Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space

Article

Abstract

In this paper, some improved regularity criteria for the 3D magneto-micropolar fluid equations are established in Morrey–Campanato spaces. It is proved that if the velocity field satisfies
$$\quad u\in L^{\frac{2}{1-r}}\left(0,T;\overset{.}{\mathcal{M}}_{p,\frac{3}{r}}( \mathbb{R}^{3})\right)\quad\text{with} \;r\in \left( 0,1\right)\;\text{or}\;u\in C\left(0,T;\overset{.}{\mathcal{M}}_{p,\frac{3}{r}}(\mathbb{R} ^{3})\right)$$
or the gradient field of velocity satisfies
$$\nabla u\in L^{\frac{2}{2-r}}\left(0,T;\overset{.}{\mathcal{M}}_{p,\frac{3}{ r}}(\mathbb{R}^{3})\right)\text{ \ with \ }r\in \left( 0,2\right),$$
then the solution remains smooth on [ 0, T] . By the embedding \({ L^{\frac{3}{r}} \subset \overset{.}{\mathcal{M}}_{p,\frac{3}{r}}}\) , we see that our result is an improvement of (Yuan in Acta Mathematica Scientia, to appear).

Mathematics Subject Classification (2000)

35Q35 76W05 35B65 

Keywords

Magneto micropolar fluid equations Regularity criterion Morrey–Campanato spaces 

References

  1. 1.
    Caflish R.E., Klapper I., Steel G.: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Comm. Math. Phys. 184, 443–455 (1997)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Cannone M., Miao C.X., Prioux N., Yuan B.Q.: The Cauchy Problem for the Magneto-hydrodynamic System. Self-similar Solutions of Nonlinear PDE, vol. 74, pp. 59–93. Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Warszawa (2006)Google Scholar
  3. 3.
    Duvant G., Lions J.L.: In équations en thermoélasticite et magnetohydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279 (1972)Google Scholar
  4. 4.
    Eringen A.C.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1996)MathSciNetGoogle Scholar
  5. 5.
    Ferreira L.C.F., Villamizar-Roa E.J.: On the existence and stability of solutions for the micropolar fluids in exterior domains. Math. Meth. Appl. Sci. 30, 1185–1208 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gala, S., Yuan, B.: Remarks on the regularity of weak solutions to magneto-micropolar fluid equations (2009)Google Scholar
  7. 7.
    Galdi G.P., Rionero S.: A note on the existence and uniqueness of solutions of the micropolar fluid equations. Internat. J. Eng. Sci. 15, 105–108 (1997)CrossRefMathSciNetGoogle Scholar
  8. 8.
    He C., Wang Y.: On the regularity criteria for weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 238, 1–17 (2007)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    He C., Xin Z.P.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Eq. 213, 235–254 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kato T.: Strong L p solutions of the Navier–Stokes equations in Morrey spaces. Bol. Soc. Bras. Mat. 22, 127–155 (1992)MATHCrossRefGoogle Scholar
  11. 11.
    Kozono H.: Weak solutions of the Navier–Stokes equations with test functions in the weak-L n space. Tohoku Math. J. 53, 55–79 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kozono H., Yamazaki M.: Semilinear Heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data. Comm. P. D. E. 19, 959–1014 (1994)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ladyzhenskaya O.: The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, New York (1969)Google Scholar
  14. 14.
    Lemarié-Rieusset P.G.: Recent Developments in the Navier–Stokes Problem. Chapman & Hall/CRC, London (2002)MATHGoogle Scholar
  15. 15.
    Lions P.L.: Mathematical Topics in Fluid Mechanics. Oxford University Press Inc., New York (1996)MATHGoogle Scholar
  16. 16.
    Lions P.L., Masmoudi N.: Uniqueness of mild solutions of the Navier–Stokes system in L N. Comm. P. D. E. 26, 2211–2226 (2001)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ortega-Torres E.E., Rojas-Medar M.A.: Magneto-micropolar fluid motion: global existence of strong solutions. Abstr. Appl. Anal. 4, 109–125 (1999)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Rojas-Medar M.A.: Magneto-micropolar fluid motion: existence and uniqueness of strong solutions. Math. Nachr. 188, 301–319 (1997)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rojas-Medar M.A., Boldrini J.L.: Magneto-micropolar fluid motion: existence of weak solutions. Rev. Mat. Complut. 11, 443–460 (1998)MATHMathSciNetGoogle Scholar
  20. 20.
    Sermange M., Temam R.: Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36, 635–664 (1983)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Taylor M.E.: Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations. Comm. P. D. E. 17, 1407–1456 (1992)MATHCrossRefGoogle Scholar
  22. 22.
    Villamizar-Roa E.J., Rodríguez-Bellido M.A.: Global existence and exponential stability for the micropolar fluid system. Z. Angew. Math. Phys. 59, 790–809 (2008)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Yamaguchi N.: Existence of global strong solution to the micropolar fluid system in a bounded domain. Math. Meth. Appl. Sci. 28, 1507–1526 (2005)MATHCrossRefGoogle Scholar
  24. 24.
    Yuan, B.Q.: Regularity of weak solutions to magneto-micropolar fluid equations. Acta Math. Sci. (in press)Google Scholar
  25. 25.
    Zhou Y.: Remarks on regularities for the 3D MHD equations. Discrete Continuous Dyn. Syst. 12, 881–886 (2005)MATHCrossRefGoogle Scholar
  26. 26.
    Zhou Y.: Regularity criteria for the 3D MHD equations in terms of the pressure. Intl. J. Non-Linear Mech. 41, 1174–1180 (2006)MATHCrossRefGoogle Scholar
  27. 27.
    Zhou Y.: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 24, 491–505 (2007)MATHCrossRefGoogle Scholar
  28. 28.
    Zhou, Y., Gala, S.: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. ZAMP (in press)Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MostaganemMostaganemAlgeria

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