The small Deborah number limit of the Doi–Onsager equation without hydrodynamics

Abstract

We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals without hydrodynamics. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position x ∈ R d (d = 2, 3) and orientation vector m ∈ S 2 (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into S 2 . This flow is a special case of the gradient flow to the Oseen-Frank energy functional for nematic liquid crystals. The key ingredient is to show the strong compactness of the family of number density functions and the proof relies on the strong compactness of the corresponding second moment (or the Q-tensor), a spectral decomposition of the linearized operator near the limit local equilibrium distribution, as well as the energy dissipation estimate.Date: September 17, 2018. 1 We refer to the book by de Gennes-Prost [5] for physics of this theory.1.2. From microscopic theories to macroscopic theories for liquid crystals. Exploring the connections between different theories for liquid crystal flow is a fundamental issue in liquid crystal studies. first derived the Ericksen-Leslie equations and determined the Leslie coefficients from the Doi-Onsager equation under the small Deborah number limit. However, the Ericksen stress was missing. E-Zhang [9] extended Kuzuu and Doi's formal derivation to the inhomogeneous case and the Ericksen stress was obtained from an extra introduced body force. Roughly speaking, E and Zhang showed that the solution (f, v) of (1.5)-(1.6) with De = ǫ has a formal expansionIt can be verified using |n(x, t)| ≡ 1 that, if a weak solution n(x, t) is smooth, then it fulfills (∂ t n − Λ∆n) ∧ n = 0 and this is equivalent to (1.9).Remark 1.2. The first part of Theorem 1.1 is concerned with the wellposedness of (1.10), which is proved in the beginning of Section 4. Although these issues can be discussed under much more relaxed assumptions on the interaction potential (1.3) as well as the initial data, for the sake of investigating the scaling limit, we restrict ourselves to the inhomogeneous Maier-Saupe potential defined by (1.11) and initial data near the local equilibria, which include local equilibrium distributions as especial cases. More precisely, if n ǫ (x) : R d → S 2 fulfills n ǫ − e 0 H 1 (R d ) ≤ C for some C independent of ǫ and for some e 0 ∈ S 2 , then f in ǫ (m, x) = 1 Z e η(m·nǫ(x)) 2 satisfies (1.22). Remark 1.3. We will give a more detailed discussion on assumptions (1.21) in Section 2.1.

Modelling and computation of liquid crystals

“…To resolve this problem, one needs to study the relations between weak solutions. Liu and Wang (2018c)…”

Mathematical problems of nematic liquid crystals: between dynamical and stationary problems

“…The study of the similar limit in the dynamical problems has generated several notable results such as [52] (QS to EL), [106] (LDG to EL), [107] (LDG to harmonic maps), [103,[108][109][110][111] (molecular theories to OF). However, the presence of the flow adds formidable difficulties to the study of this limit for weak solution, see in particular in the relaxed EL problem context the Open Problem 2.4 in [58].…”

Modeling and Computation of Liquid Crystals

“…To resolve this problem, one needs to study the relations between weak solutions. Liu and Wang (2018c) gives an attempt on this issue, where it is proved that, the solutions to the Doi-Onsager equation without hydrodynamics converges to the weak solution of the harmonic map heat flow-the gradient flow of the one-constant Oseen-Frank energy.…”