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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 |
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. |
show all references
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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