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Bifurcation and stability of stationary shear flows of Ericksen-Leslie model for nematic liquid crystals

Weishi Liu, Majed Sofiani

TL;DR

The paper analyzes stationary shear flows in the parabolic Ericksen-Leslie model for nematic liquid crystals under ν=0, focusing on the cusp regime with $γ_1=|γ_2|$ and $θ_0=θ_1$. It establishes a one-to-one correspondence between stationary states and solutions of a cusp algebraic equation through a function $D(β)$, showing that large imposed shear $ā$ yields countably many saddle-node bifurcations and multiple steady states. Spectral analysis via the Evans function reveals that zero eigenvalues occur precisely when $D'(β)=0$, and a nondegeneracy condition on $E_λ(0,β^*)$ implies that the zero eigenvalue bifurcates with opposite signs for the two emergent branches, determining stability. Additionally, the paper proves linear stability for small $ā$ by an energy estimate, demonstrating exponential decay of perturbations. The results provide a rigorous, quantitative description of bifurcation and stability in nematic shear flows and yield explicit criteria for the existence and stability of multiple stationary states.

Abstract

In this work, focusing on a critical case for shear flows of nematic liquid crystals, we investigate multiplicity and stability of stationary solutions via the parabolic Ericksen-Leslie system. We establish a one-to-one correspondence between the set of the stationary solutions with the set of the solutions of an algebraic equation for a cusp case. This one-to-one correspondence is established essentially based on the treatment in the work of Jiao, et. al. [{\em J. Diff. Dyn. Syst. {\bf 34} (2022), 239-269}] for a different case, and the relation gives directly parameter ranges for existence of multiple stationary solutions; in particular, multiple stationary solutions are created through countably many saddle-node bifurcations for the algebraic equation at critical shear speeds. The main result of the paper is on the stability of stationary solutions associated to the bifurcations; more precisely, (i) for each critical shear speed, there is a unique stationary solution and, for smaller shear speed, the stationary solution disappears but, for larger shear speed, two stationary solutions nearby bifurcate; (ii) more importantly, under a generic condition, there is a simple zero eigenvalue for the linearization of the shear flow at the critical stationary solution and, for larger shear speed, the zero eigenvalue bifurcates to a negative eigenvalue for one of the two stationary solutions and to a positive eigenvalue for the other stationary solution.

Bifurcation and stability of stationary shear flows of Ericksen-Leslie model for nematic liquid crystals

TL;DR

The paper analyzes stationary shear flows in the parabolic Ericksen-Leslie model for nematic liquid crystals under ν=0, focusing on the cusp regime with and . It establishes a one-to-one correspondence between stationary states and solutions of a cusp algebraic equation through a function , showing that large imposed shear yields countably many saddle-node bifurcations and multiple steady states. Spectral analysis via the Evans function reveals that zero eigenvalues occur precisely when , and a nondegeneracy condition on implies that the zero eigenvalue bifurcates with opposite signs for the two emergent branches, determining stability. Additionally, the paper proves linear stability for small by an energy estimate, demonstrating exponential decay of perturbations. The results provide a rigorous, quantitative description of bifurcation and stability in nematic shear flows and yield explicit criteria for the existence and stability of multiple stationary states.

Abstract

In this work, focusing on a critical case for shear flows of nematic liquid crystals, we investigate multiplicity and stability of stationary solutions via the parabolic Ericksen-Leslie system. We establish a one-to-one correspondence between the set of the stationary solutions with the set of the solutions of an algebraic equation for a cusp case. This one-to-one correspondence is established essentially based on the treatment in the work of Jiao, et. al. [{\em J. Diff. Dyn. Syst. {\bf 34} (2022), 239-269}] for a different case, and the relation gives directly parameter ranges for existence of multiple stationary solutions; in particular, multiple stationary solutions are created through countably many saddle-node bifurcations for the algebraic equation at critical shear speeds. The main result of the paper is on the stability of stationary solutions associated to the bifurcations; more precisely, (i) for each critical shear speed, there is a unique stationary solution and, for smaller shear speed, the stationary solution disappears but, for larger shear speed, two stationary solutions nearby bifurcate; (ii) more importantly, under a generic condition, there is a simple zero eigenvalue for the linearization of the shear flow at the critical stationary solution and, for larger shear speed, the zero eigenvalue bifurcates to a negative eigenvalue for one of the two stationary solutions and to a positive eigenvalue for the other stationary solution.

Paper Structure

This paper contains 12 sections, 10 theorems, 123 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Ericksen-Leslie model for nematics
  3. Shear flows and informal statement of the results
  4. Multiplicity of steady states for $\gamma_1=|\gamma_2|$ with $\theta_0=\theta_1$
  5. Characteristic of bifurcation moment of stead states
  6. Linearization and eigenvalue problems
  7. Evans function and an explicit criterion for zero eigenvalue
  8. Spectral stability/instability of stationary solutions
  9. A motivation for the bifurcation of the zero eigenvalue
  10. Computation of $E_\lambda(0,\beta^*)$
  11. Bifurcation of zero eigenvalue
  12. Linear stability for steady states with small $\bar{u}$

Key Result

Lemma 2.1

For $\gamma_1= |\gamma_2|,$ the function $G$ is invertible.

Figures (4)

  • Figure 1: (a) Spaly: $\nabla\cdot {\bf n}\neq 0$ (b) Twist: ${\bf n}\cdot\hbox{curl,}{\bf n}\neq 0$ (c) Bend: $|{\bf n}\times\hbox{curl,}{\bf n}|\neq 0$
  • Figure 2: The phase portrait for the case $\gamma_1=-\gamma_2.$ For any $n\in \mathbb{N}\cup \{0\},$ the point $(\pm n\pi,0)$ is an equilibrium point (cf. equations \ref{['storder']}).
  • Figure 3: A sketch of the graph of $D(\beta)$ for $\gamma_1=|\gamma_2|$ (See Prop. \ref{['lemmaD']}).
  • Figure 4: A saddle-node bifurcation for the equation $D(\beta)=\bar{u}/2$ takes place as $\bar{u}$ crosses a local minimum of $D(\beta)$, say at $\beta^*.$

Theorems & Definitions (17)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 3.1
  • ...and 7 more