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Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in ℝ3

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Abstract

We consider the regularity problem under the critical condition to some liquid crystal models and the Landau-Lifshitz equations. The Serrin type reularity criteria are obtained in the terms of the Besov spaces.

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References

  1. Lin F H. Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena. Comm Pure Appl Math, 42: 789–814 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  2. Lin F H, Liu C. Nonparabolic dissipative systems modelling the flow of liquid crystals. Comm Pure Appl Math, 48: 501–537 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Lin F H, Liu C. Existence of solutions for the Ericksen-Leslie system. Arch Ration Mech Anal, 154: 135–156 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Coutand D, Shkoller S. Well posedness of the full Ericksen-Leslye model of nematic liquid crystals. C R Acad Sci Ser I, 333: 919–924 (2001)

    MATH  MathSciNet  Google Scholar 

  5. Chen Y, Struwe M. Existence and partial regularity results for the heat flow of harmonic maps. Math Z, 201: 83–103 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Landau L D, Lifshitz E M. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys Z Sowj, 8: 153–169 (1935)

    MATH  Google Scholar 

  7. Chen Y, Ding S, Guo B L. Partial regularity for two-dimensional Landau-Lifshitz equations. Acta Math Sin, 14(3): 423–432 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Guo B L, Hong M. The Landau-Lifshitz equations of the ferromagnetic spin chain and harmonic maps. Calc Var PDE, 1: 311–334 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Liu X. Partial regularity for the Landau-Lifshitz systems. Calc Var PDE, 20(2): 153–173 (2003)

    Article  Google Scholar 

  10. Zhou Y, Guo B L. Existence of weak solution for boundary problems of systems of ferromagnetic chain. Sci China Ser A-Math, 27: 799–811 (1984)

    MATH  MathSciNet  Google Scholar 

  11. Zhou Y, Guo B L. Weak solution systems of ferromagnetic chain with several variables. Sci China Ser A-Math, 30: 1251–1266 (1987)

    MATH  MathSciNet  Google Scholar 

  12. Zhou Y, Guo B L, Tan S. Existence and uniqueness of smooth solution of systems of ferromagnetic chain. Sci China Ser A-Math, 34: 257–266 (1991)

    MATH  MathSciNet  Google Scholar 

  13. Zhou Y, Sun H, Guo B L. Multidimensional system of ferromagnetic chain type. Sci China Ser A-Math, 36: 1422–1434 (1993)

    MATH  MathSciNet  Google Scholar 

  14. Alouges F, Soyeur A. On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness. Nonlinear Analysis, 18(11): 1084–1096 (1992)

    Article  MathSciNet  Google Scholar 

  15. Carbou G, Fabrie P. Regular solutions for Landau-Lifshitz equation in a bounded domain. Differential Inteqrals Equations, 14: 213–229 (2001)

    MATH  MathSciNet  Google Scholar 

  16. Leray J. Sur le mouvement d’un liquide visqeux emplissant 1’espace. Acta Math, 63: 193–248 (1934)

    Article  MATH  MathSciNet  Google Scholar 

  17. Prodi G. Un teorema di unicità per le equazioni di Navier-Stokes. Ann Mat Pura Appl, 48: 173–182 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  18. Serrin J. On the interior regularity of weak solutions of the Navier-Stokes equations. Arch Ration Mech Anal, 9: 187–195 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  19. Giga Y. Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J Differential Equations, 62: 186–212 (1986)

    Article  MathSciNet  Google Scholar 

  20. da Veiga H B. A new regularity class for the Navier-Stoekes equations in ℝn. Chinese Ann Math Ser B, 16: 407–412 (1995)

    MATH  MathSciNet  Google Scholar 

  21. Kozono H, Ogawa T, Taniuchi Y. The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations. Math Z, 242: 251–278 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Miao C X. Harmonic Analysis and its Applications in PDE (in Chinese), 2nd ed. Beijing: Science Press, 2004

    Google Scholar 

  23. Triebel H. Theory of function Spaces II. Basel: Birkhäuser, 1992

    MATH  Google Scholar 

  24. Kozono H, Shimada Y. Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations. Math Nachr, 276: 63–74 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lemarié-Rieusset P G. Recent Developments in the Navier-Stokes Problem. London: Chapman & Hall/CRC, 2002

    MATH  Google Scholar 

  26. David G, Journé J L. A boundedness criterion for generalized Calderón Zygmund operators. Ann of Math, 120: 371–397 (1984)

    Article  MathSciNet  Google Scholar 

  27. Kato T, Ponce G. Commutator estimates and the Euler and Navier-Stokes equations. Comm Pure Appl Math, 41: 891–907 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Information Science and Technology, Nanjing Forestry University, Nanjing, 210037, China

    JiShan Fan

  2. Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, 100088, China

    BoLing Guo

Authors
  1. JiShan Fan
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  2. BoLing Guo
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Correspondence to JiShan Fan.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10301014)

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Fan, J., Guo, B. Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in ℝ3 . Sci. China Ser. A-Math. 51, 1787–1797 (2008). https://doi.org/10.1007/s11425-008-0013-3

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  • DOI: https://doi.org/10.1007/s11425-008-0013-3

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