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

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

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 $n_0\in H^1$.
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

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