Singularity Formation for the General Poiseuille Flow of Nematic Liquid Crystals
Tóm tắt
We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model. The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation. In this paper, we will construct an example with a finite time cusp singularity due to the quasilinearity of the wave equation, extended from an earlier result on a special case.
Tài liệu tham khảo
Bressan, A., Chen, G.: Lipschitz metric for a class of nonlinear wave equations. Arch. Ration. Mech. Anal. 226(3), 1303–1343 (2017)
Bressan, A., Chen, G.: Generic regularity of conservative solutions to a nonlinear wave equation. Ann. I. H. Poincaré-AN 34(2), 335–354 (2017)
Bressan, A., Chen, G., Zhang, Q.: Unique conservative solutions to a variational wave equation. Arch. Ration. Mech. Anal. 217(3), 1069–1101 (2015)
Bressan, A., Huang, T.: Representation of dissipative solutions to a nonlinear variational wave equation. Commun. Math. Sci. 14, 31–53 (2016)
Bressan, A., Zheng, Y.: Conservative solutions to a nonlinear variational wave equation. Commun. Math. Phys. 266, 471–497 (2006)
Cai, H., Chen, G., Du, Y.: Uniqueness and regularity of conservative solution to a wave system modeling nematic liquid crystal. J. Math. Pures Appl. 9(117), 185–220 (2018)
Chen, G., Huang, T., Liu, W.: Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model. Arch. Ration. Mech. Anal. 236, 839–891 (2020)
Chen, G., Sofiani, M., Liu, W.: Global existence of Hölder continuous solution for Poiseuille flow of nematic liquid crystals. Submitted
Chen, G., Zhang, P., Zheng, Y.: Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Commun. Pure Appl. Anal. 12(3), 1445–1468 (2013)
Chen, G., Zheng, Y.: Singularity and existence to a wave system of nematic liquid crystals. J. Math. Anal. Appl. 398, 170–188 (2013)
Ericksen, J.L.: Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal. 9, 371–378 (1962)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall Inc, Hoboken (1964)
Glassey, R.T., Hunter, J.K., Zheng, Y.: Singularities in a nonlinear variational wave equation. J. Differ. Equ. 129, 49–78 (1996)
Holden, H., Raynaud, X.: Global semigroup of conservative solutions of the nonlinear variational wave equation. Arch. Ration. Mech. Anal. 201, 871–964 (2011)
Leslie, F.M.: Some thermal effects in cholesteric liquid crystals. Proc. R. Soc. A 307, 359–372 (1968)
Leslie, F.M.: Theory of flow phenomena in liquid crystals. In: Brown, G.H. (ed) Advances in Liquid Crystals, vol. 4, pp. 1–81. Academic Press, New York (1979)
Parodi, O.: Stress tensor for a nematic liquid crystal. J. Phys. 31, 581–584 (1970)
Zhang, P., Zheng, Y.: Weak solutions to a nonlinear variational wave equation. Arch. Ration. Mech. Anal. 166, 303–319 (2003)
Zhang, P., Zheng, Y.: Conservative solutions to a system of variational wave equations of nematic liquid crystals. Arch. Ration. Mech. Anal. 195, 701–727 (2010)
Zhang, P., Zheng, Y.: Energy conservative solutions to a one-dimensional full variational wave system. Commun. Pure Appl. Math. 55, 582–632 (2012)