Blowup Rate Estimates of a Singular Potential and Its Gradient in the Landau-de Gennes Theory

Xin Yang Lu; Xiang Xu; Wujun Zhang

doi:10.1007/s00332-021-09761-x

Blowup Rate Estimates of a Singular Potential and Its Gradient in the Landau-de Gennes Theory

Xin Yang Lu, Xiang Xu, Wujun Zhang

School of Arts and Sciences, Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

In this paper, we revisit a singular bulk potential in the Landau-de Gennes free energy that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. This Maier–Saupe type singular potential was originally introduced in Katriel et al. (Mol Cryst Liquid Cryst 1:337–355, 1986), which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. Specifically, we establish blowup rates of both this singular potential and its gradient as Q approaches its physical boundary. All of the proofs are elementary.

Original language	English (US)
Article number	6
Journal	Journal of Nonlinear Science
Volume	32
Issue number	1
DOIs	https://doi.org/10.1007/s00332-021-09761-x
State	Published - Feb 2022

All Science Journal Classification (ASJC) codes

Modeling and Simulation
Engineering(all)
Applied Mathematics

Keywords

35B44
35Q82
Blowup rate
Liquid crystals
Q-tensor
Singular potential

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10.1007/s00332-021-09761-x

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title = "Blowup Rate Estimates of a Singular Potential and Its Gradient in the Landau-de Gennes Theory",

abstract = "In this paper, we revisit a singular bulk potential in the Landau-de Gennes free energy that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. This Maier–Saupe type singular potential was originally introduced in Katriel et al. (Mol Cryst Liquid Cryst 1:337–355, 1986), which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. Specifically, we establish blowup rates of both this singular potential and its gradient as Q approaches its physical boundary. All of the proofs are elementary.",

keywords = "35B44, 35Q82, Blowup rate, Liquid crystals, Q-tensor, Singular potential",

author = "Lu, {Xin Yang} and Xiang Xu and Wujun Zhang",

note = "Funding Information: We would like to thank Professor John Ball for his kind discussions, especially pointing out several useful references to us. We thank the anonymous referees for their suggestions to improve the quality of our paper. X. Y. Lu{\textquoteright}s work is supported by his NSERC Discovery Grant “Regularity of minimizers and pattern formation in geometric minimization problems.” X. Xu{\textquoteright}s work is supported by the NSF Grant DMS-2007157 and the Simons Foundation Grant No. 635288. W. J. Zhang{\textquoteright}s work is supported by NSF Grant DMS-1818861. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.",

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doi = "10.1007/s00332-021-09761-x",

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AU - Lu, Xin Yang

AU - Xu, Xiang

AU - Zhang, Wujun

N1 - Funding Information: We would like to thank Professor John Ball for his kind discussions, especially pointing out several useful references to us. We thank the anonymous referees for their suggestions to improve the quality of our paper. X. Y. Lu’s work is supported by his NSERC Discovery Grant “Regularity of minimizers and pattern formation in geometric minimization problems.” X. Xu’s work is supported by the NSF Grant DMS-2007157 and the Simons Foundation Grant No. 635288. W. J. Zhang’s work is supported by NSF Grant DMS-1818861. Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

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N2 - In this paper, we revisit a singular bulk potential in the Landau-de Gennes free energy that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. This Maier–Saupe type singular potential was originally introduced in Katriel et al. (Mol Cryst Liquid Cryst 1:337–355, 1986), which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. Specifically, we establish blowup rates of both this singular potential and its gradient as Q approaches its physical boundary. All of the proofs are elementary.

AB - In this paper, we revisit a singular bulk potential in the Landau-de Gennes free energy that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. This Maier–Saupe type singular potential was originally introduced in Katriel et al. (Mol Cryst Liquid Cryst 1:337–355, 1986), which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. Specifically, we establish blowup rates of both this singular potential and its gradient as Q approaches its physical boundary. All of the proofs are elementary.

KW - 35B44

KW - 35Q82

KW - Blowup rate

KW - Liquid crystals

KW - Q-tensor

KW - Singular potential

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Blowup Rate Estimates of a Singular Potential and Its Gradient in the Landau-de Gennes Theory

Abstract

All Science Journal Classification (ASJC) codes

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