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On the General Ericksen–Leslie System: Parodi’s Relation, Well-Posedness and Stability

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Abstract

In this paper we investigate the role of Parodi’s relation in the well-posedness and stability of the general Ericksen–Leslie system modeling nematic liquid crystal flows. First, we give a formal physical derivation of the Ericksen–Leslie system through an appropriate energy variational approach under Parodi’s relation, in which we can distinguish the conservative/dissipative parts of the induced elastic stress. Next, we prove global well-posedness and long-time behavior of the Ericksen–Leslie system under the assumption that the viscosity μ 4 is sufficiently large. Finally, under Parodi’s relation, we show the global well-posedness and Lyapunov stability for the Ericksen–Leslie system near local energy minimizers. The connection between Parodi’s relation and linear stability of the Ericksen–Leslie system is also discussed.

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Authors and Affiliations

  1. School of Mathematical Sciences and Shanghai Key, Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

    Hao Wu

  2. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Xiang Xu

  3. Department of Mathematics, Penn State University, State College, PA, 16802, USA

    Chun Liu

Authors
  1. Hao Wu
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  2. Xiang Xu
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  3. Chun Liu
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Corresponding author

Correspondence to Xiang Xu.

Additional information

Communicated by F. Lin

Hao Wu was partially supported by the NSF of China 11001058, Specialized Research Fund for the Doctoral Program of Higher Education and the “Chen Guang” project supported by the Shanghai Municipal Education Commission and the Shanghai Education Development Foundation. Chun Liu and Xiang Xu were partially supported by NSF grants DMS-0707594 and DMS-1109107.

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Wu, H., Xu, X. & Liu, C. On the General Ericksen–Leslie System: Parodi’s Relation, Well-Posedness and Stability. Arch Rational Mech Anal 208, 59–107 (2013). https://doi.org/10.1007/s00205-012-0588-2

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