Abstract
The present paper is dedicated to the study of the Cauchy problems for the three-dimensional compressible nematic liquid crystal flow. We obtain the global existence and the optimal decay rates of smooth solutions to the system under the condition that the initial data in lower regular spaces are close to the constant equilibrium state. Our main method is based on the spectral analysis and the smooth effect of dissipative operator.
Similar content being viewed by others
References
Dai, M., Schonbek, M.: Asymptotic bahavior of solutions to the liquid crystals systems in \(H^{m}(\mathbb{R}^{3})\). SIAM J. Math. Anal. 46, 3131–3150 (2014)
Dai, M., Qing, J., Schonbek, M.: Asymptotic bahavior of solutions to the liquid crystals systems in \(\mathbb{R}^{3}\). Commun. Partial Differ. Equ. 37, 2138–2164 (2012)
Deckelnick, K.: Decay estimates for the compressible Navier-Stokes equations in unbounded domains. Math. Z. 209, 115–130 (1992)
Ding, S., Huang, J., Wen, H., Zi, R.: Incompressible limit of the compressible hydrodynamic flow of liquid crystals. J. Funct. Anal. 264, 1711–1756 (2013)
Duan, R., Ukai, S., Yang, T., Zhao, H.J.: Optimal convergence rates for the compressible Navier-Stokes equations with potential forces. Math. Models Methods Appl. Sci. 17, 737–758 (2007)
Ericksen, J.L.: Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9, 371–378 (1962)
Gao, J., Tang, Q., Yao, Z.: Long-time behavior of solution for the compressible nematic liquid crystal flows in \(\mathbb{R}^{3}\) (2015). arXiv:1503.02865v1
Hu, X., Wu, H.: Global solutions to the three-dimensional compressible flow of liquid crystal. SIAM J. Math. Anal. 45, 2678–2699 (2013)
Huang, T., Wang, C., Wen, H.: Strong solutions of the compressible nematic liquid crystal flow. J. Differ. Equ. 252, 2222–2265 (2012)
Huang, T., Wang, C., Wen, H.: Bow up criterion for compressible nematic liquid crystal flows in dimension three. Arch. Ration. Mech. Anal. 204, 285–311 (2012)
Kawashima, S.: Smooth global solutions for two-dimensional equations of electromagneto-fluid dynamics. Jpn. J. Appl. Math. 1, 207–222 (1984)
Kobayashi, T.: Some estimates of solutions for the equations of motion of compressible viscous fluid in the three-dimensional exterior domain. J. Differ. Equ. 184, 587–619 (2002)
Kobayashi, T., Shibata, Y.: Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in \(\mathbb{R}^{3}\). Commun. Math. Phys. 200, 621–659 (1999)
Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968)
Matsumura, A., Nishida, T.: The initial value problems for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)
Morro, A.: Modelling of nematic liquid crystals in electromagnetic fields. Adv. Theor. Appl. Mech. 2(1), 43–58 (2009)
Ponce, G.: Global existence of small solution to a class of nonlinear evolution equations. Nonlinear Anal. 9, 399–418 (1985)
Umeda, T., Kawashima, S., Shizuta, Y.: On the decay of solutions to the linearized equations of electromagnetofluid dynamics. Jpn. J. Appl. Math. 1, 435–457 (1984)
Wang, Y., Tan, Z.: Global existence and optimal decay rate for the strong solutions in \(H^{2}\) to the compressible Navier-Stokes equations. Appl. Math. Lett. 11, 1778–1784 (2011)
Xu, F., Liu, L., Wu, Y.: On the well-posedness for the compressible nematic liquid crystal flow (in Chinese). Sci. Sin., Math. 45, 331–348 (2015)
Zakharov, A.V., Vakulenko, A.A.: Orientational dynamics of the compressible nematic liquid crystals induced by a temperature gradient. Phys. Rev. E 79, 011708 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by the National Natural Science Foundation of China (11501332, 11171034, 11371221), the Natural Science Foundation of Shandong Province (ZR2015AL007), China Postdoctoral Science Foundation funded project (2014M561893), Postdoctoral innovation fund of Shandong Province, the Open Fund of State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research Fund (IWHR-SKL-201407), and the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), and Young Scholars Research Fund of Shandong University of Technology.
Rights and permissions
About this article
Cite this article
Xu, F., Zhang, X., Wu, Y. et al. Global Existence and the Optimal Decay Rates for the Three Dimensional Compressible Nematic Liquid Crystal Flow. Acta Appl Math 150, 67–80 (2017). https://doi.org/10.1007/s10440-017-0094-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-017-0094-5