# American Institute of Mathematical Sciences

February  2012, 32(2): 539-563. doi: 10.3934/dcds.2012.32.539

## Compressible hydrodynamic flow of liquid crystals in 1-D

 1 Department of Mathematics, South China Normal University, Guangzhou, Guangdong 510631 2 Department of Mathematics, South China University of Technology, Guangzhou, 510640, China 3 Department of Mathematics, University of Kentucky, Lexington, KY 40513 4 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631

Received  November 2010 Revised  May 2011 Published  September 2011

We consider a simplified version of Ericksen-Leslie equation modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one. If the initial data $(\rho_0, u_0,n_0)\in C^{1,\alpha}(I)\times C^{2,\alpha}(I)\times C^{2,\alpha}(I, S^2)$ and $\rho_0\ge c_0>0$, then we obtain both existence and uniqueness of global classical solutions. For $0\le\rho_0\in H^1(I)$ and $(u_0, n_0)\in H^1(I)\times H^2(I,S^2)$, we obtain both existence and uniqueness of global strong solutions.
Citation: Shijin Ding, Junyu Lin, Changyou Wang, Huanyao Wen. Compressible hydrodynamic flow of liquid crystals in 1-D. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 539-563. doi: 10.3934/dcds.2012.32.539
##### References:
 [1] H. Beirão da Veiga, Long time behavior for one-dimensional motion of a general barotropic viscous fluid, Arch. Ration. Mech. Anal., 108 (1989), 141-160. [2] Blanca Climent-Ezquerra, Francisco Guillén-González and Marko Rojas-Medar, Reproductivity for a nematic liquid crystal model, Z. angew. Math. Phys., 57 (2006), 984-998. doi: 10.1007/s00033-005-0038-1. [3] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. [4] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Diff. Equa., 190 (2003), 504-523. [5] H. J. Choe and H. Kim, Global existence of the radially symmetric solutions of the Navier-Stokes equations for the isentropic compressible fuids, Math. Meth. Appl. Sci., 28 (2005), 1-28. doi: 10.1002/mma.545. [6] S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Continuous Dynam. Systems B, 15 (2011), 357-371. [7] J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358. [8] P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech. doi: 10.1007/s00021-009-0006-1. [9] S. Jiang, On initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Diff. Equa., 110 (1994), 157-181. doi: 10.1006/jdeq.1994.1064. [10] A. V. Kazhikhov, Stabilization of solutions of an initial-boundary-value problem for the equations of motion of a barotropic viscous fluid, Differ. Equ., 15 (1979), 463-467. [11] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, 23, Amer. Math. Soc., Providence RI, 1967. [12] F. Leslie, Some constitutive equations for anisotropic fluids, Q. J. Mech. Appl. Math., 19 (1966), 357-370. doi: 10.1093/qjmam/19.3.357. [13] F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. [14] F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503. [15] F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Rational Mech. Anal., 154 (2000), 135-156. doi: 10.1007/s002050000102. [16] F.-H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems, 2 (1996), 1-22. [17] F.-H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. [18] C. Liu and N. J. Walkington, Mixed methods for the approximation of liquid crystal flows, Math. Modeling and Numer. Anal., 36 (2002), 205-222. doi: 10.1051/m2an:2002010. [19] T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math. Anal., 31 (2000), 1175-1191. doi: 10.1137/S0036141097331044. [20] M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscous gas, Japan J. Appl. Math., 6 (1989), 161-177. doi: 10.1007/BF03167921. [21] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of vecocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061.

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##### References:
 [1] H. Beirão da Veiga, Long time behavior for one-dimensional motion of a general barotropic viscous fluid, Arch. Ration. Mech. Anal., 108 (1989), 141-160. [2] Blanca Climent-Ezquerra, Francisco Guillén-González and Marko Rojas-Medar, Reproductivity for a nematic liquid crystal model, Z. angew. Math. Phys., 57 (2006), 984-998. doi: 10.1007/s00033-005-0038-1. [3] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. [4] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Diff. Equa., 190 (2003), 504-523. [5] H. J. Choe and H. Kim, Global existence of the radially symmetric solutions of the Navier-Stokes equations for the isentropic compressible fuids, Math. Meth. Appl. Sci., 28 (2005), 1-28. doi: 10.1002/mma.545. [6] S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one, Discrete Continuous Dynam. Systems B, 15 (2011), 357-371. [7] J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358. [8] P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech. doi: 10.1007/s00021-009-0006-1. [9] S. Jiang, On initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Diff. Equa., 110 (1994), 157-181. doi: 10.1006/jdeq.1994.1064. [10] A. V. Kazhikhov, Stabilization of solutions of an initial-boundary-value problem for the equations of motion of a barotropic viscous fluid, Differ. Equ., 15 (1979), 463-467. [11] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, 23, Amer. Math. Soc., Providence RI, 1967. [12] F. Leslie, Some constitutive equations for anisotropic fluids, Q. J. Mech. Appl. Math., 19 (1966), 357-370. doi: 10.1093/qjmam/19.3.357. [13] F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. [14] F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. doi: 10.1002/cpa.3160480503. [15] F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Rational Mech. Anal., 154 (2000), 135-156. doi: 10.1007/s002050000102. [16] F.-H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems, 2 (1996), 1-22. [17] F.-H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. [18] C. Liu and N. J. Walkington, Mixed methods for the approximation of liquid crystal flows, Math. Modeling and Numer. Anal., 36 (2002), 205-222. doi: 10.1051/m2an:2002010. [19] T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum, SIAM J. Math. Anal., 31 (2000), 1175-1191. doi: 10.1137/S0036141097331044. [20] M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscous gas, Japan J. Appl. Math., 6 (1989), 161-177. doi: 10.1007/BF03167921. [21] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of vecocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061.
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