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On the Eringen model for nematic liquid crystals
[Sur le modèle d’Eringen pour les cristaux liquides nématiques]
Gregory A. Chechkin1, 2, 3  ; Tudor S. Ratiu4, 5, 6 ; Maxim S. Romanov7
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.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmeca.67
Keywords: Liquid crystals, Eringen equations, Nematodynamics, Existence and uniqueness, Conservation laws, Micromomentum of molecules, Local solution
Mot clés : Cristaux liquides, Équations d’Eringen, Nematodynamique, Existence et unicité, Lois de conservation, Micromomentum de molécules, Solution locale
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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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/

[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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