Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

Laux, Tim; Liu, Yuning

doi:10.1007/s00205-021-01681-0

Cited by 12 publications

(12 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is something never exists in the scalar case or several vector-valued cases studied by various authors before. It also illustrate why there are different types of boundary conditions along the interface in earlier works [16,17,8,19,18], for examples. Let us describe it roughly below.…”

mentioning

confidence: 78%

“…Obviously, F is fully minimally paired. For the isotropic-nematic phase transition problem in liquid crystals [8,18,19], the energy F : Q Ñ R (Q denotes the space of 3 ˆ3 symmetric trace free matrices) takes the form:…”

Section: 2mentioning

confidence: 99%

“…In [8], without the radially symmetric assumption, Fei-Wang-Zhang-Zhang rigorously justified the asymptotic limit for a physical R 3ˆ3 -valued model, which describes the isotropicnematic phase transition for liquid crystals (the radially symmetric case is studied in [22]). For the latter problem, we refer to [18,19] for more recent progresses. There are several interesting studies for related problems, see for examples, [4,7].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem

Fei¹,

Lin²,

Wang³

et al. 2021

Preprint

View full text Add to dashboard Cite

In this paper, we consider the sharp interface limit of a matrix-valued Allen-Cahn equation, which takes the form:We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in O `pnq and O ´pnq (represent the sets of n ˆn orthogonal matrices with determinant `1 and ´1 respectively); on the interface, the phase matrices in two sides satisfy a novel mixed boundary condition.The above result provides a solution to the Keller-Rubinstein-Sternberg's (conjecture) problem in this Opnq setting. Our proof relies on two key ingredients. First, in order to construct the approximate solution by matched asymptotic expansions as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible "degenerations" or higher dimensional kernels for the linearized operators due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which is accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.

show abstract

mentioning

confidence: 78%

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem

Fei¹,

Lin²,

Wang³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…) dx (see also [19] for a subsequent application of the relative energy method to a problem in the context of liquid crystals).…”

Section: Strategy Of the Proofmentioning

confidence: 99%

Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

Fischer¹,

Marveggio²

2022

Preprint

View full text Add to dashboard Cite

Phase-field models such as the Allen-Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen-Cahn equation with a potential with N ≥ 3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow.In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: As long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen-Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter ε 0. We even establish the rate of convergence O(ε 1/2 ). Our approach is based on the gradient flow structure of the Allen-Cahn equation and its limiting motion: Building on the recent concept of "gradient flow calibrations" for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen-Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e. g. avoiding the need for a stability analysis of the Allen-Cahn operator or additional convergence hypotheses for the energy at positive times.

show abstract

“…One of the main advantages of the method is its simplicity and its applicability in vectorial problems as it does not require a spectral analysis of the linearized Allen-Cahn operator and is not based on the comparison principle. Liu and the author [21] combined the relative entropy method with weak convergence methods to derive the sharp-interface dynamics of isotropic-nematic phase transitions in liquid crystals. Most recently, Fischer and Marveggio [9] extended the result [8] to the vector-valued Allen-Cahn equation and proved its convergence to multiphase mean curvature flow.…”

Section: Introductionmentioning

confidence: 99%

Weak-strong uniqueness for volume-preserving mean curvature flow

Laux¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

In this note, we derive a stability and weak-strong uniqueness principle for volume-preserving mean curvature flow. The proof is based on a new notion of volumepreserving gradient flow calibrations, which is a natural extension of the concept in the case without volume preservation recently introduced by Fischer et al. [arXiv:2003.05478]. The first main result shows that any strong solution with certain regularity is calibrated. The second main result consists of a stability estimate in terms of a relative entropy, which is valid in the class of distributional solutions to volume-preserving mean curvature flow.

show abstract

Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

Cited by 12 publications

References 24 publications

Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem

Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem

Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

Weak-strong uniqueness for volume-preserving mean curvature flow

Contact Info

Product

Resources

About