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Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density
1. | Institute of Applied Physics and Computational Mathematics, Beijing 100088, China |
2. | School of Mathematical Sciences, Xiamen University, Xiamen 361005, China |
References:
[1] | |
[2] |
H. Brezis and S. Wainger, A note on the limiting cases of Sobolev embedding and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.
doi: 10.1080/03605308008820154. |
[3] |
K. Chang, Heat flow and boudnary value problem for harmonic maps, Annales de l'Institut Henri Poincaré (C) Analyse non éarire, 6 (1989), 363-395. |
[4] |
K. Chang, W. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differ. Geom., 36 (1992), 507-515. |
[5] |
Y. Chen and W. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math., 99 (1990), 567-578.
doi: 10.1007/BF01234431. |
[6] |
Y. Chen and M. Struwe, Existence and partial regularity for heat flow for harmonic maps, Math. Z., 201 (1989), 83-103.
doi: 10.1007/BF01161997. |
[7] |
H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.
doi: 10.1081/PDE-120021191. |
[8] |
M. M. Dai, J. Qing and M. Schonbek, Regularity of solutions to the liquid crystals systems in $R^2$ and $R^3$, Nonlinearity, 25 (2012), 513-532.
doi: 10.1088/0951-7715/25/2/513. |
[9] |
J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34.
doi: 10.1122/1.548883. |
[10] |
J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. |
[11] |
G. P. Galdi, An introduction to the Mathematical Theory of Navier-Stokes Equations, Vol. I: Linearized Steady Problems, Springer Verlag, New York, 1994.
doi: 10.1007/978-1-4612-5364-8. |
[12] |
P. G. de Gennes, The Physics of Liquid Crystals, $2^{nd}$ edition, Oxford University Press, Oxford, 1995. |
[13] |
M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.
doi: 10.1007/s00526-010-0331-5. |
[14] |
T. Huang and C. Y. Wang, Blow up criterion for nematic liquid crystal flows, Commun. Partial Differntial Equations, 37 (2012), 875-884.
doi: 10.1080/03605302.2012.659366. |
[15] |
X. D. Huang and Y. Wang, Global strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.
doi: 10.1016/j.jde.2012.08.029. |
[16] |
F. Jiang and Z. Tan, Global weak soutions to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266.
doi: 10.1002/mma.1132. |
[17] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, $2^{nd}$ edition, Gordon and Breach, New York, 1969. |
[18] |
O. A. Ladyzhenskaya and T. N. Shilkin, On coercive estimates for solutions of linear systems of hydrodynamic type, J. Math. Sci., 123 (2004), 4580-4596.
doi: 10.1023/B:JOTH.0000041476.69749.31. |
[19] |
Z. Lei, D. Li and X. Y. Zhang, Remarks of global well posedness of liquid crystal flows and heat flows of harmonic maps in two dimensions, Proc. Amer. Math. Soc., 142 (2014), 3801-3810.
doi: 10.1090/S0002-9939-2014-12057-0. |
[20] |
F. M. Leslie, Some contitutive equations for liquid crystals, Arch Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[21] |
F. M. Leslie, Theory of flow phenomena in liquid crystals, Advances in Liquid Crystals, 4 (1979), 1-81.
doi: 10.1016/B978-0-12-025004-2.50008-9. |
[22] |
X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767.
doi: 10.1016/j.jde.2011.08.045. |
[23] |
F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.
doi: 10.1002/cpa.3160420605. |
[24] |
F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flow in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[25] |
F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.
doi: 10.1002/cpa.3160480503. |
[26] |
F. H. Lin and C. Liu, Partial regularity of the nonlinear dissipative system modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-22. |
[27] |
P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Oxford Science Publication, Oxford, 1996. |
[28] |
X. Liu and Z. Zhang, Existence of the flow of liquid crystal system, Chinese Ann. Math., 30 (2009), 1-20. |
[29] |
H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278.
doi: 10.1007/s002090100332. |
[30] |
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-1116-7. |
[31] |
T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012. |
[32] |
M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv., 60 (1985), 558-581.
doi: 10.1007/BF02567432. |
[33] |
M. Wang and W. D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Cal. Var. Partial Differ. Equ., 51 (2014), 915-962.
doi: 10.1007/s00526-013-0700-y. |
[34] |
W. Wang, P. W. Zhang and Z. F. Zhang, Well-Posedness of the Ericksen-Leslie System, Arch. Rational Mech. Anal., 210 (2013), 837-855.
doi: 10.1007/s00205-013-0659-z. |
[35] |
C. Y. Wang, Well-posedeness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Rational Mech. Anal., 200 (2011), 1-19.
doi: 10.1007/s00205-010-0343-5. |
[36] |
H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. RWA, 12 (2011), 1510-1531.
doi: 10.1016/j.nonrwa.2010.10.010. |
[37] |
X. Xu and Z. F. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181.
doi: 10.1016/j.jde.2011.08.028. |
show all references
References:
[1] | |
[2] |
H. Brezis and S. Wainger, A note on the limiting cases of Sobolev embedding and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.
doi: 10.1080/03605308008820154. |
[3] |
K. Chang, Heat flow and boudnary value problem for harmonic maps, Annales de l'Institut Henri Poincaré (C) Analyse non éarire, 6 (1989), 363-395. |
[4] |
K. Chang, W. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differ. Geom., 36 (1992), 507-515. |
[5] |
Y. Chen and W. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math., 99 (1990), 567-578.
doi: 10.1007/BF01234431. |
[6] |
Y. Chen and M. Struwe, Existence and partial regularity for heat flow for harmonic maps, Math. Z., 201 (1989), 83-103.
doi: 10.1007/BF01161997. |
[7] |
H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.
doi: 10.1081/PDE-120021191. |
[8] |
M. M. Dai, J. Qing and M. Schonbek, Regularity of solutions to the liquid crystals systems in $R^2$ and $R^3$, Nonlinearity, 25 (2012), 513-532.
doi: 10.1088/0951-7715/25/2/513. |
[9] |
J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34.
doi: 10.1122/1.548883. |
[10] |
J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. |
[11] |
G. P. Galdi, An introduction to the Mathematical Theory of Navier-Stokes Equations, Vol. I: Linearized Steady Problems, Springer Verlag, New York, 1994.
doi: 10.1007/978-1-4612-5364-8. |
[12] |
P. G. de Gennes, The Physics of Liquid Crystals, $2^{nd}$ edition, Oxford University Press, Oxford, 1995. |
[13] |
M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.
doi: 10.1007/s00526-010-0331-5. |
[14] |
T. Huang and C. Y. Wang, Blow up criterion for nematic liquid crystal flows, Commun. Partial Differntial Equations, 37 (2012), 875-884.
doi: 10.1080/03605302.2012.659366. |
[15] |
X. D. Huang and Y. Wang, Global strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.
doi: 10.1016/j.jde.2012.08.029. |
[16] |
F. Jiang and Z. Tan, Global weak soutions to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266.
doi: 10.1002/mma.1132. |
[17] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, $2^{nd}$ edition, Gordon and Breach, New York, 1969. |
[18] |
O. A. Ladyzhenskaya and T. N. Shilkin, On coercive estimates for solutions of linear systems of hydrodynamic type, J. Math. Sci., 123 (2004), 4580-4596.
doi: 10.1023/B:JOTH.0000041476.69749.31. |
[19] |
Z. Lei, D. Li and X. Y. Zhang, Remarks of global well posedness of liquid crystal flows and heat flows of harmonic maps in two dimensions, Proc. Amer. Math. Soc., 142 (2014), 3801-3810.
doi: 10.1090/S0002-9939-2014-12057-0. |
[20] |
F. M. Leslie, Some contitutive equations for liquid crystals, Arch Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[21] |
F. M. Leslie, Theory of flow phenomena in liquid crystals, Advances in Liquid Crystals, 4 (1979), 1-81.
doi: 10.1016/B978-0-12-025004-2.50008-9. |
[22] |
X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767.
doi: 10.1016/j.jde.2011.08.045. |
[23] |
F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.
doi: 10.1002/cpa.3160420605. |
[24] |
F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flow in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[25] |
F. H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537.
doi: 10.1002/cpa.3160480503. |
[26] |
F. H. Lin and C. Liu, Partial regularity of the nonlinear dissipative system modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-22. |
[27] |
P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Oxford Science Publication, Oxford, 1996. |
[28] |
X. Liu and Z. Zhang, Existence of the flow of liquid crystal system, Chinese Ann. Math., 30 (2009), 1-20. |
[29] |
H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278.
doi: 10.1007/s002090100332. |
[30] |
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-1116-7. |
[31] |
T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012. |
[32] |
M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv., 60 (1985), 558-581.
doi: 10.1007/BF02567432. |
[33] |
M. Wang and W. D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Cal. Var. Partial Differ. Equ., 51 (2014), 915-962.
doi: 10.1007/s00526-013-0700-y. |
[34] |
W. Wang, P. W. Zhang and Z. F. Zhang, Well-Posedness of the Ericksen-Leslie System, Arch. Rational Mech. Anal., 210 (2013), 837-855.
doi: 10.1007/s00205-013-0659-z. |
[35] |
C. Y. Wang, Well-posedeness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Rational Mech. Anal., 200 (2011), 1-19.
doi: 10.1007/s00205-010-0343-5. |
[36] |
H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. RWA, 12 (2011), 1510-1531.
doi: 10.1016/j.nonrwa.2010.10.010. |
[37] |
X. Xu and Z. F. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181.
doi: 10.1016/j.jde.2011.08.028. |
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