Dynamics of nematic liquid crystal flows: The quasilinear approach

Hieber, Matthias; Nesensohn, Manuel; Prüss, Jan; Schade, Katharina

doi:10.1016/j.anihpc.2014.11.001

Hieber, Matthias ; Nesensohn, Manuel ; Prüss, Jan ; Schade, Katharina

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 397-408.

Résumé
Abstract

On considère le modèle de Leslie–Ericksen pour les cristaux liquides nématiques dans un domaine borné $Ω \subset R^{n}$ . On obtient une théorie dynamique complète pour ce système, analysé comme une équation d'évolution quasi-linéare dans le cadre $L^{p} - L^{q}$ . En particulier, on démontre l' existence et l'unicité locales d'une solution forte, qui s'étend en un solution forte globale si les conditions initiales sont près d'un équilibre. De plus, on montre que la solution est analytique réelle en espace et temps.

Consider the (simplified) Leslie–Ericksen model for the flow of nematic liquid crystals in a bounded domain $Ω \subset R^{n}$ for $n > 1$ . This article develops a complete dynamic theory for these equations, analyzing the system as a quasilinear parabolic evolution equation in an $L_{p} - L_{q}$ -setting. First, the existence of a unique local strong solution is proved. This solution extends to a global strong solution, provided the initial data are close to an equilibrium or the solution is eventually bounded in the natural norm of the underlying state space. In this case the solution converges exponentially to an equilibrium. Moreover, the solution is shown to be real analytic, jointly in time and space.

MR Zbl

DOI : 10.1016/j.anihpc.2014.11.001

Classification : 35Q35, 76A15, 76D03, 35K59
Mots clés : Nematic liquid crystals, Quasilinear parabolic evolution equations, Regularity, Global solutions, Convergence to equilibria

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     title = {Dynamics of nematic liquid crystal flows: {The} quasilinear approach},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {397--408},
     publisher = {Elsevier},
     volume = {33},
     number = {2},
     year = {2016},
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     zbl = {1334.35235},
     mrnumber = {3465380},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.11.001/}
}

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Hieber, Matthias; Nesensohn, Manuel; Prüss, Jan; Schade, Katharina. Dynamics of nematic liquid crystal flows: The quasilinear approach. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 397-408. doi : 10.1016/j.anihpc.2014.11.001. http://www.numdam.org/articles/10.1016/j.anihpc.2014.11.001/

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