Global well-posedness for the dynamical Q-tensor model of liquid crystals
We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is...
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| Published in | Science China. Mathematics Vol. 58; no. 6; pp. 1349 - 1366 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Beijing
Science China Press
01.06.2015
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1674-7283 1869-1862 |
| DOI | 10.1007/s11425-015-4990-8 |
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| Abstract | We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions. |
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| AbstractList | We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic
Q
-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions. We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions. |
| Author | HUANG JinRui DING ShiJin |
| AuthorAffiliation | School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China |
| Author_xml | – sequence: 1 givenname: JinRui surname: Huang fullname: Huang, JinRui organization: School of Mathematics and Computational Science, Wuyi University – sequence: 2 givenname: ShiJin surname: Ding fullname: Ding, ShiJin email: dingsj@scnu.edu.cn organization: School of Mathematical Sciences, South China Normal University |
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| Cites_doi | 10.1007/s11425-013-4620-2 10.1007/s00205-009-0249-2 10.3934/dcds.2009.23.455 10.1007/s00205-012-0530-7 10.1007/s00205-012-0588-2 10.1137/13095015X 10.1002/mana.200610776 10.1007/s00205-011-0443-x 10.1093/oso/9780195076943.001.0001 10.1007/s11425-013-4761-3 10.1016/j.na.2014.09.011 10.1103/PhysRevE.58.7475 10.1007/s00526-011-0460-5 10.1016/j.jde.2014.11.008 10.1137/10079224X 10.4310/CMS.2015.v13.n1.a3 10.1007/s00220-014-2079-9 10.1007/s11425-013-4746-2 10.1137/130945405 |
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| DocumentTitleAlternate | Global well-posedness for the dynamical Q-tensor model of liquid crystals |
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| Keywords | nematic liquid crystals global solutions uniqueness 35R35 35Q30 tensor 76N10 |
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| Notes | dynamical tensor Stokes parabolic nematic viscosity Navier estimates uniqueness proof We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor system. We first prove the global existence of weak solutions in dimension three. Furthermore, the global well-posedness of strong solutions is studied with sufficiently large viscosity of fluid. Finally, we show a continuous dependence result on the initial data which directly yields the weak-strong uniqueness of solutions. 11-1787/N ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
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| References | Guillén-GonzálezFRodríguez-BlellidoM AWeak solutions for a initial-boundary Q-tensor problem related to liquid crystalsNonlinear Anal TMA20151128410410.1016/j.na.2014.09.0111304.35547 ChenHZhangP WA tensor model for liquid crystals on a spherical surfaceSci China Math2013562549255910.1007/s11425-013-4746-2 DingS JHuangJ RLinJ YGlobal existence for slightly compressible hydrodynamic flow of liquid crystals in two dimensionsSci China Math2013562233225010.1007/s11425-013-4620-2062721143123568 XuJZhangP WFrom microscopic theory to macroscopic theory-symmetries and order parameters of rigid moleculesSci China Math20145744346810.1007/s11425-013-4761-31299.820163166230 HuangJ RDingS JCompressible hydrodynamic flow of nematic liquid crystals with vacuumJ Differential Equations20152581653168410.1016/j.jde.2014.11.008063971213295596 BaumanPParkJPhillipsDAnalysis of nematic liquid crystals with disclination linesArch Rational Mech Anal201220579582610.1007/s00205-012-0530-71281.760202960033 QianTShengPGeneralized hydrodynamic equations for nematic liquid crystlasPhysical Review E1998587475748510.1103/PhysRevE.58.7475 HuangJ RLinF HWangC YRegularity and existence of global solutions to the Ericksen-Leslie system in R2Comm Math Phys201433180585010.1007/s00220-014-2079-91298.351473238531 WangD HXuXYuCGlobal weak solution for a coupled compressible Navier-Stokes and Q-tensor systemCommum Math Sci201513498210.4310/CMS.2015.v13.n1.a3064082933238138 Guillén-GonzálezFRodríguez-BlellidoM ARojas-MedarM ASufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal modelMa Math Nachr200928284686710.1002/mana.2006107761173.35033 De GennesP GThe Physics of Liquid Crystals1974OxfordClarendon Press PaicuMZarnescuAGlobal existence and regularity for the full coupled Navier-Stokes and Q-tensor systemSIAM J Math Anal2011432009204910.1137/10079224X1233.351602837493 PaicuMZarnescuAEnergy dissipation and regularity for a coupled Navier-Stokes and Q-tensor systemArch Ration Mech Anal2012203456710.1007/s00205-011-0443-x061019802864407 WuHXuXLiuCAsymptotic behavior for a nematic liquid crystal model with different kinematic transport propertiesCalc Var Partial Differ Equ20124531934510.1007/s00526-011-0460-51259.350402984135 FeireislERoccaESchimpernaGNonisothermal nematic liquid crystal flows with the Ball-Majumdar free energyAnn Mat Pura Appl, in press2015 AbelsHDolzmannGLiuY NWell-posedness of a fully-coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary dataSIAM J Math Anal2014463050307710.1137/130945405063658253252809 HanJ QLuoYWangWFrom microscopic theory to macroscopic theory: a systematic study on static modeling for liquid crystalsArch Rational Mech Anal2014 Mottram N J, Newton C. Introduction to Q-tensor theory. University of Strathclyde, Department of Mathematics, Research Report, 10, 2004 Guillén-GonzálezFRodríguez-BlellidoM AWeak time regularity and uniqueness for a Q-tensor modelSIAM J Math Anal2014463540356710.1137/13095015X064057153272623 WuHXuXLiuCOn the General Ericksen-Leslie System: Parodis Relation, Well-Posedness and StabilityArch Ration Mech Anal20132085910710.1007/s00205-012-0588-2061490093021544 SunHLiuCOn energetic variational approaches in modelling the nematic liquid crystal flowsDiscrete Contin Dyn Syst20092345547510.3934/dcds.2009.23.4551156.760072449088 MajumdarAZarnescuALandau-De Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyondArch Rational Mech Anal201019622728010.1007/s00205-009-0249-21304.760072601074 BerisA NEdwardsB JThermodynamics of Flowing Systems with Internal Microstructure1994New YorkOxford University Press WangWZhangP WZhangZ FFrom microscopic theory to macroscopic theory: dynamics of the rod-like liquid crystal molecules2013 AbelsHDolzmannGLiuY NStrong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions2013 WangWZhangP WZhangZ FThe small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equationComm Pure Appl Math2015 WangWZhangP WZhangZ FRigorous derivation from Landau-de Gennes theory to Ericksen-Leslie theorySIAM J Math Anal2015 J Q Han (4990_CR12) 2014 H Wu (4990_CR25) 2012; 45 D H Wang (4990_CR21) 2015; 13 J R Huang (4990_CR14) 2015; 258 M Paicu (4990_CR18) 2012; 203 H Abels (4990_CR1) 2013 H Abels (4990_CR2) 2014; 46 P Bauman (4990_CR3) 2012; 205 W Wang (4990_CR22) 2013 W Wang (4990_CR24) 2015 F Guillén-González (4990_CR9) 2009; 282 4990_CR16 H Chen (4990_CR5) 2013; 56 S J Ding (4990_CR7) 2013; 56 M Paicu (4990_CR17) 2011; 43 T Qian (4990_CR19) 1998; 58 H Sun (4990_CR20) 2009; 23 E Feireisl (4990_CR8) 2015 P G Gennes De (4990_CR6) 1974 J R Huang (4990_CR13) 2014; 331 F Guillén-González (4990_CR10) 2014; 46 A N Beris (4990_CR4) 1994 F Guillén-González (4990_CR11) 2015; 112 W Wang (4990_CR23) 2015 A Majumdar (4990_CR15) 2010; 196 H Wu (4990_CR26) 2013; 208 J Xu (4990_CR27) 2014; 57 |
| References_xml | – reference: FeireislERoccaESchimpernaGNonisothermal nematic liquid crystal flows with the Ball-Majumdar free energyAnn Mat Pura Appl, in press2015 – reference: WuHXuXLiuCAsymptotic behavior for a nematic liquid crystal model with different kinematic transport propertiesCalc Var Partial Differ Equ20124531934510.1007/s00526-011-0460-51259.350402984135 – reference: BaumanPParkJPhillipsDAnalysis of nematic liquid crystals with disclination linesArch Rational Mech Anal201220579582610.1007/s00205-012-0530-71281.760202960033 – reference: AbelsHDolzmannGLiuY NWell-posedness of a fully-coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary dataSIAM J Math Anal2014463050307710.1137/130945405063658253252809 – reference: Guillén-GonzálezFRodríguez-BlellidoM AWeak time regularity and uniqueness for a Q-tensor modelSIAM J Math Anal2014463540356710.1137/13095015X064057153272623 – reference: HanJ QLuoYWangWFrom microscopic theory to macroscopic theory: a systematic study on static modeling for liquid crystalsArch Rational Mech Anal2014 – reference: DingS JHuangJ RLinJ YGlobal existence for slightly compressible hydrodynamic flow of liquid crystals in two dimensionsSci China Math2013562233225010.1007/s11425-013-4620-2062721143123568 – reference: Mottram N J, Newton C. Introduction to Q-tensor theory. University of Strathclyde, Department of Mathematics, Research Report, 10, 2004 – reference: QianTShengPGeneralized hydrodynamic equations for nematic liquid crystlasPhysical Review E1998587475748510.1103/PhysRevE.58.7475 – reference: De GennesP GThe Physics of Liquid Crystals1974OxfordClarendon Press – reference: XuJZhangP WFrom microscopic theory to macroscopic theory-symmetries and order parameters of rigid moleculesSci China Math20145744346810.1007/s11425-013-4761-31299.820163166230 – reference: Guillén-GonzálezFRodríguez-BlellidoM AWeak solutions for a initial-boundary Q-tensor problem related to liquid crystalsNonlinear Anal TMA20151128410410.1016/j.na.2014.09.0111304.35547 – reference: MajumdarAZarnescuALandau-De Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyondArch Rational Mech Anal201019622728010.1007/s00205-009-0249-21304.760072601074 – reference: AbelsHDolzmannGLiuY NStrong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions2013 – reference: WangWZhangP WZhangZ FThe small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equationComm Pure Appl Math2015 – reference: WangWZhangP WZhangZ FFrom microscopic theory to macroscopic theory: dynamics of the rod-like liquid crystal molecules2013 – reference: PaicuMZarnescuAGlobal existence and regularity for the full coupled Navier-Stokes and Q-tensor systemSIAM J Math Anal2011432009204910.1137/10079224X1233.351602837493 – reference: WangWZhangP WZhangZ FRigorous derivation from Landau-de Gennes theory to Ericksen-Leslie theorySIAM J Math Anal2015 – reference: BerisA NEdwardsB JThermodynamics of Flowing Systems with Internal Microstructure1994New YorkOxford University Press – reference: WuHXuXLiuCOn the General Ericksen-Leslie System: Parodis Relation, Well-Posedness and StabilityArch Ration Mech Anal20132085910710.1007/s00205-012-0588-2061490093021544 – reference: HuangJ RDingS JCompressible hydrodynamic flow of nematic liquid crystals with vacuumJ Differential Equations20152581653168410.1016/j.jde.2014.11.008063971213295596 – reference: Guillén-GonzálezFRodríguez-BlellidoM ARojas-MedarM ASufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal modelMa Math Nachr200928284686710.1002/mana.2006107761173.35033 – reference: PaicuMZarnescuAEnergy dissipation and regularity for a coupled Navier-Stokes and Q-tensor systemArch Ration Mech Anal2012203456710.1007/s00205-011-0443-x061019802864407 – reference: ChenHZhangP WA tensor model for liquid crystals on a spherical surfaceSci China Math2013562549255910.1007/s11425-013-4746-2 – reference: SunHLiuCOn energetic variational approaches in modelling the nematic liquid crystal flowsDiscrete Contin Dyn Syst20092345547510.3934/dcds.2009.23.4551156.760072449088 – reference: WangD HXuXYuCGlobal weak solution for a coupled compressible Navier-Stokes and Q-tensor systemCommum Math Sci201513498210.4310/CMS.2015.v13.n1.a3064082933238138 – reference: HuangJ RLinF HWangC YRegularity and existence of global solutions to the Ericksen-Leslie system in R2Comm Math Phys201433180585010.1007/s00220-014-2079-91298.351473238531 – volume: 56 start-page: 2233 year: 2013 ident: 4990_CR7 publication-title: Sci China Math doi: 10.1007/s11425-013-4620-2 – volume: 196 start-page: 227 year: 2010 ident: 4990_CR15 publication-title: Arch Rational Mech Anal doi: 10.1007/s00205-009-0249-2 – volume-title: Arch Rational Mech Anal year: 2014 ident: 4990_CR12 – ident: 4990_CR16 – volume: 23 start-page: 455 year: 2009 ident: 4990_CR20 publication-title: Discrete Contin Dyn Syst doi: 10.3934/dcds.2009.23.455 – volume: 205 start-page: 795 year: 2012 ident: 4990_CR3 publication-title: Arch Rational Mech Anal doi: 10.1007/s00205-012-0530-7 – volume-title: SIAM J Math Anal year: 2015 ident: 4990_CR23 – volume: 208 start-page: 59 year: 2013 ident: 4990_CR26 publication-title: Arch Ration Mech Anal doi: 10.1007/s00205-012-0588-2 – volume-title: Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions year: 2013 ident: 4990_CR1 – volume: 46 start-page: 3540 year: 2014 ident: 4990_CR10 publication-title: SIAM J Math Anal doi: 10.1137/13095015X – volume: 282 start-page: 846 year: 2009 ident: 4990_CR9 publication-title: Ma Math Nachr doi: 10.1002/mana.200610776 – volume: 203 start-page: 45 year: 2012 ident: 4990_CR18 publication-title: Arch Ration Mech Anal doi: 10.1007/s00205-011-0443-x – volume-title: Thermodynamics of Flowing Systems with Internal Microstructure year: 1994 ident: 4990_CR4 doi: 10.1093/oso/9780195076943.001.0001 – volume-title: Ann Mat Pura Appl, in press year: 2015 ident: 4990_CR8 – volume: 57 start-page: 443 year: 2014 ident: 4990_CR27 publication-title: Sci China Math doi: 10.1007/s11425-013-4761-3 – volume: 112 start-page: 84 year: 2015 ident: 4990_CR11 publication-title: Nonlinear Anal TMA doi: 10.1016/j.na.2014.09.011 – volume-title: From microscopic theory to macroscopic theory: dynamics of the rod-like liquid crystal molecules year: 2013 ident: 4990_CR22 – volume: 58 start-page: 7475 year: 1998 ident: 4990_CR19 publication-title: Physical Review E doi: 10.1103/PhysRevE.58.7475 – volume: 45 start-page: 319 year: 2012 ident: 4990_CR25 publication-title: Calc Var Partial Differ Equ doi: 10.1007/s00526-011-0460-5 – volume: 258 start-page: 1653 year: 2015 ident: 4990_CR14 publication-title: J Differential Equations doi: 10.1016/j.jde.2014.11.008 – volume-title: Comm Pure Appl Math year: 2015 ident: 4990_CR24 – volume-title: The Physics of Liquid Crystals year: 1974 ident: 4990_CR6 – volume: 43 start-page: 2009 year: 2011 ident: 4990_CR17 publication-title: SIAM J Math Anal doi: 10.1137/10079224X – volume: 13 start-page: 49 year: 2015 ident: 4990_CR21 publication-title: Commum Math Sci doi: 10.4310/CMS.2015.v13.n1.a3 – volume: 331 start-page: 805 year: 2014 ident: 4990_CR13 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| Snippet | We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q-tensor... We consider a complex fluid modeling nematic liquid crystal flows, which is described by a system coupling Navier-Stokes equations with a parabolic Q -tensor... |
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| SubjectTerms | Applications of Mathematics Computational fluid dynamics Dynamical systems Dynamics Fluid flow Fluids Liquid crystals Mathematical models Mathematics Mathematics and Statistics Navier-Stokes equations Navier-Stokes方程 向列液晶 张量 模型 流体模拟 解的存在性 连续依赖性 适定性 |
| Title | Global well-posedness for the dynamical Q-tensor model of liquid crystals |
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