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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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
Ericksen J. Conservation laws for liquid crystals. Trans Soc Rheol, 1961, 5: 22–34
Ericksen J. Hydrostatic theory of liquid crystals. Arch Ration Mech Anal, 1962, 9(1): 371–378
Ericksen J. Equilibrium theory of liquid crystals. In: Brown G, ed. Advances in Liquid Crystals, Vol 2. New York: Academic Press, 1976, 233–298
Ericksen J. Continuum theory of nematic liquid crystals. Res Mechanica, 1987, 22: 381–392
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
Fan J, Gao H, Guo B. Regularity criteria for the Navier-Stokes-Landau-Lifshitz system. J Math Anal Appl, 2009, 363(1): 29–37
Feireisl E. Dynamics of Viscous Compressible Fluids. Oxford: Oxford Univ Press, 2004
Greenberg J M, Maccamy R C, Coffman C V. On the long-time behavior of ferroelectric systems. Phys D, 1999, 134(3): 362–383
Kim H. A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations. SIAM J Math Anal, 2006, 37: 1417–1434
Ladyzhenskaya O A, Ural’Tseva N N, Solonnikov N A. Linear and Quasilinear Elliptic Equations. New York: Academic Press, 1968
Leslie F M. Some constitutive equations for liquid crystals. Arch Ration Mech Anal, 1968, 28(4): 265–283
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
Lin F. A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm Pure Appl Math, 1998, 51(3): 241–257
Lin F, Lin J, Wang C. Liquid crystal flows in two dimensions. Arch Ration Mech Anal, 2010, 197(1): 297–336
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
Lin F, Liu C. Nonparabolic dissipative systems modeling the flow of liquid crystals. Comm Pure Appl Math, 2010, 48(5): 501–537
Schein B M. Techniques of semigroup theory. Semigroup Forum, 1994, 49(1): 397–402
Simon J. Nonhomogeneous viscous incompressible fluids: existence of viscosity, density and pressure. SIAM J Math Anal, 1990, 20: 1093–1117
Temam R. Navier-Stokes Equations. Studies in Mathematics and Its Applications, Vol 2. Amsterdam: North-Holland, 1977
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11731014, 11571254).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-019-0800-x