Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-31T07:43:14.525Z Has data issue: false hasContentIssue false
  • English
  • Français

The solvability for a nonlinear degenerate hyperbolic–parabolic coupled system arising from nematic liquid crystals

Part of: Equations of mathematical physics and other areas of application Hyperbolic equations and systems Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  13 October 2023

Yanbo Hu Open the ORCID record for Yanbo Hu [Opens in a new window]
Yanbo Hu*
Affiliation:
Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou 310023, P.R. China and School of Mathematics, Hangzhou Normal University, Hangzhou 311121, P.R. China
*
e-mail: yanbo.hu@hotmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper focuses on the Cauchy problem for a one-dimensional quasilinear hyperbolic–parabolic coupled system with initial data given on a line of parabolicity. The coupled system is derived from the Poiseuille flow of full Ericksen–Leslie model in the theory of nematic liquid crystals, which incorporates the crystal and liquid properties of the materials. The main difficulty comes from the degeneracy of the hyperbolic equation, which makes that the system is not continuously differentiable and then the classical methods for the strictly hyperbolic–parabolic coupled systems are invalid. With a choice of a suitable space for the unknown variable of the parabolic equation, we first solve the degenerate hyperbolic problem in a partial hodograph plane and express the smooth solution in terms of the original variables. Based on the smooth solution of the hyperbolic equation, we then construct an iterative sequence for the unknown variable of the parabolic equation by the fundamental solution of the heat equation. Finally, we verify the uniform convergence of the iterative sequence in the selected function space and establish the local existence and uniqueness of classical solutions to the degenerate coupled problem.

Keywords

Nematic liquid crystalshyperbolic–parabolic coupled systemdegenerate hyperbolicCauchy problemclassical solution

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics 35L80: Degenerate hyperbolic equations 35K05: Heat equation 35A09: Classical solutions
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 68
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The work was supported by the NSF of China (Grant Nos. grants 12171130 and 12071106) and the NSF of Zhejiang province of China (Grant No. LY21A010017).

References

Adlem, K., Čopiv, M., Luckhurst, G., et al., Chemically induced twist-bend nematic liquid crystals, liquid crystal dimers, and negative elastic constants . Phys. Rev. E. 88(2013), no. 2, 022503.CrossRefGoogle ScholarPubMed
Bressan, A. and Chen, G., Lipschitz metrics for a class of nonlinear wave equations . Arch. Rat. Mech. Anal. 226(2017), no. 3, 1303–1343.CrossRefGoogle Scholar
Bressan, A. and Chen, G., Generic regularity of conservative solutions to a nonlinear wave equation . Ann. I. H. Poincaré-An. 34(2017), no. 2, 335–354.CrossRefGoogle Scholar
Bressan, A., Chen, G., and Zhang, Q., Unique conservative solutions to a variational wave equation . Arch. Rat. Mech. Anal. 217(2015), no. 3, 1069–1101.CrossRefGoogle Scholar
Bressan, A. and Zheng, Y., Conservative solutions to a nonlinear variational wave equation . Commun. Math. Phys. 266(2006), no. 2, 471–497.CrossRefGoogle Scholar
Cai, H., Chen, G., and Du, Y., Uniqueness and regularity of conservative solution to a wave system modeling nematic liquid crystal . J. Math. Pures Appl. 117(2018), 185–220.CrossRefGoogle Scholar
Calderer, M. and Liu, C., Liquid crystal flow: dynamic and static configurations . SIAM J. Appl. Math. 60(2000), no. 6, 1925–1949.Google Scholar
Chanderasekhar, S., Liquid crystals, Cambridge University Press, Cambridge, 1992.CrossRefGoogle Scholar
Chen, G., Huang, T., and Liu, W., Poiseuille flow of nematic liquid crystals via the full Ericksen–Leslie model . Arch. Rat. Mech. Anal. 236(2020), no. 2, 839–891.CrossRefGoogle Scholar
Chen, G., Liu, W., and Sofiani, M., The Poiseuille flow of the full Ericksen–Leslie model for nematic liquid crystals: the general case. Preprint, 2023. arXiv:2302.08616 CrossRefGoogle Scholar
De Gennes, P. and Prost, J., The physics of liquid crystals, Oxford University Press, Oxford, 1993.CrossRefGoogle Scholar
Dozov, I., On the spontaneous symmetry breaking in the mesophases of achiral banana-shaped molecules . Europhys. Lett. 56(2001), no. 2, 247–253.CrossRefGoogle Scholar
Ericksen, J., Conservation laws for liquid crystals . Trans. Soc. Rheology 5(1961), no. 1, 23–34.CrossRefGoogle Scholar
Evans, L., Partial differential equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.Google Scholar
Friedman, A., Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.Google Scholar
Glassey, R., Hunter, J., and Zheng, Y., Singularities of a variational wave equation . J. Differential Equations 129(1996), no. 1, 49–78.CrossRefGoogle Scholar
Hsiao, L. and Jiang, S., Nonlinear hyperbolic–parabolic coupled systems . In: Dafermos, C. M. and Feireisl, E. (eds.), Handbook of differential equations: evolutionary equations, Vol. 1, North-Holland, Amsterdam, 2002, pp. 287–384.CrossRefGoogle Scholar
Hu, Y., Conservative solutions to a system of variational wave equations . J. Differential Equations 252(2012), no. 6, 4002–4026.CrossRefGoogle Scholar
Hu, Y., Singularity for a nonlinear degenerate hyperbolic–parabolic coupled system arising from nematic liquid crystals . Adv. Nonlinear Analysis 12(2023), no. 1, 20220268.CrossRefGoogle Scholar
Hu, Y. and Li, J., Sonic-supersonic solutions for the two-dimensional steady full Euler equations . Arch. Rat. Mech. Anal. 235(2020), no. 3, 1819–1871.CrossRefGoogle Scholar
Hu, Y. and Wang, G., On the Cauchy problem for a nonlinear variational wave equation with degenerate initial data, nonlinear. Analysis 176(2018), 192–208.Google Scholar
Hunter, J. and Saxton, R., Dynamics of director fields . SIAM J. Appl. Math. 51(1991), no. 6, 1498–1521.CrossRefGoogle Scholar
Jiang, N. and Luo, Y., On well-posedness of Ericksen–Leslie’s hyperbolic incompressible liquid crystal model . SIAM J. Math. Anal. 51(2019), no. 1, 403–434.CrossRefGoogle Scholar
Jiang, N. and Luo, Y., The zero inertia limit from hyperbolic to parabolic Ericksen–Leslie system of liquid crystal flow . J. Funct. Anal. 282(2022), no. 1, 109280.CrossRefGoogle Scholar
Jiang, N., Luo, Y., and Tang, S., On well-posedness of Ericksen–Leslie’s parabolic–hyperbolic liquid crystal model in compressible flow . Math. Mod. Meth. Appl. Sci. 29(2019), no. 1, 121–183.CrossRefGoogle Scholar
Kato, K. and Sugiyama, Y., Local existence and uniqueness theory for the second sound equation in one space dimension . J. Hyperbolic Differ. Equ. 9(2012), no. 1, 177–193.CrossRefGoogle Scholar
Kato, K. and Sugiyama, Y., Blow up of solutions to the second sound equation in one space dimension . Kyushu J. Math. 67(2013), no. 1, 129–142.CrossRefGoogle Scholar
Kawashima, S., Systems of a hyperbolic–parabolic composite type, with applications to the equations of magnetohydrodynamics. Ph.D. thesis, Kyoto University, 1984.Google Scholar
Leslie, F., Some constitutive equations for liquid crystals . Arch. Rat. Mech. Anal. 28(1968), no. 4, 265–283.CrossRefGoogle Scholar
Leslie, F., Theory of flow phenomena in liquid crystals . In: Advances in liquid crystals. Vol. 4, Academic Press, New York, 1979, pp. 1–81.Google Scholar
Li, T., Yu, W., and Shen, W., Cauchy’s problem for quasilinear hyperbolic–parabolic coupled systems (in Chinese) . Acta Math. Appl. Sin. 4(1981), no. 4, 321–338.Google Scholar
Li, T., Yu, W., and Shen, W., Second initial-boundary value problems for quasi-linear hyperbolic–parabolic coupled systems (in Chinese) . Chinese Ann. of Math. 2(1981), no. 1, 65–90.Google Scholar
Li, T., Yu, W., and Shen, W., First initial-boundary value problems for quasilinear hyperbolic–parabolic coupled systems . Chinese Ann. of Math. 5B(1984), no. 1, 77–90.Google Scholar
Lin, F., Nonlinear theory of defects in nematic liquid crystals; phase transition and phenomena . Commun. Pure Appl. Math. 42(1989), no. 6, 789–814.CrossRefGoogle Scholar
Lin, F. and Liu, C., Static and dynamic theories of liquid crystals . J. Partial Differ. Equ. 14(2001), no. 4, 289–330.Google Scholar
Panov, V., Nagaraj, M., Vij, J., et al., Spontaneous periodic deformations in nonchiral planar-aligned bimesogens with a nematic–nematic transition and a negative elastic constant . Phys. Rev. Lett. 105(2010), no. 3., 167801.CrossRefGoogle Scholar
Protter, M., The Cauchy problem for a hyperbolic second order equation with data on the parabolic line . Canad. J. Math. 6(1954), 542–553.CrossRefGoogle Scholar
Saxton, R., Finite time boundary blowup for a degenerate, quasilinear Cauchy problem . In: Hale, J. and Wiener, J. (eds.), Partial differential equations, Pitman Research Notes in Mathematics Series, 273, Longman, New York, 1992, pp. 212–215.Google Scholar
Slemrod, M., Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoelasticity . Arch. Rat. Mech. Anal. 76(1981), no. 2, 97–133.CrossRefGoogle Scholar
Vertogen, G. and De Jeu, W., Thermotropic liquid crystals, fundamentals, Springer, Berlin, 1988.CrossRefGoogle Scholar
Vol’pert, A. and Hudjaev, S., On the Cauchy problem for composite systems of nonlinear differential equations . Math. USSR Sbornik 16(1972), no. 4, 517–544.CrossRefGoogle Scholar
Zhang, P. and Zheng, Y., Weak solutions to a nonlinear variational wave equation . Arch. Rat. Mech. Anal. 166(2003), no. 4, 303–319.CrossRefGoogle Scholar
Zhang, P. and Zheng, Y., Weak solutions to a nonlinear variational wave equation with general data . Ann. I. H. Poincaré-An. 22(2005), no. 2, 207–226.CrossRefGoogle Scholar
Zhang, P. and Zheng, Y., Conservative solutions to a system of variational wave equations of nematic liquid crystals . Arch. Rat. Mech. Anal. 195(2010), no. 3, 701–727.CrossRefGoogle Scholar
Zhang, P. and Zheng, Y., Energy conservative solutions to a one-dimensional full variational wave system . Commun. Pure Appl. Math. 65(2012), no. 5, 683–726.CrossRefGoogle Scholar
Zhang, T. and Zheng, Y., Sonic-supersonic solutions for the steady Euler equations . Indiana Univ. Math. J. 63(2014), no. 6, 1785–1817.CrossRefGoogle Scholar
Zheng, S., Initial boundary value problems for quasilinear hyperbolic–parabolic coupled systems in higher dimensional spaces . Chin. Ann. Math. Ser. B 4(1983), no. 4, 443–462.Google Scholar
Zheng, S., Global solutions and applications to a class of quasilinear hyperbolic--parabolic coupled systems . Sci. China Ser. A 27(1984), no. 12, 1274–1286.Google Scholar
Zheng, S., Global smooth solutions to the Cauchy problem of quasilinear hyperbolic--parabolic coupled system in higher dimensional spaces (in Chinese) . Chinese Ann. Math. Ser. A 5(1984), no. 11, 681–690.Google Scholar
Zheng, S., Nonlinear parabolic equations and hyperbolic–parabolic coupled systems, CRC Press, Boca Raton, FL, 1995.Google Scholar
Zheng, S. and Shen, W., Global solutions to the Cauchy problem of quasilinear hyperbolic--parabolic coupled system . Science in China Ser. A 30(1987), no. 11, 1133–1149.Google Scholar
Zorski, H. and Infeld, E., New soliton equation for dipole chains . Phys. Rev. Lett. 68(1992), no. 8, 1180–1183.CrossRefGoogle ScholarPubMed