Time Evolution of Nematic Liquid Crystals with Variable Degree of Orientation

Calderer, M. Carme; Golovaty, Dmitry; Lin, Fucai; Li, Chun

doi:10.1137/s0036141099362086

Cited by 29 publications

(37 citation statements)

References 10 publications

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“…ε (x, t) and s (2) ε (x, t). Choosing v =s ε in (2.29a) and η =φ ε in (2.29b), and noting the monotonicity of |a|…”

Section: Theorem 25 For Any Fixedmentioning

confidence: 99%

“…Here we extend this regularization to (P δ ), and consider for any given constant ε ∈ (0, 1]: (P δ,ε ) Find {s ε , ϕ ε } such that (1.2b) subject to the same initial and boundary conditions as (P δ ). Calderer, Golovaty, Lin and Liu [2] showed that as for the original system (P 0 ), the regularized system (P 0,ε ) also satisfies a dissipative energy law, which in turn implies some uniform (in ε) estimates for {s ε , ϕ ε }. Based on these uniform estimates, they then showed global existence of weak solutions to the original system (P 0 ), for a smooth domain Ω, using compactness arguments.…”

Section: Introductionmentioning

confidence: 99%

“…This ensures that the perfect alignments s = − 1 2 and s = 1 are excluded from being equilibrium states, which is physically reasonable; see ([4], p. 109). In order to overcome the degeneracy in (1.1b) at s = 0, Calderer, Golovaty, Lin and Liu [2] considered a regularization (P 0,ε ) of (P 0 ). Here we extend this regularization to (P δ ), and consider for any given constant ε ∈ (0, 1]: (P δ,ε ) Find {s ε , ϕ ε } such that (1.2b) subject to the same initial and boundary conditions as (P δ ).…”

Section: Introductionmentioning

confidence: 99%

“…Global existence of weak solutions to the system (P 0 ) was considered in Theorem 4.3 of [2]. However, there are a number of gaps in the proof given there for passing to the limit ε → 0 in the term |∇ϕ ε | 2 s ε on the right hand side of (1.2a).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Barrett¹,

Feng

Prohl

2006

ESAIM: M2AN

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Abstract. 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 fully discrete finite element method for this regularized system, and we establish the (subsequence) convergence of this finite element approximation to the solution of the regularized system as the mesh parameters tend to zero; and to a solution of the original degenerate parabolic system when the the mesh and regularization parameters all approach zero. Finally, numerical experiments are included which show the formation, annihilation and evolution of line singularities/defects in such models.Mathematics Subject Classification. 35K55, 35K65, 35Q35, 65M12, 65M60, 76A15.

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“…ε (x, t) and s (2) ε (x, t). Choosing v =s ε in (2.29a) and η =φ ε in (2.29b), and noting the monotonicity of |a|…”

Section: Theorem 25 For Any Fixedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Barrett¹,

Feng

Prohl

2006

ESAIM: M2AN

View full text Add to dashboard Cite

show abstract

“…There is a vast literature on the hydrodynamic of liquid crystal systems. For background we list a few, with no intention to be complete: [9,12,13,18,19,17,1,2,3,4,24,20,11]. In particular, the asymptotic behavior of solutions to the flow of nematic liquid crystals was studied for bounded domains in [18,24].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Behavior of Solutions to the Liquid Crystal System in $H^m(\mathbb{R}^3)$

Dai¹,

Schonbek²

2014

SIAM J. Math. Anal.

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Large‐time behavior of solutions to a 1D liquid crystal system

Qin

Feng

2017

Math Methods in App Sciences

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In this paper, we prove the large-time behavior, as time tends to infinity, of solutions in H i × H i 0 × H i+1 (i = 1, 2) and H 4 × H 4 0 × H 4 for a system modeling the nematic liquid crystal flow, which consists of a subsystem of the compressible Navier-Stokes equations coupling with a subsystem including a heat flow equation for harmonic maps. KEYWORDS large-time behavior, liquid crystal system Math Meth Appl Sci. 2017;40:7077-7103.wileyonlinelibrary.com/journal/mma

show abstract

Time Evolution of Nematic Liquid Crystals with Variable Degree of Orientation

Cited by 29 publications

References 10 publications

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

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

Asymptotic Behavior of Solutions to the Liquid Crystal System in $H^m(\mathbb{R}^3)$

Large‐time behavior of solutions to a 1D liquid crystal system

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