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

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

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.
Citation: Meng Wang, Wendong Wang, Zhifei Zhang. On the uniqueness of weak solution for the 2-D Ericksen--Leslie system. Discrete & 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.   Google Scholar

[2]

J. Y. Chemin, Perfect Incompressible Fluids,, Oxford Lecture series in Mathematics and its Applications, (1998).   Google Scholar

[3]

J. Ericksen, Conservation laws for liquid crystals,, Trans. Soc. Rheol., 5 (1961), 23.  doi: 10.1122/1.548883.  Google Scholar

[4]

M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations,, part II, (1998).  doi: 10.1007/978-3-662-06218-0.  Google Scholar

[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.  doi: 10.1007/s00526-010-0331-5.  Google Scholar

[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.  doi: 10.1016/j.aim.2012.06.009.  Google Scholar

[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.  doi: 10.1080/03605302.2013.871026.  Google Scholar

[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.  doi: 10.1007/s00220-014-2079-9.  Google Scholar

[9]

T. Huang and C.-Y. Wang, Blow up criterion for nematic liquid crystal flows,, Comm. Partial Differential Equations, 37 (2012), 875.  doi: 10.1080/03605302.2012.659366.  Google Scholar

[10]

F. Leslie, Some constitutive equations for anisotropic fluids,, Quart. J. Mech. Appl. Math., 19 (1966), 357.  doi: 10.1093/qjmam/19.3.357.  Google Scholar

[11]

F. Leslie, Some constitutive equations for liquid crystals,, Arch. Ration. Mech. Anal., 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[12]

F. Leslie, Theory of flow phenomena in liquid crystals,, The Theory of Liquid Crystals, 4 (1979), 1.  doi: 10.1016/B978-0-12-025004-2.50008-9.  Google Scholar

[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 $\mathbbR^2$,, , ().   Google Scholar

[14]

F.-H. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789.  doi: 10.1002/cpa.3160420605.  Google Scholar

[15]

F.-H. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions,, Arch. Ration. Mech. Anal., 197 (2010), 297.  doi: 10.1007/s00205-009-0278-x.  Google Scholar

[16]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar

[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.   Google Scholar

[18]

F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, Arch. Ration. Mech. Anal., 154 (2000), 135.  doi: 10.1007/s002050000102.  Google Scholar

[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.  doi: 10.1007/s11401-010-0612-5.  Google Scholar

[20]

O. Parodi, Stress tensor for a nematic liquid crystal,, Journal de Physique, 31 (1970), 581.   Google Scholar

[21]

M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces,, Comm. Math. Helv., 60 (1985), 558.  doi: 10.1007/BF02567432.  Google Scholar

[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.  doi: 10.1007/s00205-010-0343-5.  Google Scholar

[23]

C. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $S^2$,, Jounal of Functional Analysis, 266 (2014), 5360.  doi: 10.1016/j.jfa.2014.02.023.  Google Scholar

[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.  doi: 10.1002/cpa.21549.  Google Scholar

[25]

W. Wang, P. Zhang and Z. Zhang, Well-posedness of the Ericksen-Leslie system,, Arch. Ration. Mech. Anal., 210 (2013), 837.  doi: 10.1007/s00205-013-0659-z.  Google Scholar

[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.  doi: 10.1007/s00526-013-0700-y.  Google Scholar

[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.  doi: 10.1007/s00205-012-0588-2.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.08.028.  Google Scholar

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.   Google Scholar

[2]

J. Y. Chemin, Perfect Incompressible Fluids,, Oxford Lecture series in Mathematics and its Applications, (1998).   Google Scholar

[3]

J. Ericksen, Conservation laws for liquid crystals,, Trans. Soc. Rheol., 5 (1961), 23.  doi: 10.1122/1.548883.  Google Scholar

[4]

M. Giaquinta, G. Modica and J. Soucek, Cartesian Currents in the Calculus of Variations,, part II, (1998).  doi: 10.1007/978-3-662-06218-0.  Google Scholar

[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.  doi: 10.1007/s00526-010-0331-5.  Google Scholar

[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.  doi: 10.1016/j.aim.2012.06.009.  Google Scholar

[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.  doi: 10.1080/03605302.2013.871026.  Google Scholar

[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.  doi: 10.1007/s00220-014-2079-9.  Google Scholar

[9]

T. Huang and C.-Y. Wang, Blow up criterion for nematic liquid crystal flows,, Comm. Partial Differential Equations, 37 (2012), 875.  doi: 10.1080/03605302.2012.659366.  Google Scholar

[10]

F. Leslie, Some constitutive equations for anisotropic fluids,, Quart. J. Mech. Appl. Math., 19 (1966), 357.  doi: 10.1093/qjmam/19.3.357.  Google Scholar

[11]

F. Leslie, Some constitutive equations for liquid crystals,, Arch. Ration. Mech. Anal., 28 (1968), 265.  doi: 10.1007/BF00251810.  Google Scholar

[12]

F. Leslie, Theory of flow phenomena in liquid crystals,, The Theory of Liquid Crystals, 4 (1979), 1.  doi: 10.1016/B978-0-12-025004-2.50008-9.  Google Scholar

[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 $\mathbbR^2$,, , ().   Google Scholar

[14]

F.-H. Lin, Nonlinear theory of defects in nematic liquid crystal: Phase transition and flow phenomena,, Comm. Pure Appl. Math., 42 (1989), 789.  doi: 10.1002/cpa.3160420605.  Google Scholar

[15]

F.-H. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions,, Arch. Ration. Mech. Anal., 197 (2010), 297.  doi: 10.1007/s00205-009-0278-x.  Google Scholar

[16]

F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals,, Comm. Pure Appl. Math., 48 (1995), 501.  doi: 10.1002/cpa.3160480503.  Google Scholar

[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.   Google Scholar

[18]

F.-H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system,, Arch. Ration. Mech. Anal., 154 (2000), 135.  doi: 10.1007/s002050000102.  Google Scholar

[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.  doi: 10.1007/s11401-010-0612-5.  Google Scholar

[20]

O. Parodi, Stress tensor for a nematic liquid crystal,, Journal de Physique, 31 (1970), 581.   Google Scholar

[21]

M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces,, Comm. Math. Helv., 60 (1985), 558.  doi: 10.1007/BF02567432.  Google Scholar

[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.  doi: 10.1007/s00205-010-0343-5.  Google Scholar

[23]

C. Wang and X. Xu, On the rigidity of nematic liquid crystal flow on $S^2$,, Jounal of Functional Analysis, 266 (2014), 5360.  doi: 10.1016/j.jfa.2014.02.023.  Google Scholar

[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.  doi: 10.1002/cpa.21549.  Google Scholar

[25]

W. Wang, P. Zhang and Z. Zhang, Well-posedness of the Ericksen-Leslie system,, Arch. Ration. Mech. Anal., 210 (2013), 837.  doi: 10.1007/s00205-013-0659-z.  Google Scholar

[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.  doi: 10.1007/s00526-013-0700-y.  Google Scholar

[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.  doi: 10.1007/s00205-012-0588-2.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.08.028.  Google Scholar

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