JournalsaihpcVol. 33, No. 2pp. 397–408

Dynamics of nematic liquid crystal flows: The quasilinear approach

  • Matthias Hieber

    Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgarten-Strasse 7, D-64289 Darmstadt, Germany, University of Pittsburgh, 607 Benedum Engineering Hall, Pittsburgh, PA 15261, USA
  • Manuel Nesensohn

    Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgarten-Strasse 7, D-64289 Darmstadt, Germany
  • Jan Prüss

    Martin-Luther-Universität Halle-Wittenberg, Institut für Mathematik, Theodor-Lieser-Strasse 5, D-06120 Halle, Germany
  • Katharina Schade

    Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgarten-Strasse 7, D-64289 Darmstadt, Germany
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Abstract

Consider the (simplified) Leslie–Ericksen model for the flow of nematic liquid crystals in a bounded domain ΩRn\Omega \subset \mathbb{R}^{n} for n>1n > 1. This article develops a complete dynamic theory for these equations, analyzing the system as a quasilinear parabolic evolution equation in an LpLqL_{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.

Résumé

On considère le modèle de Leslie–Ericksen pour les cristaux liquides nématiques dans un domaine borné ΩRn\Omega \subset \mathbb{R}^{n}. On obtient une théorie dynamique complète pour ce système, analysé comme une équation d'évolution quasi-linéare dans le cadre LpLqL^{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.

Cite this article

Matthias Hieber, Manuel Nesensohn, Jan Prüss, Katharina Schade, Dynamics of nematic liquid crystal flows: The quasilinear approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 2, pp. 397–408

DOI 10.1016/J.ANIHPC.2014.11.001