The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations

Communications in Mathematical Physics

Volumn 275, Issue 3, 2007, Pages 861-872

  Chen, Qionglei ^a   Miao, Changxing ^a   Zhang, Zhifei ^b  

^a INSTITUTE OF APPLIED PHYSICS AND COMPUTATIONAL MATHEMATICS   (China)

^b PEKING UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34548453989     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-007-0319-y     Document Type: Article

References (25)

1. Beale J.T., Kato T. and Majda A.J. (1984). Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94: 61-66
2. Caflisch R.E., Klapper I. and Steele G. (1997). Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Commun. Math. Phys. 184: 443-455
3. Cannone M., Chen Q. and Miao C. (2007). A losing estimate for the Ideal MHD equations with application to Blow-up criterion. SIAM J. Math. Anal. 38: 1847-1859
4. Chemin J.-Y. (1998). Perfect Incompressible Fluids. Oxford University Press, New York
5. Chemin J.-Y. and Lerner N. (1992). Flot de champs de vecteurs non lipschitziens et equations de Navier-Stokes. J. Diff. Eq. 121: 314-328
6. Constantin P. and Fefferman C. (1993). Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42: 775-788
7. Hasegawa A. (1985). Self-organization processed in continuous media. Adv. in Phys. 34: 1-42
8. He C. and Xin Z. (2005). On the regularity of weak solutions to the magnetohydrodynamic equations. J. Diff. Eq. 213: 235-254
9. Kato T. and Ponce G. (1988). Commutator estimates and Euler and Navier-Stokes equations. Comm. Pure. Appl. Math. 41: 891-907
10. Kozono H. and Taniuchi Y. (2000). Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235: 173-194
11. Kozono H., Ogawa T. and Taniuchi Y. (2002). The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z. 242: 251-278
12. Majda A.J. and Bertozzi A.L. (2002). Vorticity and Incompressible Flow. Cambridge University Press, Cambridge
13. Meyer Y. (1992). Wavelets and operators. Cambridge University Press, Cambridge
14. Ogawa T. (2003). Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow. SIAM J. Math. Anal. 34: 1318-1330
15. Planchon F. (2003). An extension of the Beale-Kato-Majda criterion for the Euler equations. Commun. Math. Phys. 232: 319-326
16. Politano H., Pouquet A. and Sulem P.L. (1995). Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 2: 2931-2939
17. Sermange M. and Temam R. (1983). Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36: 635-664
18. Stein E.M. (1971). Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, NJ
19. Triebel, H.: Theory of Function Spaces. Monograph in Mathematics, Vol. 78. Basel: Birkhauser Verlag, 1983
20. Wu J. (2000). Analytic results related to magneto-hydrodynamics turbulence. Phys. D 136: 353-372
21. Wu J. (2002). Bounds and new approaches for the 3D MHD equations. J. Nonlinear Sci. 12: 395-413
22. Wu J. (2004). Regularity results for weak solutions of the 3D MHD equations. Discrete. Contin. Dynam. Syst. 10: 543-556
23. Wu, J.: Regularity criteria for the generalized MHD equations. Preprint
24. Zhang Z. and Liu X. (2004). On the blow-up criterion of smooth solutions to the 3D Ideal MHD equations. Acta Math. Appl. Sinica E 20: 695-700
25. Zhou Y. (2005). Remarks on regularities for the 3D MHD equations. Discrete. Contin. Dynam. Syst. 12: 881-886