American Institute of Mathematical Sciences

Advanced
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

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 & 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

[1]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[2]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326
[3]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[4]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384
[5]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[6]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[7]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[8]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[9]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[10]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[11]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320
[12]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353
[13]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[14]

Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331
[15]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325
[16]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[17]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[18]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[19]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[20]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences