Compressible hydrodynamic flow of liquid crystals in 1-D

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

Compressible hydrodynamic flow of liquid crystals in 1-D

Shijin Ding ^1, , Junyu Lin ^2, , Changyou Wang ^3, and Huanyao Wen ^4,

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

Abstract
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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.

Keywords: Nematic liquid crystals, compressible hydrodynamic flow, classical and strong solutions..

Mathematics Subject Classification: Primary: 35K55, 76N1.

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.

show all references

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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