Global m-Equivariant Solutions of Nematic Liquid Crystal Flows in Dimension Two

Chen, Yuan; Yu, Yong

doi:10.1007/s00205-017-1144-x

Cited by 5 publications

(9 citation statements)

References 16 publications

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“…We refer readers to the survey article [11] by Lin-Wang and references therein. Most recently solutions of (1.1) with finite time singularity have also been constructed by authors in [8], where the spatial domain is a bounded open set in R 3 . In contrast to [8], in the current work, we are concerned with global solutions of (1.1) which become singular at t " 8.…”

Section: I1 Background and Motivationmentioning

confidence: 99%

“…Proof of Theorem 1.1. The local existence of (1.8) with initial data satisfying (1.12)-(1.14) can be obtained by methods in [3]- [4]. We omit it here.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…When µ " 0 and V 3 " 0, the system (1.3)- (1.4) is then reduced to a 2D version of (1.1). Its vorticity formulation has been studied by authors in [3]. In fact a global weak solution is obtained in [3] under the assumption that the initial vorticity of fluid lies in L 1 pR 2 q and the initial director field has finite Dirichlet energy.…”

Section: Introduction I1 Background and Motivationmentioning

confidence: 99%

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Twisted Solutions to a Simplified Ericksen–Leslie Equation

Chen

Kim

2018

Arch Rational Mech Anal

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In this article we construct global solutions to a simplified Ericksen-Leslie system on R 3 . The constructed solutions are twisted and periodic along the x3-axis with period d " 2π L µ. Here µ ą 0 is the twist rate. d is the distance between two planes which are parallel to the x1x2 -plane. Liquid crystal material is placed in the region enclosed by these two planes. Given a well-prepared initial data, our solutions exist classically for all t P r0, 8q. However these solutions become singular at all points on the x3-axis and escape into third dimension exponentially while t Ñ 8. An optimal blow up rate is also obtained.

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Section: I1 Background and Motivationmentioning

confidence: 99%

“…Proof of Theorem 1.1. The local existence of (1.8) with initial data satisfying (1.12)-(1.14) can be obtained by methods in [3]- [4]. We omit it here.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Section: Introduction I1 Background and Motivationmentioning

confidence: 99%

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Twisted Solutions to a Simplified Ericksen–Leslie Equation

Chen

Kim

2018

Arch Rational Mech Anal

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“…Based on the finite time singularities of the 2D heat flow of harmonic maps [6], solutions of (1.1)-(1.4) with finite time singularities have been recently constructed in [18], where the spatial domain is a bounded open set in R 3 . The long time behavior of solutions of (1.1)-(1.4) is concerned in [9,10]. The behavior of defects is also an important subject in the study of liquid crystals.…”

Section: Introductionmentioning

confidence: 99%

Constraint-Preserving Energy-Stable Scheme for the 2D Simplified Ericksen-Leslie System

Bao¹

2021

JCM

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Here we consider the numerical approximations of the 2D simplified Ericksen-Leslie system. We first rewrite the system and get a new system. For the new system, we propose an easy-to-implement time discretization scheme which preserves the sphere constraint at each node, enjoys a discrete energy law, and leads to linear and decoupled elliptic equations to be solved at each time step. A discrete maximum principle of the scheme in the finite element form is also proved. Some numerical simulations are performed to validate the scheme and simulate the dynamic motion of liquid crystals.

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“…Moreover, Huang, Wang and Wen [39] established a blow up criterion for compressible nematic liquid crystal flows in dimension three. Recently, Chen and Yu [6] constructed global m-equivariant solutions in R 2 that the orientation field blows up logarithmically as t → +∞. See also Lin-Wang [51] for a survey of some important developments of mathematical studies of nematic liquid crystals.…”

Section: Introductionmentioning

confidence: 99%