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Critical points and bifurcations of the three-dimensional onsager model for liquid crystals

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Title: Critical points and bifurcations of the three-dimensional onsager model for liquid crystals
Authors: Vollmer, MAC
Item Type: Journal Article
Abstract: We study the bifurcation diagram of the Onsager free-energy functional for liquid crystals with orientation parameter on the sphere. In particular, we concentrate on the bifurcations from the isotropic solution for a general class of two-body interaction potentials including the Onsager kernel. Reformulating the problem as a non-linear eigenvalue problem for the kernel operator, we prove that spherical harmonics are the corresponding eigenfunctions and we present a direct relationship between the coefficients of the Taylor expansion of this class of interaction potentials and their eigenvalues. We find explicit expressions for all bifurcation points corresponding to bifurcations from the isotropic state of the Onsager free-energy functional equipped with the Onsager interaction potential. A substantial amount of our analysis is based on the use of spherical harmonics and a special algorithm for computing expansions of products of spherical harmonics in terms of spherical harmonics is presented. Using a Lyapunov–Schmidt reduction, we derive a bifurcation equation depending on five state variables. The dimension of this state space is further reduced to two dimensions by using the rotational symmetry of the problem and the invariant theory of groups. On the basis of these results, we show that the first bifurcation from the isotropic state of the Onsager interaction potential is a transcritical bifurcation and that the corresponding solution is uniaxial. In addition, we prove some global properties of the bifurcation diagram such as the fact that the trivial solution is the unique local minimiser if the bifurcation parameter is high, that it is not a local minimiser if the bifurcation parameter is small, the boundedness of all equilibria of the functional and that the bifurcation branches are either unbounded or that they meet another bifurcation branch.
Issue Date: 1-Nov-2017
Date of Acceptance: 24-Jun-2017
URI: http://hdl.handle.net/10044/1/80834
DOI: 10.1007/s00205-017-1146-8
ISSN: 0003-9527
Publisher: Springer
Start Page: 851
End Page: 922
Journal / Book Title: Archive for Rational Mechanics and Analysis
Volume: 226
Issue: 2
Copyright Statement: © 2017 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Keywords: Science & Technology
Physical Sciences
Technology
Mathematics, Applied
Mechanics
Mathematics
STATIONARY SOLUTIONS
ORDERED STATES
SYMMETRY
EQUATION
SPHERE
Science & Technology
Physical Sciences
Technology
Mathematics, Applied
Mechanics
Mathematics
STATIONARY SOLUTIONS
ORDERED STATES
SYMMETRY
EQUATION
SPHERE
math.AP
math.AP
37G40, 76A15, 35B32, 33C55
General Physics
0101 Pure Mathematics
0102 Applied Mathematics
Publication Status: Published
Online Publication Date: 2017-07-14
Appears in Collections:School of Public Health