American Institute of Mathematical Sciences

Advanced
February  2013, 33(2): 757-788. doi: 10.3934/dcds.2013.33.757

Globally weak solutions to the flow of compressible liquid crystals system

Xian-Gao Liu 1, and  Jie Qing 2,

1. 

Institute of Mathematics, Fudan University, Shanghai, China
2. 

Mathematics Department, UC at Santa Cruz, United States

Received  July 2011 Revised  October 2011 Published  September 2012

We study a simplified system for the compressible fluid of Nematic Liquid Crystals in a bounded domain in three Euclidean space and prove the global existence of the finite energy weak solutions.
Keywords: global existence., compressible, Nematic liquid crystals.
Mathematics Subject Classification: 76N10, 35Q35, 35Q3.
Citation: Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757
References:
[1]

M. C. Calderer and C. Liu, Liquid crystal flow: Dynamic and staic configurations,, SIAM J. Appl. Math., 60 (2000), 1925.   Google Scholar

[2]

D. Coutand and S. Shkoller, Well posedness of the full Ericksen-Leslye model of nematic liquid crystals,, Note C. R. A. S, 333 (2001), 919.   Google Scholar

[3]

J. L. Ericksen, Hydrostatic theory of liquid crystals,, Arch. Rational Mech. Anal., 9 (1962), 371.   Google Scholar

[4]

J. L. Ericksen and D. Kinderlehrer, "Theory and Applications of Liquid Crystals,", IMA Vol. 5, (1986).   Google Scholar

[5]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (2004).   Google Scholar

[6]

D. Forster, T. Lubensky, P. Martin, J. Swift and P. Pershan, Hydrodynamics of liquid crystals,, Phys. Rev. Lett., 26 (1971), 1016.  doi: 10.1103/PhysRevLett.26.1016.  Google Scholar

[7]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, J. Math. Fluid Mech., 3 (2001), 358.   Google Scholar

[8]

de Gennes, "The Physics of Liquid Crystals,", Claredon Press, (1993).   Google Scholar

[9]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data,, J. Diff. equns, 120 (1995), 215.  doi: 10.1006/jdeq.1995.1111.  Google Scholar

[10]

D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data,, Arch. Rational Mech. Anal., 132 (1995), 1.   Google Scholar

[11]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,, Comm. Math. Phys., 215 (2001), 559.   Google Scholar

[12]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Gordon and Breach, (1969).   Google Scholar

[13]

O. A. Ladyzhenskaya, N. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations Of Parabolic Type,", Transl. Math. Monographs, 23 (1968).   Google Scholar

[14]

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

[15]

F. M. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[16]

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

[17]

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

[18]

F. H. Lin and C. Liu, Static ang dynamic theories of liquid crystals,, Journal of Partial Differential Equations, 14 (2001), 289.   Google Scholar

[19]

F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie System,, Arch. Rational Mech. Aanl., 154 (2000), 135.   Google Scholar

[20]

F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals,, Discrete and Continuous Dynamic Systems, 2 (1996), 1.   Google Scholar

[21]

J. L. Lions, "Quelques Méthodes Derésolution des Problèms aux Limites Nonlinéaires,", Dunod, (1960).   Google Scholar

[22]

P. L. Lions, "Mathematical Topics in Fluid Dynamics,", Vol.1. Compressible models. Oxford Science Publication, (1998).   Google Scholar

[23]

P. L. Lions, "Mathematical Topics in Fluid Dynamics,", Vol.2. Compressible models. Oxford Science Publication, (1998).   Google Scholar

[24]

X. Liu and Z. Zhang, Global existence of weak solutions for the incompressible liquid crystals,, Chinese Anna. Math., 30 (2009), 1.   Google Scholar

[25]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhäuser, (1995).   Google Scholar

[26]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible and heat-conductives fluids,, Proc. Japan Acad., 55 (1979), 337.  doi: 10.3792/pjaa.55.337.  Google Scholar

[27]

J. Phillips, "Liquid Crystals,", mcgill. ca, (2005).   Google Scholar

[28]

S. V. Pasechnik, V. G. Chigrinov, D. V. Shmeliova, "Liquid Crystals: Viscous and Elastic Properties,", Wiley-VCH, (2009).   Google Scholar

[29]

D. Serre and J.L. Lions(Rapporteur), Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible,, C. R. Acad. Sci. Paris, 303 (1986), 639.   Google Scholar

[30]

M. J. Stephen, Hydrodynamics of liquid crystals,, Phys. Rev. A, 2 (1970), 1558.   Google Scholar

[31]

R. Temam, "Navier-Stokes Equations,", rev. ed., (1977).   Google Scholar

[32]

W. Stewart, "The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction,", Liquid crystals bookseries, (2004).   Google Scholar

[33]

Y. Z. Xie, "The Physics of Liquid Crystals,", Scientific Press, (1988).   Google Scholar

[34]

D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals,, , ().   Google Scholar

show all references

References:
[1]

M. C. Calderer and C. Liu, Liquid crystal flow: Dynamic and staic configurations,, SIAM J. Appl. Math., 60 (2000), 1925.   Google Scholar

[2]

D. Coutand and S. Shkoller, Well posedness of the full Ericksen-Leslye model of nematic liquid crystals,, Note C. R. A. S, 333 (2001), 919.   Google Scholar

[3]

J. L. Ericksen, Hydrostatic theory of liquid crystals,, Arch. Rational Mech. Anal., 9 (1962), 371.   Google Scholar

[4]

J. L. Ericksen and D. Kinderlehrer, "Theory and Applications of Liquid Crystals,", IMA Vol. 5, (1986).   Google Scholar

[5]

E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (2004).   Google Scholar

[6]

D. Forster, T. Lubensky, P. Martin, J. Swift and P. Pershan, Hydrodynamics of liquid crystals,, Phys. Rev. Lett., 26 (1971), 1016.  doi: 10.1103/PhysRevLett.26.1016.  Google Scholar

[7]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, J. Math. Fluid Mech., 3 (2001), 358.   Google Scholar

[8]

de Gennes, "The Physics of Liquid Crystals,", Claredon Press, (1993).   Google Scholar

[9]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data,, J. Diff. equns, 120 (1995), 215.  doi: 10.1006/jdeq.1995.1111.  Google Scholar

[10]

D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data,, Arch. Rational Mech. Anal., 132 (1995), 1.   Google Scholar

[11]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,, Comm. Math. Phys., 215 (2001), 559.   Google Scholar

[12]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,", Gordon and Breach, (1969).   Google Scholar

[13]

O. A. Ladyzhenskaya, N. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations Of Parabolic Type,", Transl. Math. Monographs, 23 (1968).   Google Scholar

[14]

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

[15]

F. M. Leslie, Some constitutive equations for liquid crystals,, Arch. Rational Mech. Anal., 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[16]

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

[17]

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

[18]

F. H. Lin and C. Liu, Static ang dynamic theories of liquid crystals,, Journal of Partial Differential Equations, 14 (2001), 289.   Google Scholar

[19]

F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie System,, Arch. Rational Mech. Aanl., 154 (2000), 135.   Google Scholar

[20]

F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals,, Discrete and Continuous Dynamic Systems, 2 (1996), 1.   Google Scholar

[21]

J. L. Lions, "Quelques Méthodes Derésolution des Problèms aux Limites Nonlinéaires,", Dunod, (1960).   Google Scholar

[22]

P. L. Lions, "Mathematical Topics in Fluid Dynamics,", Vol.1. Compressible models. Oxford Science Publication, (1998).   Google Scholar

[23]

P. L. Lions, "Mathematical Topics in Fluid Dynamics,", Vol.2. Compressible models. Oxford Science Publication, (1998).   Google Scholar

[24]

X. Liu and Z. Zhang, Global existence of weak solutions for the incompressible liquid crystals,, Chinese Anna. Math., 30 (2009), 1.   Google Scholar

[25]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Birkhäuser, (1995).   Google Scholar

[26]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible and heat-conductives fluids,, Proc. Japan Acad., 55 (1979), 337.  doi: 10.3792/pjaa.55.337.  Google Scholar

[27]

J. Phillips, "Liquid Crystals,", mcgill. ca, (2005).   Google Scholar

[28]

S. V. Pasechnik, V. G. Chigrinov, D. V. Shmeliova, "Liquid Crystals: Viscous and Elastic Properties,", Wiley-VCH, (2009).   Google Scholar

[29]

D. Serre and J.L. Lions(Rapporteur), Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible,, C. R. Acad. Sci. Paris, 303 (1986), 639.   Google Scholar

[30]

M. J. Stephen, Hydrodynamics of liquid crystals,, Phys. Rev. A, 2 (1970), 1558.   Google Scholar

[31]

R. Temam, "Navier-Stokes Equations,", rev. ed., (1977).   Google Scholar

[32]

W. Stewart, "The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction,", Liquid crystals bookseries, (2004).   Google Scholar

[33]

Y. Z. Xie, "The Physics of Liquid Crystals,", Scientific Press, (1988).   Google Scholar

[34]

D. Wang and C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals,, , ().   Google Scholar

[1]

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
[2]

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
[3]

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
[4]

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
[5]

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
[6]

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
[7]

Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591
[14]

Patricia Bauman, Daniel Phillips, Jinhae Park. Existence of solutions to boundary value problems for smectic liquid crystals. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 243-257. doi: 10.3934/dcdss.2015.8.243
[15]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[16]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[17]

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 - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
[18]

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 - A, 2020  doi: 10.3934/dcds.2020304
[19]

Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165
[20]

Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences