On the energy instability of liquid crystals

V. A. Vladimirov, K. I. Ilin

Research output: Contribution to journalArticle

Abstract

The direct Lyapunov method is used to investigate the stability of general equilibria of a nematic liquid crystal. First, we prove the converse Lagrange theorem stating that an equilibrium is unstable to small perturbations if the distortion energy has no minimum at this equilibrium (i.e. if the second variation of the distortion energy evaluated at the equilibrium is not positive definite). The proof is constructive rather than abstract: we explicitly construct a functional that grows exponentially with time by virtue of linearized equations of motion provided the condition of the theorem is satisfied. We obtain an explicit formula that gives the dependence of the perturbation growth rate upon the equilibrium considered and the initial data for the perturbation. Secondly, we obtain the upper and lower bounds for growing solutions of the linearized problem, and we identify the initial data corresponding to the most unstable mode (i.e. to the perturbation with maximal growth rate). All results are obtained in quite a general formulation: a nematic is inside a three-dimensional domain of an arbitrary shape and strong anchoring on the boundary is supposed; the standard equations of nematodynamics are employed as the governing equations.

Original languageEnglish
Pages (from-to)23-36
Number of pages14
JournalEuropean Journal of Applied Mathematics
Volume9
Issue number1
Publication statusPublished - Feb 1998

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Liquid Crystal
Liquid crystals
Lyapunov methods
Nematic liquid crystals
Energy
Perturbation
Equations of motion
Unstable
Lagrange's theorem
Lyapunov Direct Method
Second Variation
General Equilibrium
Nematic Liquid Crystal
Small Perturbations
Converse
Positive definite
Explicit Formula
Upper and Lower Bounds
Governing equation
Equations of Motion

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

On the energy instability of liquid crystals. / Vladimirov, V. A.; Ilin, K. I.

In: European Journal of Applied Mathematics, Vol. 9, No. 1, 02.1998, p. 23-36.

Research output: Contribution to journalArticle

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