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Dynamics of the Ericksen–Leslie equations with general Leslie stress I: the incompressible isotropic case

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Abstract

The Ericksen–Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress tensor is studied in the case of a non-isothermal and incompressible fluid. This system is shown to be locally strongly well-posed in the \(L_p\)-setting, and a dynamical theory is developed. The equilibria are identified and shown to be normally stable. In particular, a local solution extends to a unique, global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold. In this case, the solution converges exponentially to an equilibrium, in the topology of the state manifold. The above results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi’s relation.

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Authors and Affiliations

  1. Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgarten-Strasse 7, 64289, Darmstadt, Germany

    Matthias Hieber

  2. 607 Benedum Engineering Hall, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Matthias Hieber

  3. Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, Theodor-Lieser-Strasse 5, 06120, Halle, Germany

    Jan Prüss

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  1. Matthias Hieber
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Correspondence to Matthias Hieber.

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Hieber, M., Prüss, J. Dynamics of the Ericksen–Leslie equations with general Leslie stress I: the incompressible isotropic case. Math. Ann. 369, 977–996 (2017). https://doi.org/10.1007/s00208-016-1453-7

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  • DOI: https://doi.org/10.1007/s00208-016-1453-7

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