A note on a global strong solution to the 2D Cauchy problem of density-dependent nematic liquid crystal flows with vacuum

Xin Zhong

doi:10.1016/j.crma.2018.04.011

Partial differential equations

A note on a global strong solution to the 2D Cauchy problem of density-dependent nematic liquid crystal flows with vacuum
[Sur la solution forte globale du problème de Cauchy pour l'écoulement d'un cristal liquide nématique bidimensionnel, dépendant de la densité et avec vide]

Présenté par : the Editorial Board

Xin Zhong ¹

¹ School of Mathematics and Statistics, Southwest University, Chongqing 400715, People's Republic of China

Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 503-508.

Résumé
Abstract

Dans Li–Liu–Zhong (Nonlinearity 30 (2017) 4062–4088), les auteurs démontrent l'existence d'une unique solution forte globale au problème de Cauchy pour l'écoulement d'un cristal liquide nématique, incompressible, non homogène, bidimensionnel, avec vide. Ce résultat est valide dans la mesure où la densité initiale donnée et le gradient de dérive d'orientation ne sont pas trop lents à l'infini et l'énergie de base ${‖ \sqrt{ρ_{0}} u_{0} ‖}_{L^{2}}^{2} + {‖ \nabla d_{0} ‖}_{L^{2}}^{2}$ est petite. Le but de la présente Note est d'expliciter précisément cette dernière condition de petitesse.

In Li–Liu–Zhong (Nonlinearity 30 (2017) 4062–4088), the authors proved the existence of a unique global strong solution to the Cauchy problem of 2D nonhomogeneous incompressible nematic liquid crystal flows with vacuum as far-field density provided the initial data density and the gradient of orientation decay not too slow at infinity, and the basic energy ${‖ \sqrt{ρ_{0}} u_{0} ‖}_{L^{2}}^{2} + {‖ \nabla d_{0} ‖}_{L^{2}}^{2}$ is small. In this note, we aim at precisely describing this smallness condition.

Reçu le : 2018-01-10
Accepté le : 2018-04-03
Publié le : 2018-04-16

DOI : 10.1016/j.crma.2018.04.011

Affiliations des auteurs :

Xin Zhong ¹

¹ School of Mathematics and Statistics, Southwest University, Chongqing 400715, People's Republic of China

@article{CRMATH_2018__356_5_503_0,
     author = {Xin Zhong},
     title = {A note on a global strong solution to the {2D} {Cauchy} problem of density-dependent nematic liquid crystal flows with vacuum},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {503--508},
     publisher = {Elsevier},
     volume = {356},
     number = {5},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.011},
     language = {en},
}

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Xin Zhong. A note on a global strong solution to the 2D Cauchy problem of density-dependent nematic liquid crystal flows with vacuum. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 503-508. doi : 10.1016/j.crma.2018.04.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.011/

Bibliographie
Cité par

[1] J.L. Ericksen Hydrostatic theory of liquid crystal, Arch. Ration. Mech. Anal., Volume 9 (1962), pp. 371-378

[2] F. Leslie Theory of flow phenomenum in liquid crystals, The Theory of Liquid Crystals, vol. 4, Academic Press, London–New York, 1979, pp. 1-81

[3] L. Li; Q. Liu; X. Zhong Global strong solutions to the two-dimensional density-dependent nematic liquid crystal flows with vacuum, Nonlinearity, Volume 30 (2017), pp. 4062-4088

[4] X. Li; D. Wang Global strong solution to the density-dependent incompressible flow of liquid crystals, Trans. Amer. Math. Soc., Volume 367 (2015), pp. 2301-2338

[5] F. Lin; J. Lin; C. Wang Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., Volume 197 (2010), pp. 297-336

[6] F. Lin; C. Wang Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, Commun. Pure Appl. Math., Volume 69 (2016), pp. 1532-1571

[7] Q. Liu; S. Liu; W. Tan; X. Zhong Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows with vacuum, J. Differ. Equ., Volume 261 (2016), pp. 6521-6569

Cité par Sources :

^☆ Supported by the Postdoctoral Science Foundation of Chongqing (No. xm2017015), China Postdoctoral Science Foundation (No. 2017M610579), Fundamental Research Funds for the Central Universities (No. XDJK2017C050), and the Doctoral Fund of Southwest University (No. SWU116033).

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