Twisted Solutions to a Simplified Ericksen–Leslie Equation

Chen, Yuan; Kim, Soojung; Yu, Yong

doi:10.1007/s00205-018-1321-6

Cited by 3 publications

(2 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Bao¹

2021

JCM

View full text Add to dashboard Cite

show abstract

“…Nevertheless, it should be pointed out that important progresses have been achieved for the original Ericksen-Leslie model (as well as its simplified versions), in which the penalisation energy density W (d) is removed and instead, a Lagrangian multiplier corresponding to the unitary constraint |d(t, x)| = 1 has to be introduced in the director equation. For instance, we refer the interested readers to [7,23,24,27,29,33,39,44,49] for the simplified liquid crystal system involving the transported heat flow of harmonic map, to [21,26,32,45] for the general Ericksen-Leslie system as well as its non-isothermal extension [12], and to [2,28] for the case with inertial term in the director equation (i.e., the so-called hyperbolic Ericksen-Leslie system).…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of weak solutions for the general Ericksen-Leslie system with Ginzburg-Landau penalization in T^2

De Anna,

2021

Preprint

View full text Add to dashboard Cite

The Ericksen-Leslie system is a fundamental hydrodynamic model that describes the evolution of incompressible liquid crystal flows of nematic type. In this paper, we prove the uniqueness of global weak solutions to the general Ericksen-Leslie system with a Ginzburg-Landau type approximation in a two dimensional periodic domain. The proof is based on some delicate energy estimates for the difference of two weak solutions within a suitable functional framework that is less regular than the usual one at the natural energy level, combined with the Osgood lemma involving a specific double-logarithmic type modulus of continuity. We overcome the essential mathematical difficulties arising from those highly nonlinear terms in the Leslie stress tensor and in particular, the lack of maximum principle for the director equation due to the stretching effect of the fluid on the director field. Our argument makes full use of the coupling structure as well as the dissipative nature of the system, and relies on some techniques from harmonic analysis and paradifferential calculus in the periodic setting.

show abstract

Uniqueness of global weak solutions for the general Ericksen–Leslie system with Ginzburg–Landau penalization in $${\mathbb {T}}^2$$

Anna

2023

Calc. Var.

View full text Add to dashboard Cite

The Ericksen–Leslie system is a fundamental hydrodynamic model that describes the evolution of incompressible liquid crystal flows of nematic type. In this paper, we prove the uniqueness of global weak solutions to the general Ericksen–Leslie system with a Ginzburg–Landau type approximation in a two dimensional periodic domain. The proof is based on some delicate energy estimates for the difference of two weak solutions within a suitable functional framework that is less regular than the usual one at the natural energy level, combined with the Osgood lemma involving a specific double-logarithmic type modulus of continuity. We overcome the essential mathematical difficulties arising from those highly nonlinear terms in the Leslie stress tensor and in particular, the lack of maximum principle for the director equation due to the stretching effect of the fluid on the director field. Our argument makes full use of the coupling structure as well as the dissipative nature of the system, and relies on some techniques from harmonic analysis and paradifferential calculus in the periodic setting.

show abstract

Twisted Solutions to a Simplified Ericksen–Leslie Equation

Cited by 3 publications

References 16 publications

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

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

Uniqueness of weak solutions for the general Ericksen-Leslie system with Ginzburg-Landau penalization in T^2

Uniqueness of global weak solutions for the general Ericksen–Leslie system with Ginzburg–Landau penalization in $${\mathbb {T}}^2$$

Contact Info

Product

Resources

About