefficients), and satisfies Let $Q_0(t,\mathbf{x})=s(\mathbf{n}(t,\mathbf{x})\mathbf{n}(t,\mathbf{x})-\frac{1}{3}\mathbf{I})$ be the uniaxial minimizer of the bulk energy $f_b$, and the functions $(Q_

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 4.1: see LW
  • Proposition 4.2
  • Lemma 4.3
  • proof
  • ...and 28 more