Annali di Matematica Pura ed Applicata

, Volume 191, Issue 2, pp 293–309 | Cite as

Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces

  • Qiao Liu
  • Jihong Zhao
  • Shangbin Cui


We study existence and asymptotic stability of solutions to a n-dimensional generalized incompressible magneto-hydrodynamic system with initial value \({(u_{0},b_{0}) \in (PM^{a})^{2n}}\) , where PM a is the pseudomeasure space, and \({a \in \mathbb{R}}\) is a given parameter. Some regularizing rate estimates for the β-th spatial derivatives of solution are also established, which particularly imply that the solution is analytic in the spatial variable.


Generalized magneto-hydrodynamic system Pseudomeasure space Regularizing rate Spatial analyticity 

Mathematics Subject Classification (2000)

35B65 35Q35 76W05 


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  1. 1.
    Cannone M., Karch G.: Smooth or singlar solutions to the Navier-Stokes system. J. Differ. Equ. 197, 247–274 (2004)MathSciNetMATHCrossRefGoogle 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, Banach Center Publications, Institute of Mathematics, Polish Academy of Scuences, Warszawa, vol. 74, pp. 59–93 (2006)Google Scholar
  3. 3.
    Cao, C., Wu, J.: Two regularity for the 3D MHD equation, J. Differ. Equ. (2009). doi: 10.1016/j.jde.2009.09.020
  4. 4.
    Ferreira B.L.C.F, Roa E.J.V.: of the convection problem in a pseudomeasure-type space. Proc. R. Soc. A 464, 1983–1999 (2008)MATHCrossRefGoogle Scholar
  5. 5.
    Giga Y., Sawada O.: On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem. Nonlinear Anal. Appl. 1(2), 549–562 (2003)MathSciNetGoogle Scholar
  6. 6.
    Kahane C.: On the spatial analyticity of solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 33, 386–405 (1969)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Karch G., Prioux N.: In viscous Boussinesq equations. Proc. Am. Math. Soc. 136, 879–888 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lemarié-Rieusset P.G.: Recent Developments in the Navier-Stokes Problem. Chapman and Hall/CRC, London (2002)MATHCrossRefGoogle Scholar
  9. 9.
    Liu Q., Cui S.: Regularizing rate estimates for mild solutions of the incompressible magneto- hydrodynamic system (submitted)Google Scholar
  10. 10.
    Miura H., Sawada O.: On the regularizing rate estimates of Koch-Tataru’s solution to the Navier-Stokes equations. Asymptot. Anal. 49, 1–15 (2006)MathSciNetMATHGoogle Scholar
  11. 11.
    Sawada O.: On analyticity rate estimates of the solutions to the Navier-Stokes equations in Bessel-potential spaces. J. Math. Anal. Appl. 312, 1–13 (2005)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Secchi P.: On the equations of ideal incompressible magneto-hydrodynamics. Rend. Sem. Math. Univ. Padova 90, 103–119 (1993)MathSciNetMATHGoogle Scholar
  13. 13.
    Sermange M., Temam R.: Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36, 635–664 (1983)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Stein E.M.: Singular Intergrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)Google Scholar
  15. 15.
    Wu J.: Generalized MHD equations. J. Differ. Equ. 195, 284–312 (2003)MATHCrossRefGoogle Scholar
  16. 16.
    Wu J.: Regularity results for weak solutions of the 3D MHD equations. Dis. Continuous Dyn. Syst. 10, 543–556 (2004)MATHCrossRefGoogle Scholar
  17. 17.
    Wu J.: Regularity criteria for the generalized MHD equations. Commun. Partial Differ. Equ. 33, 285–306 (2008)MATHCrossRefGoogle Scholar

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsSun Yat-sen UniversityGuangzhouPeople’s Republic of China

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