\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space

Abstract Related Papers Cited by
  • In this paper, some improved regularity criteria for the 3D magneto-micropolar fluid equations are established in critical Morrey-Campanato spaces. It is proved that if the velocity field satisfies

    $u\in L^{\frac{2}{1-r}}(0,T; M_{2,\frac{3}{r}}(R^3)) $ with $r\in (0, 1)$ or $u\in C(0, T; M_{2,3}(R^3))$

    or the gradient field of velocity satisfies

    $ \nabla u\in L^{\frac{2}{2-r}}(0, T; M_{2,\frac{3}{ r}}(R^3))$ with $r\in (0,1], $

    then the solution remains smooth on $[0,T] $.

    Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76D05.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    R. E. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455.doi: doi:10.1007/s002200050067.

    [2]

    Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations, Comm. Math. Phys., 275 (2007), 861-872.doi: doi:10.1007/s00220-007-0319-y.

    [3]

    G. Duvaut and J. L. Lions, Inéquations en thermoé lasticité et magnéto-hydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.

    [4]

    S. Gala, A note on the uniqueness of mild solutions to the Navier-Stokes equations, Arch. Math.(Basel), 84 (2007), 448-454.

    [5]

    G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluids equations, Internat. J. Engrg. Sci., 15 (1977), 105-108.doi: doi:10.1016/0020-7225(77)90025-8.

    [6]

    C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.doi: doi:10.1016/j.jde.2004.07.002.

    [7]

    T. Kato, Strong $L^p$ solutions of the Navier-Stokes equations in Morrey spaces, Bol. Soc. Bras. Mat.(N.S.), 22 (1992), 127-155.

    [8]

    P. G. Lemarié-Rieusset, "Recent Developments in the Navier-Stokes Problem," Research Notes in Mathematics, Chapman & Hall, CRC, 2002.

    [9]

    P. G. Lemarié-Rieusset, The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam., 23 (2007), 897-930.

    [10]

    G. Lukaszewicz, "Micropolar Fluids: Theory and Applications," Birkhauser, Berlin 1998.

    [11]

    S. Machihara and T. Ozawa, Interpolation inequalities in Besov spaces, Proc. Amer. Math. Soc., 131 (2003), 1553-1556.doi: doi:10.1090/S0002-9939-02-06715-1.

    [12]

    E. Ortega-Torres and M. A. Rojas-Medar, On the uniqueness and regularity of the weak solutions for magneto-micropolar equations, Rev. Mat. Apl., 17 (1996), 75-90.

    [13]

    M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.doi: doi:10.1002/mana.19971880116.

    [14]

    M. A. Rojas-Medar and J. L. Boldrini, Magneto-micropolar fluid motion: existence of weak solutions, Rev. Mat. Complut., 11 (1998), 443-460.

    [15]

    M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.doi: doi:10.1002/cpa.3160360506.

    [16]

    M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407-1456.doi: doi:10.1080/03605309208820892.

    [17]

    J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.doi: doi:10.1080/03605300701382530.

    [18]

    B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Mathematica Scientia, 30 (2010), 1469-1480.doi: doi:10.1016/S0252-9602(10)60139-7.

    [19]

    Y. Zhou, Remarks on regularities for the 3D MHD equations, Disc. Cont. Dyna. Sys., 12 (2005), 881-886.doi: doi:10.3934/dcds.2005.12.881.

    [20]

    Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41 (2006), 1174-1180.doi: doi:10.1016/j.ijnonlinmec.2006.12.001.

    [21]

    Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.

    [22]

    Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, To appear in Forum Math (2010), DOI 10.1515/FORM.2011.079.

    [23]

    Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199.doi: doi:10.1007/s00033-009-0023-1.

    [24]

    Y. Zhou and S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field. Nonlinear Anal, Nonlinear Anal., 72 (2010), 3643-3648.doi: doi:10.1016/j.na.2009.12.045.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(113) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return