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.
References
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
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
Chen, Y., Yu, Y.: Global Solutions of Nematic Crystal Flows in Dimension Two (preprint)
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
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
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
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
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
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
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
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
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
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
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)
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The third author is partially supported by RGC Grants Nos. 14306414 and 409613.
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Communicated by N. Masmoudi
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Chen, Y., Kim, S. & Yu, Y. Twisted Solutions to a Simplified Ericksen–Leslie Equation. Arch Rational Mech Anal 232, 303–336 (2019). https://doi.org/10.1007/s00205-018-1321-6
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DOI: https://doi.org/10.1007/s00205-018-1321-6