On the Eringen model for nematic liquid crystals

Gregory A. Chechkin; Tudor S. Ratiu; Maxim S. Romanov

doi:10.5802/crmeca.67

Short paper

On the Eringen model for nematic liquid crystals

Gregory A. Chechkin ^1,^2,³ ; Tudor S. Ratiu ^4,^5,⁶; Maxim S. Romanov ⁷

¹ Institute of Mathematics and Mathematical Modeling, Pushkin st. 125, Almaty, 050010, Kazakhstan
² Institute of Mathematics with Computing Center—Subdivision of the Ufa Federal Research Center of Russian Academy of Science, Chernyshevskogo St., 112, Ufa, 450008, Russia
³ M. V. Lomonosov Moscow State University, Leninskie Gory, 1, Moscow, 119991, Russia
⁴ School of Mathematical Sciences and Ministry of Education Laboratory of Scientific Computing (MOE-LSC), Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang, Shanghai, 200240, China
⁵ Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland
⁶ Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
⁷ M.V. Lomonosov Moscow State University, Leninskie Gory, 1, Moscow, 119991, Russia

Comptes Rendus. Mécanique, Volume 349 (2021) no. 1, pp. 21-27.

Abstract
Résumé

We introduce the three-dimensional Eringen system of equations for the nematodynamics of liquid crystals, announce the short time existence and uniqueness of strong solutions for the one-dimensional problem in the periodic case, and show the continuous dependence of the solution on the initial data.

Nous présentons le système tridimensionnel d’équations d’Eringen pour la nématodynamique des cristaux liquides, annonçons l’existence en temps et l’unicité de solutions fortes pour le problème unidimensionnel dans le cas périodique et montrons la dépendance continue de la solution sur les données initiales.

Received: 2020-12-13
Revised: 2020-12-14
Accepted: 2020-12-14
Published online: 2021-02-05

DOI: 10.5802/crmeca.67

Keywords: Liquid crystals, Eringen equations, Nematodynamics, Existence and uniqueness, Conservation laws, Micromomentum of molecules, Local solution
Mots-clés : Cristaux liquides, Équations d’Eringen, Nematodynamique, Existence et unicité, Lois de conservation, Micromomentum de molécules, Solution locale

Author's affiliations:

Gregory A. Chechkin ^{1, 2, 3}; Tudor S. Ratiu ^{4, 5, 6}; Maxim S. Romanov ⁷

¹ Institute of Mathematics and Mathematical Modeling, Pushkin st. 125, Almaty, 050010, Kazakhstan
² Institute of Mathematics with Computing Center—Subdivision of the Ufa Federal Research Center of Russian Academy of Science, Chernyshevskogo St., 112, Ufa, 450008, Russia
³ M. V. Lomonosov Moscow State University, Leninskie Gory, 1, Moscow, 119991, Russia
⁴ School of Mathematical Sciences and Ministry of Education Laboratory of Scientific Computing (MOE-LSC), Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang, Shanghai, 200240, China
⁵ Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland
⁶ Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
⁷ M.V. Lomonosov Moscow State University, Leninskie Gory, 1, Moscow, 119991, Russia

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Gregory A. Chechkin and Tudor S. Ratiu and Maxim S. Romanov},
     title = {On the {Eringen} model for nematic liquid crystals},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {21--27},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {349},
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     year = {2021},
     doi = {10.5802/crmeca.67},
     language = {en},
}

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Gregory A. Chechkin; Tudor S. Ratiu; Maxim S. Romanov. On the Eringen model for nematic liquid crystals. Comptes Rendus. Mécanique, Volume 349 (2021) no. 1, pp. 21-27. doi : 10.5802/crmeca.67. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.67/

References
Cited by

[1] G. A. Chechkin; T. S. Ratiu; M. S. Romanov; V. N. Samokhin On the existence and uniqueness theorem in 2D nematodynamics. Finite speed of propagation of perturbations, Russ. Acad. Sci. Dokl. Math., Volume 91 (2015) no. 3, pp. 354-358 (Translated from Dokl. Akad. Nauk.462 (2015), no. 5, p. 519-523) | Zbl

[2] G. A. Chechkin; T. S. Ratiu; M. S. Romanov; V. N. Samokhin On unique solvability of the full three-dimensional Ericksen–Leslie system, C. R. Méc., Volume 344 (2016) no. 7, pp. 459-463 | DOI

[3] G. A. Chechkin; T. S. Ratiu; M. S. Romanov; V. N. Samokhin Existence and uniqueness theorems for the two-dimensional Ericksen–Leslie system, J. Math. Fluid Mech., Volume 18 (2016), pp. 571-589 | DOI | MR | Zbl

[4] G. A. Chechkin; T. P. Chechkina; T. S. Ratiu; M. S. Romanov Nematodynamics and random homogenization, Appl. Anal., Volume 95 (2016) no. 10, pp. 2243-2253 | DOI | MR | Zbl

[5] G. A. Chechkin; T. S. Ratiu; M. S. Romanov; V. N. Samokhin Existence and uniqueness theorems for the full three-dimensional Ericksen–Leslie system, Math. Models Methods Appl. Sci. (M $^{3}$ AS), Volume 27 (2017) no. 5, pp. 807-843 | DOI | MR | Zbl

[6] A. C. Eringen A unified continuum theory of liquid crystals, ARI - Int. J. Phys. Eng. Sci., Volume 50 (1997), pp. 73-84 | MR | Zbl

[7] A. C. Eringen Microcontinium Field Theories. Volume II. Fluent Media, Springer-Verlag, New York, 2001

[8] F. Gay-Balmaz; T. S. Ratiu The geometric structure of complex fluids, Adv. Appl. Math., Volume 42 (2009), pp. 176-275 | DOI | MR | Zbl

[9] O. A. Ladyzhenskaja; V. A. Solonnikov; N. N. Ural’ceva Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968 | MR

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