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Weak and smooth solutions to incompressible Navier-Stokes-Landau-Lifshitz-Maxwell equations

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Abstract

Considering the Navier-Stokes-Landau-Lifshitz-Maxwell equations, in dimensions two and three, we use Galerkin method to prove the existence of weak solution. Then combine the a priori estimates and induction technique, we obtain the existence of smooth solution.

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References

  1. Caffarelli L, Kohn R, Nirenberg L. Partial regularity of suitable weak solutions of the navier-stokes equations. Comm Pure Appl Math, 2010, 35(6): 771–831

    Article  MathSciNet  Google Scholar 

  2. Ericksen J. Conservation laws for liquid crystals. Trans Soc Rheol, 1961, 5: 22–34

    MathSciNet  Google Scholar 

  3. Ericksen J. Hydrostatic theory of liquid crystals. Arch Ration Mech Anal, 1962, 9(1): 371–378

    Article  MathSciNet  Google Scholar 

  4. Ericksen J. Equilibrium theory of liquid crystals. In: Brown G, ed. Advances in Liquid Crystals, Vol 2. New York: Academic Press, 1976, 233–298

    Google Scholar 

  5. Ericksen J. Continuum theory of nematic liquid crystals. Res Mechanica, 1987, 22: 381–392

    Google Scholar 

  6. Ericksen J L, Kinderlehrer D. Theory and Applications of Liquid Crystals. The IMA volumes in Mathematics and Its Applications, Vol 5. New York: Springer-Verlag, 1986

    MATH  Google Scholar 

  7. Fan J, Gao H, Guo B. Regularity criteria for the Navier-Stokes-Landau-Lifshitz system. J Math Anal Appl, 2009, 363(1): 29–37

    Article  MathSciNet  Google Scholar 

  8. Feireisl E. Dynamics of Viscous Compressible Fluids. Oxford: Oxford Univ Press, 2004

    MATH  Google Scholar 

  9. Greenberg J M, Maccamy R C, Coffman C V. On the long-time behavior of ferroelectric systems. Phys D, 1999, 134(3): 362–383

    Article  MathSciNet  Google Scholar 

  10. Kim H. A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J Math Anal, 2006, 37: 1417–1434

    Article  MathSciNet  Google Scholar 

  11. Ladyzhenskaya O A, Ural’Tseva N N, Solonnikov N A. Linear and Quasilinear Elliptic Equations. New York: Academic Press, 1968

    MATH  Google Scholar 

  12. Leslie F M. Some constitutive equations for liquid crystals. Arch Ration Mech Anal, 1968, 28(4): 265–283

    Article  MathSciNet  Google Scholar 

  13. Leslie F M. Theory of flow phenomena in liquid crystals. In: Brown G, ed. Advances in Liquid Crystals, Vol 4. New York: Academic Press, 1979, 1–81

    Google Scholar 

  14. Lin F. A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm Pure Appl Math, 1998, 51(3): 241–257

    Article  MathSciNet  Google Scholar 

  15. Lin F, Lin J, Wang C. Liquid crystal flows in two dimensions. Arch Ration Mech Anal, 2010, 197(1): 297–336

    Article  MathSciNet  Google Scholar 

  16. Lin F, Liu C. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete Contin Dyn Syst, 1995, 2(1): 1–22

    Article  MathSciNet  Google Scholar 

  17. Lin F, Liu C. Nonparabolic dissipative systems modeling the flow of liquid crystals. Comm Pure Appl Math, 2010, 48(5): 501–537

    Article  MathSciNet  Google Scholar 

  18. Schein B M. Techniques of semigroup theory. Semigroup Forum, 1994, 49(1): 397–402

    Article  MathSciNet  Google Scholar 

  19. Simon J. Nonhomogeneous viscous incompressible fluids: existence of viscosity, density and pressure. SIAM J Math Anal, 1990, 20: 1093–1117

    Article  Google Scholar 

  20. Temam R. Navier-Stokes Equations. Studies in Mathematics and Its Applications, Vol 2. Amsterdam: North-Holland, 1977

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11731014, 11571254).

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Boling Guo

  2. The Graduate School of China Academy of Engineering Physics, Beijing, 100088, China

    Fengxia Liu

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  1. Boling Guo
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  2. Fengxia Liu
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Correspondence to Fengxia Liu.

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Guo, B., Liu, F. Weak and smooth solutions to incompressible Navier-Stokes-Landau-Lifshitz-Maxwell equations. Front. Math. China 14, 1133–1161 (2019). https://doi.org/10.1007/s11464-019-0800-x

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  • DOI: https://doi.org/10.1007/s11464-019-0800-x

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