Abstract
In this paper, we study the global existence of weak solutions to the Cauchy problem of the three-dimensional equations for compressible isentropic nematic liquid crystal flows subject to discontinuous initial data. It is assumed here that the initial energy is suitably small in L 2, and the initial density, the gradients of initial velocity/liquid crystal director field are bounded in L ∞, L 2 and H 1, respectively. This particularly implies that the initial data may contain vacuum states and the oscillations of solutions could be arbitrarily large. As a byproduct, we also prove the global existence of smooth solutions with strictly positive density and small initial energy.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Adams, R.A.: Sobolev Space. Academic Press, New York (1975)
Ericksen, J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34 (1961)
Ericksen, J.L.: Continuum theory of nematic liquid crystals. Res Mech. 21, 381–392 (1987)
Hardt, R., Kinderlehrer, D.: Mathematical Questions of Liquid Crystal Theory. The IMA Volumes in Mathematics and Its Applications, vol. 5. Springer, New York (1987)
Hoff, D.: Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120, 215–254 (1995)
Hoff, D., Serre, D.: The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51(4), 887–898 (1991)
Huang, X.D., Li, J., Xin, Z.-P.: Global well-posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible Navier-Stokes equations. Commun. Pure Appl. Math. 65(4), 549–585 (2012)
Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS, Providence (1968)
Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968)
Li, H.L., Li, J., Xin, Z.-P.: Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations. Commun. Math. Phys. 281, 401–444 (2008)
Li, J., Zhang, J., Zhao, J.: On the motion of three-dimensional compressible isentropic flows with large external potential forces and vacuum (2011). arXiv:1111.2114
Li, J., Xu, Z., Zhang, J.: Global well-posedness with large oscillations and vacuum to the three-dimensional equations of compressible nematic liquid crystal flows (2012). arXiv:1204.4966
Lin, F.H.: Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena. Commun. Pure Appl. Math. 42, 789–814 (1989)
Lions, P.L.: Mathematical Topics in Fluid Dynamics, vol. 2. Compressible Models. Oxford Science Publications, Oxford (1996)
Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)
Wu, G.C., Tan, Z.: Global low-energy weak solution and large-time behavior for the compressible flow of liquid crystals (2012). arXiv:1210.1269v1
Xin, Z.P.: Blowup of smooth solutions to the compressible Navier-Stokes equations with compact density. Commun. Pure Appl. Math. 51, 229–240 (1998)
Zlotnik, A.A.: Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations. Differ. Equ. 36, 701–716 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, Y. Global Weak Solutions of 3D Compressible Nematic Liquid Crystal Flows with Discontinuous Initial Data and Vacuum. Acta Appl Math 142, 149–171 (2016). https://doi.org/10.1007/s10440-015-0021-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-015-0021-6