A Molecular Kinetic Theory of Inhomogeneous Liquid Crystal Flow and the Small Deborah Number Limit

Weinan, E; Zhang, Pingwen

doi:10.4310/maa.2006.v13.n2.a5

Cited by 28 publications

(20 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Wang, E, Liu, and Zhang [25] set up a formalism in which the interaction between molecules is treated more directly using the positionorientation distribution function via interaction potentials. They extended the free energy (1.4) to include the effects of nonlocal intermolecular interactions through an interaction potential as follows: In this paper, we take the following form as in [7,28]:…”

Section: The Doi-onsager Theorymentioning

confidence: 99%

See 1 more Smart Citation

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Wang

Zhang

2014

Comm Pure Appl Math

Self Cite

View full text Add to dashboard Cite

Abstract. We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the EricksenLeslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional. We use x ∈ Ω ⊆ R 3 to denote the material point and f (x, m, t) to represent the number density for the number of molecules whose orientation is parallel to m at point x and time t. where De is the Deborah number, R is the rotational gradient operator(see Section 3), κ is a constant velocity gradient, and U is the mean-field interaction potential. Onsager [18]

show abstract

Section: The Doi-onsager Theorymentioning

confidence: 99%

“…Now 1 will be determined by (7.3), whose existence is ensured by Lemma 7.2. Once 1 is determined, it can be proved that equation (7.4) is equivalent to (2.11); see [7] for the details. Now we solve…”

Section: Hilbert Expansionmentioning

confidence: 99%

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Wang

Zhang

2014

Comm Pure Appl Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…All of these works contain a large number of unknown parameters which in general can not be determined rationally. At this stage, E and Zhang [12] (cf. [32]) established a new model for inhomogeneous flows of LCPs in R 3 with a few adjustable parameters which can model a variety of configurations of polymeric liquid crystal molecules.…”

mentioning

confidence: 98%

“…Subsequently, Wang, Zhang and Zhang [34] provided a rigorous derivation of the Ericksen-Leslie equations starting from the Doi-Onsager equations by the Hilbert expansion method. The model of E and Zhang [12] takes the form:…”

mentioning

confidence: 99%

Global existence of weak solutions to inhomogeneous Doi-Onsager equations

Chen¹,

Zhou²

2022

DCDS-B

View full text Add to dashboard Cite

In this paper, we study the inhomogeneous Doi-Onsager equations with a special viscous stress. We prove the global existence of weak solutions in the case of periodic regions without considering the effect of the constraint force arising from the rigidity of the rods. The key ingredient is to show the convergence of the nonlinear terms, which can be reduced to proving the strong compactness of the moment of the family of number density functions. The proof is based on the propagation of strong compactness by studying a transport equation for some defect measure, L2-estimates for a family of number density functions, and energy dissipation estimates.

show abstract

“…The first is the molecular kinetic theory, and the latter two are the continuum theory. Kuzuu & Doi[40] and Weinan & Zhang[41] have formally derived the Ericksen-Leslie equation from the Doi-Onsager equation by taking small Deborah number limit. Wang et al[42] have rigorously justified this formal derivation before the first singular time of the Ericksen-Leslie equation.…”

mentioning

confidence: 99%

Recent developments of analysis for hydrodynamic flow of nematic liquid crystals

Lin

Wang

2014

Phil. Trans. R. Soc. A.

115

View full text Add to dashboard Cite

The study of hydrodynamics of liquid crystals leads to many fascinating mathematical problems, which has prompted various interesting works recently. This article reviews the static Oseen-Frank theory and surveys some recent progress on the existence, regularity, uniqueness and large time asymptotic of the hydrodynamic flow of nematic liquid crystals. We will also propose a few interesting questions for future investigations.

show abstract

A Molecular Kinetic Theory of Inhomogeneous Liquid Crystal Flow and the Small Deborah Number Limit

Cited by 28 publications

References 21 publications

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Global existence of weak solutions to inhomogeneous Doi-Onsager equations

Recent developments of analysis for hydrodynamic flow of nematic liquid crystals

Contact Info

Product

Resources

About