-
Previous Article
Globally weak solutions to the flow of compressible liquid crystals system
- DCDS Home
- This Issue
-
Next Article
On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains
Uniqueness of harmonic map heat flows and liquid crystal flows
1. | Department of Mathematics, South China University of Technology, Guangzhou, 510640 |
References:
[1] |
Y. Chen and W. Y. Ding, Blow up and global existence for heat flows of harmonic maps, Invent. Math., 99 (1990), 567-578.
doi: 10.1007/BF01234431. |
[2] |
K. Chang, W. Ding and R. Ye, Finite time blow-up of the heat flow of harmonic maps from surfaces, JDG, 36 (1992), 507-515. |
[3] |
J. M. Coron and J. M. Ghidaglia, Explosion en temps fini pour le flot des applications harmoniques, C. R. Acad. Sci. Paris, 308 (1989), 339-344. |
[4] |
Y. Chen and M. Struwe, Existence and partial regularity results for the heat flow of harmonic maps, Math. Z., 201 (1989), 83-103.
doi: 10.1007/BF01161997. |
[5] |
J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. |
[6] |
J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160.
doi: 10.2307/2373037. |
[7] |
A. Freire, Uniqueness for the harmonic map flow from surfaces to general targets, Comm. Math. Helvetici., 70 (1995), 310-338.
doi: 10.1007/BF02566010. |
[8] |
P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," New York, Oxford University Press, 1993. |
[9] |
J. Jost, Ein existenzbeiweis fiir harmonisch Abbildungen, die ein Dirichlet problem 16sen mittels der methode des warmeflusses, Manuscripta Math., 34 (1981), 17-25.
doi: 10.1007/BF01168706. |
[10] |
H. Koch and D. Tataru, Well-posedness for theNavier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[11] |
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[12] |
J. Y. Lin and S. J. Ding, On the well-posedness for the heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals in critical spaces,, Math. Meth. Appl. Sciences, ().
doi: 10.1002/mma.1548. |
[13] |
F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. |
[14] |
F. Lin and C. Liu, Partial regularities of nonlinear disspative systems modeling the flow of liquid crystals, DCDS, 2 (1996), 1-23. |
[15] |
F. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Rational Mech. Aanl., 154 (2000), 135-156. |
[16] |
F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[17] |
F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Annals of Mathematics, 31 (2010), 921-938.
doi: 10.1007/s11401-010-0612-5. |
[18] |
H. Miura, Remark on uniqueness of mild solutions to the Navier-Stokes equations, J. Funt. Anal., 218 (2005), 110-129.
doi: 10.1016/j.jfa.2004.07.007. |
[19] |
M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Diff. Geom., 28 (1988), 485-502. |
[20] |
A. Soyeur, A global existence result for the heat flow of harmonic maps, Comm. PDE, 15 (1990), 237-244.
doi: 10.1080/03605309908820685. |
[21] |
M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv, 60 (1985), 558-581.
doi: 10.1007/BF02567432. |
[22] |
C. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Rational Mech. Anal., 200 (2011), 1-19. |
[23] |
X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, JDE, 252 (2012), 1169-1181.
doi: 10.1016/j.jde.2011.08.028. |
show all references
References:
[1] |
Y. Chen and W. Y. Ding, Blow up and global existence for heat flows of harmonic maps, Invent. Math., 99 (1990), 567-578.
doi: 10.1007/BF01234431. |
[2] |
K. Chang, W. Ding and R. Ye, Finite time blow-up of the heat flow of harmonic maps from surfaces, JDG, 36 (1992), 507-515. |
[3] |
J. M. Coron and J. M. Ghidaglia, Explosion en temps fini pour le flot des applications harmoniques, C. R. Acad. Sci. Paris, 308 (1989), 339-344. |
[4] |
Y. Chen and M. Struwe, Existence and partial regularity results for the heat flow of harmonic maps, Math. Z., 201 (1989), 83-103.
doi: 10.1007/BF01161997. |
[5] |
J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. |
[6] |
J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160.
doi: 10.2307/2373037. |
[7] |
A. Freire, Uniqueness for the harmonic map flow from surfaces to general targets, Comm. Math. Helvetici., 70 (1995), 310-338.
doi: 10.1007/BF02566010. |
[8] |
P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," New York, Oxford University Press, 1993. |
[9] |
J. Jost, Ein existenzbeiweis fiir harmonisch Abbildungen, die ein Dirichlet problem 16sen mittels der methode des warmeflusses, Manuscripta Math., 34 (1981), 17-25.
doi: 10.1007/BF01168706. |
[10] |
H. Koch and D. Tataru, Well-posedness for theNavier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[11] |
F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[12] |
J. Y. Lin and S. J. Ding, On the well-posedness for the heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals in critical spaces,, Math. Meth. Appl. Sciences, ().
doi: 10.1002/mma.1548. |
[13] |
F. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501-537. |
[14] |
F. Lin and C. Liu, Partial regularities of nonlinear disspative systems modeling the flow of liquid crystals, DCDS, 2 (1996), 1-23. |
[15] |
F. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Rational Mech. Aanl., 154 (2000), 135-156. |
[16] |
F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[17] |
F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Annals of Mathematics, 31 (2010), 921-938.
doi: 10.1007/s11401-010-0612-5. |
[18] |
H. Miura, Remark on uniqueness of mild solutions to the Navier-Stokes equations, J. Funt. Anal., 218 (2005), 110-129.
doi: 10.1016/j.jfa.2004.07.007. |
[19] |
M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Diff. Geom., 28 (1988), 485-502. |
[20] |
A. Soyeur, A global existence result for the heat flow of harmonic maps, Comm. PDE, 15 (1990), 237-244.
doi: 10.1080/03605309908820685. |
[21] |
M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv, 60 (1985), 558-581.
doi: 10.1007/BF02567432. |
[22] |
C. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Rational Mech. Anal., 200 (2011), 1-19. |
[23] |
X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows, JDE, 252 (2012), 1169-1181.
doi: 10.1016/j.jde.2011.08.028. |
[1] |
T. Tachim Medjo. On the existence and uniqueness of solution to a stochastic simplified liquid crystal model. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2243-2264. doi: 10.3934/cpaa.2019101 |
[2] |
Domenico Mucci. Maps into projective spaces: Liquid crystal and conformal energies. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 597-635. doi: 10.3934/dcdsb.2012.17.597 |
[3] |
Zhaoyang Qiu, Yixuan Wang. Martingale solution for stochastic active liquid crystal system. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2227-2268. doi: 10.3934/dcds.2020360 |
[4] |
Zhenlu Cui, M. Carme Calderer, Qi Wang. Mesoscale structures in flows of weakly sheared cholesteric liquid crystal polymers. Discrete and Continuous Dynamical Systems - B, 2006, 6 (2) : 291-310. doi: 10.3934/dcdsb.2006.6.291 |
[5] |
Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic and Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691 |
[6] |
Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure and Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341 |
[7] |
Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455 |
[8] |
Chun Liu, Jie Shen. On liquid crystal flows with free-slip boundary conditions. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 307-318. doi: 10.3934/dcds.2001.7.307 |
[9] |
Xiaoli Li, Boling Guo. Well-posedness for the three-dimensional compressible liquid crystal flows. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1913-1937. doi: 10.3934/dcdss.2016078 |
[10] |
Zhenlu Cui, Qi Wang. Permeation flows in cholesteric liquid crystal polymers under oscillatory shear. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 45-60. doi: 10.3934/dcdsb.2011.15.45 |
[11] |
Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007 |
[12] |
Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4515-4533. doi: 10.3934/dcdsb.2020110 |
[13] |
Yang Liu, Xin Zhong. On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5219-5238. doi: 10.3934/cpaa.2020234 |
[14] |
Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229 |
[15] |
Jishan Fan, Fei Jiang. Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions. Communications on Pure and Applied Analysis, 2016, 15 (1) : 73-90. doi: 10.3934/cpaa.2016.15.73 |
[16] |
Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165 |
[17] |
Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations and Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007 |
[18] |
Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161 |
[19] |
Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379 |
[20] |
Shengquan Liu, Jianwen Zhang. Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2631-2648. doi: 10.3934/dcdsb.2016065 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]