On the uniqueness of weak solution for the 2-D Ericksen--Leslie system

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May 2016, 21(3): 919-941. doi: 10.3934/dcdsb.2016.21.919

On the uniqueness of weak solution for the 2-D Ericksen--Leslie system

Meng Wang ^1, , Wendong Wang ^2, and Zhifei Zhang ^3,

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

Received October 2014 Revised September 2015 Published January 2016

Abstract
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In this paper, we prove the uniqueness of weak solutions to the two dimensional full Ericksen-Leslie system with the Leslie stress and general Ericksen stress under the physical constrains on the Leslie coefficients. This question remains unknown even in the case when the Leslie stress is vanishing. The main technique used in the proof is Littlewood-Paley analysis performed in a very delicate way. Different from the earlier result in [28], we introduce a new metric and explore the algebraic structure of the molecular field.

Keywords: Ericksen--Leslie system, Littlewood-Paley theory., uniqueness of weak solution.

Mathematics Subject Classification: Primary: 35A02, 76A15; Secondary: 35Q3.

Citation: Meng Wang, Wendong Wang, Zhifei Zhang. On the uniqueness of weak solution for the 2-D Ericksen--Leslie system. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 919-941. doi: 10.3934/dcdsb.2016.21.919

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 $\mathbb{R}^2$, 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 $\mathbb{R}^3$, 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 $\mathbb{R}^2$, 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 $\mathbb{R}^2$, arXiv:1410.1119
[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 $\mathbb{R}^2$, 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 $\mathbb{R}^3$, 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 $\mathbb{R}^2$, 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 $\mathbb{R}^2$, arXiv:1410.1119
[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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