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Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation

John W. Barrett, Xiaobing Feng, Andreas Prohl (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a degenerate parabolic system which models the evolution of nematic liquid crystal with variable degree of orientation. The system is a slight modification to that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120]. We prove the global existence of weak solutions by passing to the limit in a regularized system. Moreover, we propose a practical...

Disclinations and hedgehogs in nematic liquid crystals with variable degree of orientation

Epifanio G. Virga (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

There is enough evidence to re-examine disclinations and hedgehogs, the singularities often observed in nematic liquid crystals, in the light of a new theory that allows for local changes in the degree of orientation.

Equilibrium confocal textures in a smetic-A cell

Epifanio G. Virga, Jean-Baptiste Fournier (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We study the textures of smectic-A liquid crystals consisting in curved, but stricdy equidistant lamellae. Assuming translational symmetry, we can generate them from a single curve. The free energy is a non-trivial functional of it. We learn how to derive the equilibrium equation for this curve, when the texture is confined between two parallel plates, which exert a weak anchoring on the orientation of the lamellae, but do not interfere direcdy with their position. Finally, we describe an instability...

La scelta delle interazioni inerziali nei continui con microstruttura

Paolo Podio-Guidugli (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Dedicando speciale attenzione all’esempio significativo dei cristalli liquidi di Ericksen [6], viene presentato un apparato assiomatico che consente di dedurre rappresentazioni coerenti delle interazioni d’inerzia e dell’energia cinetica per continui con microstruttura.

Maximal regularity of the spatially periodic Stokes operator and application to nematic liquid crystal flows

Jonas Sauer (2016)

Czechoslovak Mathematical Journal

We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal L p -regularity of the periodic Laplace and Stokes operators and a local-in-time existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group G : = ℝ n - 1 × ℝ / L ℤ to obtain an ℛ -bound for the...

Mechanisms of Cell Motion in Confined Geometries

R. J. Hawkins, R. Voituriez (2010)

Mathematical Modelling of Natural Phenomena

We present a simple mechanism of cell motility in a confined geometry, inspired by recent motility assays in microfabricated channels. This mechanism relies mainly on the coupling of actin polymerisation at the cell membrane to geometric confinement. We first show analytically using a minimal model of polymerising viscoelastic gel confined in a narrow channel that spontaneous motion occurs due to polymerisation alone. Interestingly, this mechanism...

Mixed methods for the approximation of liquid crystal flows

Chun Liu, Noel J. Walkington (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen–Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve H 2 ( Ω ) norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.

Mixed Methods for the Approximation of Liquid Crystal Flows

Chun Liu, Noel J. Walkington (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen–Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve H2(Ω) norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations

Noel J. Walkington (2011)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.

Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations*

Noel J. Walkington (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.

Singularities, defects and chaos in organized fluids

Roland Ribotta, Ahmed Belaidi, Alain Joets (2003)

Banach Center Publications

The singularities occurring in any sort of ordering are known in physics as defects. In an organized fluid defects may occur both at microscopic (molecular) and at macroscopic scales when hydrodynamic ordered structures are developed. Such a fluid system serves as a model for the study of the evolution towards a strong disorder (chaos) and it is found that the singularities play an important role in the nature of the chaos. Moreover both types of defects become coupled at the onset of turbulence....

The Cauchy problem for the liquid crystals system in the critical Besov space with negative index

Sen Ming, Han Yang, Zili Chen, Ls Yong (2017)

Czechoslovak Mathematical Journal

The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space B ˙ p , 1 n / p - 1 ( ℝ n ) × B ˙ p , 1 n / p ( ℝ n ) with n < p < 2 n is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed.

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