Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one

Previous Article
Using the immersed boundary method to model complex fluids-structure interaction in sperm motility
DCDS-B Home
This Issue
Next Article
On the stochastic immersed boundary method with an implicit interface formulation

March 2011, 15(2): 357-371. doi: 10.3934/dcdsb.2011.15.357

Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one

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

1.	Department of Mathematics, South China Normal University, Guangzhou, Guangdong 510631
2.	Department of Mathematics, University of Kentucky, Lexington, KY 40513, United States
3.	School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

Received November 2009 Revised March 2010 Published December 2010

Abstract
Related Papers

We consider the equation modeling the compressible hydrodynamic flow of liquid crystals in one dimension. In this paper, we establish the existence of a weak solution $(\rho, u,n)$ of such a system when the initial density function $0\le \rho_0 \in L^\gamma$ for $\gamma>1$, $u_0\in L^2$, and $n_0\in H^1$. This extends a previous result by [12], where the existence of a weak solution was obtained under the stronger assumption that the initial density function $0$<$c\le \rho_0\in H^1$, $u_0\in L^2$, and $n_0\in H^1$.

Keywords: Compressible hydrodynamic flow, weak solution., nematic liquid crystals.

Mathematics Subject Classification: Primary: 35K55, 35D3.

Citation: 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

References:

[1]	J. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371. doi: 10.1007/BF00253358. Google Scholar
[2]	F. Leslie, Some constitute equations for anisotropic fluids,, Q. J. Mech. Appl. Math., 19 (1966), 357. doi: 10.1093/qjmam/19.3.357. Google Scholar
[3]	F. H. Lin, Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789. doi: 10.1002/cpa.3160420605. Google Scholar
[4]	F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math. Vol. \textbf{XLV III} (1995), XLV III (1995), 501. doi: 10.1002/cpa.3160480503. Google Scholar
[5]	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
[6]	F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals,, DCDS, 2 (1996), 1. Google Scholar
[7]	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
[8]	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
[9]	Blanca Climent-Ezquerra, Francisco Guillén-González and Marko Rojas-Medar, Reproductivity for a nematic liquid crystal model,, Z. angew. Math. Phys. (2006) 984-998., (2006), 984. Google Scholar
[10]	F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in dimensions two,, Arch. Rational Mech. Anal., 197 (2010), 297. doi: 10.1007/s00205-009-0278-x. Google Scholar
[11]	H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals,, Preprint (2009)., (2009). Google Scholar
[12]	S. J. Ding, J. Y. Lin, C. Y. Wang and H. Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D,, Preprint (2009)., (2009). Google Scholar
[13]	P. L. Lions, "Mathematical Topics in Fluid Mechanics,", Vol. II, (1998). Google Scholar
[14]	S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,, Commun. Math. Phys., 215 (2001), 559. doi: 10.1007/PL00005543. Google Scholar
[15]	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. doi: 10.1007/PL00000976. Google Scholar
[16]	J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure,, SIAM J. Math. Anal., 21 (1990), 1093. doi: 10.1137/0521061. Google Scholar
[17]	O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Amer. Math. Soc., (1968). Google Scholar
[18]	L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar
[19]	E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (2004). Google Scholar
[20]	P. L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. I,, Incompressible Models, (1996). Google Scholar

show all references

References:

[1]	J. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371. doi: 10.1007/BF00253358. Google Scholar
[2]	F. Leslie, Some constitute equations for anisotropic fluids,, Q. J. Mech. Appl. Math., 19 (1966), 357. doi: 10.1093/qjmam/19.3.357. Google Scholar
[3]	F. H. Lin, Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789. doi: 10.1002/cpa.3160420605. Google Scholar
[4]	F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math. Vol. \textbf{XLV III} (1995), XLV III (1995), 501. doi: 10.1002/cpa.3160480503. Google Scholar
[5]	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
[6]	F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals,, DCDS, 2 (1996), 1. Google Scholar
[7]	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
[8]	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
[9]	Blanca Climent-Ezquerra, Francisco Guillén-González and Marko Rojas-Medar, Reproductivity for a nematic liquid crystal model,, Z. angew. Math. Phys. (2006) 984-998., (2006), 984. Google Scholar
[10]	F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in dimensions two,, Arch. Rational Mech. Anal., 197 (2010), 297. doi: 10.1007/s00205-009-0278-x. Google Scholar
[11]	H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals,, Preprint (2009)., (2009). Google Scholar
[12]	S. J. Ding, J. Y. Lin, C. Y. Wang and H. Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1-D,, Preprint (2009)., (2009). Google Scholar
[13]	P. L. Lions, "Mathematical Topics in Fluid Mechanics,", Vol. II, (1998). Google Scholar
[14]	S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations,, Commun. Math. Phys., 215 (2001), 559. doi: 10.1007/PL00005543. Google Scholar
[15]	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. doi: 10.1007/PL00000976. Google Scholar
[16]	J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure,, SIAM J. Math. Anal., 21 (1990), 1093. doi: 10.1137/0521061. Google Scholar
[17]	O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Amer. Math. Soc., (1968). Google Scholar
[18]	L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar
[19]	E. Feireisl, "Dynamics of Viscous Compressible Fluids,", Oxford University Press, (2004). Google Scholar
[20]	P. L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. I,, Incompressible Models, (1996). Google Scholar

[1]	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
[2]	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
[3]	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
[4]	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
[5]	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
[6]	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
[7]	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
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379
[14]	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
[15]	Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1
[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]	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
[18]	Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[19]	Jishan Fan, Tohru Ozawa. Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 859-867. doi: 10.3934/dcds.2009.25.859
[20]	M. Gregory Forest, Hongyun Wang, Hong Zhou. Sheared nematic liquid crystal polymer monolayers. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 497-517. doi: 10.3934/dcdsb.2009.11.497

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences