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On the uniqueness of weak solution for the 2-D Ericksen--Leslie system
1. | Department of Mathematics, Zhejiang University, Hangzhou 310027, China |
2. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
3. | School of Mathematical Sciences, Peking University, Beijing 100871, China |
References:
[1] |
J. M. Bony, Calcul symbolique et propagation des singularitiés pour les équations aux dérivées partielles non linéaires, Ann. Ecole Norm. Sup., 14 (1981), 209-246. |
[2] |
J. Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture series in Mathematics and its Applications, Vol. 14, Oxford University Press, New York, 1998. |
[3] |
J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 23-34.
doi: 10.1122/1.548883. |
[4] |
M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations, part II, Variational integrals, A series of modern serveys in mathematics, Vol. 38, Springer-Verlag, 1998.
doi: 10.1007/978-3-662-06218-0. |
[5] |
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. |
[6] |
M.-C. Hong and Z.-P. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbbR^2$, Adv. Math., 231 (2012), 1364-1400.
doi: 10.1016/j.aim.2012.06.009. |
[7] |
M.-C. Hong, J.-K. Li and Z.-P. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\mathbbR^3$, Comm. Partial Differential Equations, 39 (2014), 1284-1328.
doi: 10.1080/03605302.2013.871026. |
[8] |
J.-R. Huang, F.-H. Lin and C.-Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbbR^2$, Comm. Math. Phys., 331 (2014), 805-850.
doi: 10.1007/s00220-014-2079-9. |
[9] |
T. Huang and C.-Y. Wang, Blow up criterion for nematic liquid crystal flows, Comm. Partial Differential Equations, 37 (2012), 875-884.
doi: 10.1080/03605302.2012.659366. |
[10] |
F. Leslie, Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math., 19 (1966), 357-370.
doi: 10.1093/qjmam/19.3.357. |
[11] |
F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[12] |
F. Leslie, Theory of flow phenomena in liquid crystals, The Theory of Liquid Crystals, Academic Press, London-New York, 4 (1979), 1-81.
doi: 10.1016/B978-0-12-025004-2.50008-9. |
[13] |
J.-K. Li, E. Titi and Z.-P. Xin, On the uniqueness of weak solutions to weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbbR^2$,, , ().
|
[14] |
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. |
[15] |
F.-H. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[16] |
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. |
[17] |
F.-H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems, 2 (1996), 1-22. |
[18] |
F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 154 (2000), 135-156.
doi: 10.1007/s002050000102. |
[19] |
F.-H. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938.
doi: 10.1007/s11401-010-0612-5. |
[20] |
O. Parodi, Stress tensor for a nematic liquid crystal, Journal de Physique, 31 (1970), 581-584. |
[21] |
M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comm. 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. Ration. Mech. Anal., 200 (2011), 1-19.
doi: 10.1007/s00205-010-0343-5. |
[23] |
C. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $S^2$, Jounal of Functional Analysis, 266 (2014), 5360-5376.
doi: 10.1016/j.jfa.2014.02.023. |
[24] |
W. Wang, P. Zhang and Z. Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen- Leslie equation, Comm. Pure Appl. Math., 68 (2015), 1326-1398.
doi: 10.1002/cpa.21549. |
[25] |
W. Wang, P. Zhang and Z. Zhang, Well-posedness of the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 210 (2013), 837-855.
doi: 10.1007/s00205-013-0659-z. |
[26] |
M. Wang and W.-D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differential Equations, 51 (2014), 915-962.
doi: 10.1007/s00526-013-0700-y. |
[27] |
H. Wu, X. Xu and C. Liu, On the general Ericksen Leslie system: Parodis relation, well-posedness and stability, Arch. Ration. Mech. Anal., 208 (2013), 59-107.
doi: 10.1007/s00205-012-0588-2. |
[28] |
X. Xu and Z. 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] |
J. M. Bony, Calcul symbolique et propagation des singularitiés pour les équations aux dérivées partielles non linéaires, Ann. Ecole Norm. Sup., 14 (1981), 209-246. |
[2] |
J. Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture series in Mathematics and its Applications, Vol. 14, Oxford University Press, New York, 1998. |
[3] |
J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 23-34.
doi: 10.1122/1.548883. |
[4] |
M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations, part II, Variational integrals, A series of modern serveys in mathematics, Vol. 38, Springer-Verlag, 1998.
doi: 10.1007/978-3-662-06218-0. |
[5] |
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. |
[6] |
M.-C. Hong and Z.-P. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\mathbbR^2$, Adv. Math., 231 (2012), 1364-1400.
doi: 10.1016/j.aim.2012.06.009. |
[7] |
M.-C. Hong, J.-K. Li and Z.-P. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\mathbbR^3$, Comm. Partial Differential Equations, 39 (2014), 1284-1328.
doi: 10.1080/03605302.2013.871026. |
[8] |
J.-R. Huang, F.-H. Lin and C.-Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbbR^2$, Comm. Math. Phys., 331 (2014), 805-850.
doi: 10.1007/s00220-014-2079-9. |
[9] |
T. Huang and C.-Y. Wang, Blow up criterion for nematic liquid crystal flows, Comm. Partial Differential Equations, 37 (2012), 875-884.
doi: 10.1080/03605302.2012.659366. |
[10] |
F. Leslie, Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math., 19 (1966), 357-370.
doi: 10.1093/qjmam/19.3.357. |
[11] |
F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.
doi: 10.1007/BF00251810. |
[12] |
F. Leslie, Theory of flow phenomena in liquid crystals, The Theory of Liquid Crystals, Academic Press, London-New York, 4 (1979), 1-81.
doi: 10.1016/B978-0-12-025004-2.50008-9. |
[13] |
J.-K. Li, E. Titi and Z.-P. Xin, On the uniqueness of weak solutions to weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbbR^2$,, , ().
|
[14] |
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. |
[15] |
F.-H. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[16] |
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. |
[17] |
F.-H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems, 2 (1996), 1-22. |
[18] |
F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 154 (2000), 135-156.
doi: 10.1007/s002050000102. |
[19] |
F.-H. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938.
doi: 10.1007/s11401-010-0612-5. |
[20] |
O. Parodi, Stress tensor for a nematic liquid crystal, Journal de Physique, 31 (1970), 581-584. |
[21] |
M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comm. 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. Ration. Mech. Anal., 200 (2011), 1-19.
doi: 10.1007/s00205-010-0343-5. |
[23] |
C. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $S^2$, Jounal of Functional Analysis, 266 (2014), 5360-5376.
doi: 10.1016/j.jfa.2014.02.023. |
[24] |
W. Wang, P. Zhang and Z. Zhang, The small Deborah number limit of the Doi-Onsager equation to the Ericksen- Leslie equation, Comm. Pure Appl. Math., 68 (2015), 1326-1398.
doi: 10.1002/cpa.21549. |
[25] |
W. Wang, P. Zhang and Z. Zhang, Well-posedness of the Ericksen-Leslie system, Arch. Ration. Mech. Anal., 210 (2013), 837-855.
doi: 10.1007/s00205-013-0659-z. |
[26] |
M. Wang and W.-D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differential Equations, 51 (2014), 915-962.
doi: 10.1007/s00526-013-0700-y. |
[27] |
H. Wu, X. Xu and C. Liu, On the general Ericksen Leslie system: Parodis relation, well-posedness and stability, Arch. Ration. Mech. Anal., 208 (2013), 59-107.
doi: 10.1007/s00205-012-0588-2. |
[28] |
X. Xu and Z. 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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