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Abstract: This paper proves a new regularity criterion ω:=rotu∈L1(0,T;B ?0∞,∞) for the 3D generalized MHD system with fractional diffusion terms (-Δ)αu with α>98 and zero magnetic diffusivity.Here u is the fluid velocity,ω is the vorticity and B ?0∞,∞ is the homogeneous Besov space.
Key words: Blowup criterion; generalized MHD system; fractional diffusion
CLC number: O 175.29 Document code: A Article ID: 1000-5137(2014)05-0545-05
2010 Mathematics Subject Classifications: 35Q35, 76D03, 76W05
Received date: 2014-06-01
Foundation item: Supported by NSFC (No.11171154); the Zhejiang Innovation Project (Grant No.T200905); the ZJNSF (Grant No.R6090109); the NSFC (Grant No.11171154)
References:
［1］ J. Wu.Generalized MHD equations ［J］.J Differential Equations,2003,195:284-312.
［2］ J. Wu.Regularity criteria for the generalized MHD equations ［J］.Comm Partial Differential Equations,2008,33:285-306.
［3］ J. Fan,H. Gao,G. Nakamura.Regularity criteria for the generalized magnetohydrodynamic equations and the quasi-geostrophic equations［J］.Taiwanese J Math,2011,15(3):1059-1073.
［4］ Y. Zhou.Regularity criteria for the generalized viscous MHD equations ［J］.Ann I H Poincare-AN,2007,24:491-505.
［5］ J. Wu.Global regularity for a class of generalized magnetohydrodynamic equations ［J］.J Math Fluid Mech,2011,13:295-305.
［6］ Z. Ye.Two regularity criteria to the 2D generalized MHD equations with zero magnetic diffusivity ［J］.J Math Anal Appl.(in press).
［7］ T. Kato,G. Ponce.Comutator estimates and the Euler and Navier-Stokes equations ［J］.Comm Pure Appl Math,1988,41:891-907.
［8］ H. Kozono,T. Ogawa,Y. Taniuchi.The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations ［J］.Math Z,2002,242:251-278.
(Zhenzhen Feng)
Key words: Blowup criterion; generalized MHD system; fractional diffusion
CLC number: O 175.29 Document code: A Article ID: 1000-5137(2014)05-0545-05
2010 Mathematics Subject Classifications: 35Q35, 76D03, 76W05
Received date: 2014-06-01
Foundation item: Supported by NSFC (No.11171154); the Zhejiang Innovation Project (Grant No.T200905); the ZJNSF (Grant No.R6090109); the NSFC (Grant No.11171154)
References:
［1］ J. Wu.Generalized MHD equations ［J］.J Differential Equations,2003,195:284-312.
［2］ J. Wu.Regularity criteria for the generalized MHD equations ［J］.Comm Partial Differential Equations,2008,33:285-306.
［3］ J. Fan,H. Gao,G. Nakamura.Regularity criteria for the generalized magnetohydrodynamic equations and the quasi-geostrophic equations［J］.Taiwanese J Math,2011,15(3):1059-1073.
［4］ Y. Zhou.Regularity criteria for the generalized viscous MHD equations ［J］.Ann I H Poincare-AN,2007,24:491-505.
［5］ J. Wu.Global regularity for a class of generalized magnetohydrodynamic equations ［J］.J Math Fluid Mech,2011,13:295-305.
［6］ Z. Ye.Two regularity criteria to the 2D generalized MHD equations with zero magnetic diffusivity ［J］.J Math Anal Appl.(in press).
［7］ T. Kato,G. Ponce.Comutator estimates and the Euler and Navier-Stokes equations ［J］.Comm Pure Appl Math,1988,41:891-907.
［8］ H. Kozono,T. Ogawa,Y. Taniuchi.The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations ［J］.Math Z,2002,242:251-278.
(Zhenzhen Feng)