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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  doi: 10.1007/s00033-005-0038-1.  Google Scholar

[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.  doi: 10.1002/cpa.3160350604.  Google Scholar

[4]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, J. Diff. Equa., 190 (2003), 504.   Google Scholar

[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.  doi: 10.1002/mma.545.  Google Scholar

[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.   Google Scholar

[7]

J. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371.  doi: 10.1007/BF00253358.  Google Scholar

[8]

P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system,, J. Math. Fluid Mech., ().  doi: 10.1007/s00021-009-0006-1.  Google Scholar

[9]

S. Jiang, On initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas,, J. Diff. Equa., 110 (1994), 157.  doi: 10.1006/jdeq.1994.1064.  Google Scholar

[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.   Google Scholar

[11]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, 23 (1967).   Google Scholar

[12]

F. Leslie, Some constitutive equations for anisotropic fluids,, Q. J. Mech. Appl. Math., 19 (1966), 357.  doi: 10.1093/qjmam/19.3.357.  Google Scholar

[13]

F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789.  doi: 10.1002/cpa.3160420605.  Google Scholar

[14]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar

[15]

F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, Arch. Rational Mech. Anal., 154 (2000), 135.  doi: 10.1007/s002050000102.  Google Scholar

[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.   Google Scholar

[17]

F.-H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297.  doi: 10.1007/s00205-009-0278-x.  Google Scholar

[18]

C. Liu and N. J. Walkington, Mixed methods for the approximation of liquid crystal flows,, Math. Modeling and Numer. Anal., 36 (2002), 205.  doi: 10.1051/m2an:2002010.  Google Scholar

[19]

T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum,, SIAM J. Math. Anal., 31 (2000), 1175.  doi: 10.1137/S0036141097331044.  Google Scholar

[20]

M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscous gas,, Japan J. Appl. Math., 6 (1989), 161.  doi: 10.1007/BF03167921.  Google Scholar

[21]

J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of vecocity, density, and pressure,, SIAM J. Math. Anal., 21 (1990), 1093.  doi: 10.1137/0521061.  Google Scholar

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.   Google Scholar

[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.  doi: 10.1007/s00033-005-0038-1.  Google Scholar

[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.  doi: 10.1002/cpa.3160350604.  Google Scholar

[4]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids,, J. Diff. Equa., 190 (2003), 504.   Google Scholar

[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.  doi: 10.1002/mma.545.  Google Scholar

[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.   Google Scholar

[7]

J. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371.  doi: 10.1007/BF00253358.  Google Scholar

[8]

P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system,, J. Math. Fluid Mech., ().  doi: 10.1007/s00021-009-0006-1.  Google Scholar

[9]

S. Jiang, On initial boundary value problems for a viscous, heat-conducting, one-dimensional real gas,, J. Diff. Equa., 110 (1994), 157.  doi: 10.1006/jdeq.1994.1064.  Google Scholar

[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.   Google Scholar

[11]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translations of Mathematical Monographs, 23 (1967).   Google Scholar

[12]

F. Leslie, Some constitutive equations for anisotropic fluids,, Q. J. Mech. Appl. Math., 19 (1966), 357.  doi: 10.1093/qjmam/19.3.357.  Google Scholar

[13]

F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789.  doi: 10.1002/cpa.3160420605.  Google Scholar

[14]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar

[15]

F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, Arch. Rational Mech. Anal., 154 (2000), 135.  doi: 10.1007/s002050000102.  Google Scholar

[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.   Google Scholar

[17]

F.-H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297.  doi: 10.1007/s00205-009-0278-x.  Google Scholar

[18]

C. Liu and N. J. Walkington, Mixed methods for the approximation of liquid crystal flows,, Math. Modeling and Numer. Anal., 36 (2002), 205.  doi: 10.1051/m2an:2002010.  Google Scholar

[19]

T. Luo, Z. Xin and T. Yang, Interface behavior of compressible Navier-Stokes equations with vacuum,, SIAM J. Math. Anal., 31 (2000), 1175.  doi: 10.1137/S0036141097331044.  Google Scholar

[20]

M. Okada, Free boundary value problems for the equation of one-dimensional motion of viscous gas,, Japan J. Appl. Math., 6 (1989), 161.  doi: 10.1007/BF03167921.  Google Scholar

[21]

J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of vecocity, density, and pressure,, SIAM J. Math. Anal., 21 (1990), 1093.  doi: 10.1137/0521061.  Google Scholar

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