TY - JOUR

T1 - On the energy instability of liquid crystals

AU - Vladimirov, V. A.

AU - Ilin, K. I.

PY - 1998/2

Y1 - 1998/2

N2 - 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.

AB - 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.

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U2 - 10.1017/S0956792597003288

DO - 10.1017/S0956792597003288

M3 - Article

AN - SCOPUS:0032331333

SN - 0956-7925

VL - 9

SP - 23

EP - 36

JO - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

IS - 1

ER -