Dynamics of Director Fields

Hunter, John K.; Saxton, Ralph

doi:10.1137/0151075

Cited by 439 publications

(416 citation statements)

References 5 publications

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“…The Hunter-Saxton equation was proposed as a model for the propagation of orientation waves in nematic liquid crystals in [60]. Its geodesic nature was discovered in [72].…”

Section: Geodesic Equationmentioning

confidence: 99%

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

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“…The Hunter-Saxton equation was proposed as a model for the propagation of orientation waves in nematic liquid crystals in [60]. Its geodesic nature was discovered in [72].…”

Section: Geodesic Equationmentioning

confidence: 99%

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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“…In order to test our schemes in practice, we compared them with two other schemes, the first order Engquist-Osher scheme proposed in [6] and a central scheme which is an adaptation of schemes presented in [10]. We have no convergence proofs for these schemes.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In this paper we will specialize to stationary flow, and hence focus exclusively on the dynamics of the director field, a map n : R 3 → S 3 from the Euclidean space to the unit ball; see Saxton [14]. It is common to consider the Oseen-Franck expression for the internal energy (see [14], [15], [6])…”

Section: Introductionmentioning

confidence: 99%

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Convergent difference schemes for the Hunter–Saxton equation

2007

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Abstract. We propose and analyze several finite difference schemes for the Hunter-Saxton equationThis equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of u, which is achieved by analyzing various renormalizations (in the sense of DiPerna and Lions) of the numerical schemes. Finally, we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results.

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“…The two-component Hunter-Saxton system is a generalization of the HunterSaxton equation modeling the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal (see Hunter & Saxton [22] for a derivation, and also [1-3, 43, 49]), since the former obviously reduces to the latter if the initial datum ρ 0 is chosen to vanish identically. It turns out that if this choice is made for arbitrary α ∈ R, one arrives at the generalized Proudman-Johnson equation [14,35,36,38,40,48] with parameter a = α − 1.…”

Section: Introductionmentioning

confidence: 99%

Global Existence for the Generalized Two-Component Hunter–Saxton System

Wunsch

2011

J. Math. Fluid Mech.

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We study the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting. We first prove a persistence result of the solutions. Then for some particular choices of the parameters (α, κ), we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter-Saxton system under proper assumptions on the initial data. This significantly improves recent results obtained in [46,47].

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Dynamics of Director Fields

Cited by 439 publications

References 5 publications

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Convergent difference schemes for the Hunter–Saxton equation

Global Existence for the Generalized Two-Component Hunter–Saxton System

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