American Institute of Mathematical Sciences

Advanced
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

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 & 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.  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), 984-998. 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), 771-831. 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), 504-523.  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), 1-28. 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), 357-371. Google Scholar

[7]

J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. 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), 157-181. 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), 463-467. 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, Amer. Math. Soc., Providence RI, 1967.  Google Scholar

[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.  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), 789-814. 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), 501-537. 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), 135-156. 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), 1-22.  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), 297-336. 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), 205-222. 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), 1175-1191. 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), 161-177. 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), 1093-1117. 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), 141-160.  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), 984-998. 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), 771-831. 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), 504-523.  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), 1-28. 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), 357-371. Google Scholar

[7]

J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. 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), 157-181. 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), 463-467. 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, Amer. Math. Soc., Providence RI, 1967.  Google Scholar

[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.  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), 789-814. 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), 501-537. 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), 135-156. 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), 1-22.  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), 297-336. 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), 205-222. 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), 1175-1191. 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), 161-177. 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), 1093-1117. doi: 10.1137/0521061.  Google Scholar

[1]

Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4515-4533. doi: 10.3934/dcdsb.2020110
[2]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357
[3]

Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757
[4]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304
[5]

Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165
[6]

Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445
[7]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[8]

Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007
[9]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565
[10]

Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623
[11]

Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
[12]

Tong Tang, Yongfu Wang. Strong solutions to compressible barotropic viscoelastic flow with vacuum. Kinetic & Related Models, 2015, 8 (4) : 765-775. doi: 10.3934/krm.2015.8.765
[13]

Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341
[14]

Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106
[15]

Zdzisław Brzeźniak, Erika Hausenblas, Paul André Razafimandimby. A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5785-5802. doi: 10.3934/dcdsb.2019106
[16]

Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
[17]

Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379
[18]

Apala Majumdar. The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1303-1337. doi: 10.3934/cpaa.2012.11.1303
[19]

Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1
[20]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (40)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences