Advertisement

Twisted Solutions to a Simplified Ericksen–Leslie Equation

  • Yuan Chen
  • Soojung Kim
  • Yong Yu
Article
  • 42 Downloads

Abstract

In this article we construct global solutions to a simplified Ericksen–Leslie system on \({\mathbb{R}^3}\). The constructed solutions are twisted and periodic along the x3-axis with period \({d = 2\pi \big/ \mu}\). Here \({\mu > 0}\) is the twist rate and d is the distance between two planes which are parallel to the x1x2-plane. Liquid crystal material is placed in the region enclosed by these two planes. Given a well-prepared initial data, our solutions exist classically for all \({t \in (0, \infty)}\). However, these solutions become singular at all points on the x3-axis and escape into third dimension exponentially while \({t \rightarrow \infty}\). An optimal blow up rate is also obtained.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding

The third author is partially supported by RGC Grants Nos. 14306414 and 409613.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Angenent, S., Hulshof, J.: Singularities at \(t = \infty \) in Equivariant Harmonic Map Flow, Geometric Evolution Equations. Contemporary Mathematics, vol. 367, pp. 1–15. American Mathematical Society, Providence, 2005.  https://doi.org/10.1090/conm/367 Google Scholar
  2. 2.
    Chang N.-H., Shatah J., Uhlenbeck K.: Schrödinger maps. Commun. Pure Appl. Math. 53(5), 590–602 (2000)  https://doi.org/10.1002/(SICI1097-0312(20000553:5%3c590::AID-CPA2%3e3.0.CO;2-R CrossRefGoogle Scholar
  3. 3.
    Chen, Y., Yu, Y.: Global Solutions of Nematic Crystal Flows in Dimension Two (preprint)Google Scholar
  4. 4.
    Chen Y., Yu Y.: Global m-equivariant solutions of nematic liquid crystal flows in dimension two. Arch. Ration. Mech. Anal. 226(2), 767–808 (2017)  https://doi.org/10.1007/s00205-017-1144-x MathSciNetCrossRefGoogle Scholar
  5. 5.
    Guan M., Gustafson S., Tsai T.-P.: Global existence and blow-up for harmonic map heat flow. J. Differ. Equ. 246(1), 1–20 (2009)  https://doi.org/10.1016/j.jde.2008.09.011 ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Gustafson S., Kang K., Tsai T.-P.: Asymptotic stability of harmonic maps under the Schrödinger flow. Duke Math. J. 145(3), 537–583 (2008)  https://doi.org/10.1215/00127094-2008-058 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gustafson, S., Nakanishi, K., Tsai, T.-P.: Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on \(\mathbb{R}^2\). Commun. Math. Phys. 300, 205–242, 2010.  https://doi.org/10.1007/s00220-010-1116-6 ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Huang T., Lin F.H., Liu C., Wang C.Y.: Finite time singularity of the nematic liquid crystal flow in dimension three. Arch. Ration. Mech. Anal. 221, 1223–1254 (2016)  https://doi.org/10.1007/s00205-016-0983-1 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lin F.H.:: On nematic liquid crystals with variable degree of orientation. Commun. Pure Appl. Math. 44(4), 453–468 (1991)  https://doi.org/10.1002/cpa.3160440404 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lin F.-H., Lin J., Wang C.Y.: Liquid crystal flows in two dimensions. Arch. Ration. Mech. Anal. 197(1), 297–336 (2010)  https://doi.org/10.1007/s00205-009-0278-x MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lin, F.H., Wang, C.Y.: Recent developments of analysis for hydrodynamic flow of nematic liquid crystals. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2029), 20130361, 18 pp, 2014.  https://doi.org/10.1098/rsta.2013.0361 ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Raphaël P., Schweyer R.: Stable blow-up dynamics for the 1-corotational energy critical harmonic heat flow. Commun. Pure Appl. Math. 66(3), 414–480 (2013)  https://doi.org/10.1002/cpa.21435 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Schonbek M.E.: L 2 decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal., 88(3), 209–222 (1985)  https://doi.org/10.1007/BF00752111 CrossRefGoogle Scholar
  14. 14.
    Stewart, I.W.: The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction. The Liquid Crystal Book Series. Taylor and Francis, Milton Park (2004)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.School of MathematicsKorea Institute for Advanced StudySeoulKorea
  3. 3.Department of MathematicsThe Chinese University of Hong KongShatin, N.T.Hong Kong

Personalised recommendations