American Institute of Mathematical Sciences

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 and 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), 371-378. doi: 10.1007/BF00253358. [2] F. Leslie, Some constitute equations for anisotropic fluids, Q. J. Mech. Appl. Math., 19 (1966), 357-370. doi: 10.1093/qjmam/19.3.357. [3] F. H. Lin, Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. [4] F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math. Vol. XLV III (1995), 501-537. doi: 10.1002/cpa.3160480503. [5] 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. [6] F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, DCDS, 2 (1996), 1-23. [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), 771-831. doi: 10.1002/cpa.3160350604. [8] 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. [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. [10] F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in dimensions two, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. [11] H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Preprint (2009). [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). [13] P. L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. II, Compressible Models. Clarendon Press, Oxford, 1998. [14] S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Commun. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543. [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), 358-392. doi: 10.1007/PL00000976. [16] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061. [17] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence RI, 1968. [18] L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Vol. 19, 1998. [19] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford University Press, Oxford, 2004. [20] P. L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. I, Incompressible Models, Clarendon Press, Oxford, 1996.

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References:
 [1] J. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. doi: 10.1007/BF00253358. [2] F. Leslie, Some constitute equations for anisotropic fluids, Q. J. Mech. Appl. Math., 19 (1966), 357-370. doi: 10.1093/qjmam/19.3.357. [3] F. H. Lin, Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814. doi: 10.1002/cpa.3160420605. [4] F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math. Vol. XLV III (1995), 501-537. doi: 10.1002/cpa.3160480503. [5] 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. [6] F. H. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, DCDS, 2 (1996), 1-23. [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), 771-831. doi: 10.1002/cpa.3160350604. [8] 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. [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. [10] F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in dimensions two, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. [11] H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Preprint (2009). [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). [13] P. L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. II, Compressible Models. Clarendon Press, Oxford, 1998. [14] S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Commun. Math. Phys., 215 (2001), 559-581. doi: 10.1007/PL00005543. [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), 358-392. doi: 10.1007/PL00000976. [16] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061. [17] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc., Providence RI, 1968. [18] L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Vol. 19, 1998. [19] E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford University Press, Oxford, 2004. [20] P. L. Lions, "Mathematical Topics in Fluid Mechanics," Vol. I, Incompressible Models, Clarendon Press, Oxford, 1996.
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