Regularity criteria for the generalized MHD equations

Communications in Partial Differential Equations

Volumn 33, Issue 2, 2008, Pages 285-306

  Wu, Jiahong ^a  

^a OKLAHOMA STATE UNIVERSITY   (United States)

Author keywords

Besov space; Generalized MHD equations; Regularity criteria

Indexed keywords

EID: 38949096754     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605300701382530     Document Type: Article

References (19)

1. Beale, J. T., Kato, T., Majda, A. (1984). Remarks on breakdown of smooth solutions for the 3D Euler equations. Comm. Math. Phys. 94:61-66.
2. Beirao da Veiga, H. (1987). Existence and asymptotic behavior for strong solutions of the Navier-=Stokes equations in the whole space. Indiana Univ. Math. J. 36:149-166.
3. n. Chinese Ann. Math. Ser. B 16:407-412.
4. Bergh, J., Löfström, J. (1976). Interpolation Spaces, An Introduction. Berlin-Heidelberg-New York: Springer-Verlag.
5. Caflisch, R. E., Klapper, I., Steele, G. (1997). Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Comm. Math. Phys. 184: 443-455.
6. Chen, Q., Miao, C., Zhang, Z. (2007). A new Bernstein inequality and the 2D dissipative quasi-geostrophic equation. Comm. Math. Phys. 271:821-838.
7. 3) of mild solutions of the Navier-Stokes equations]. C. R. Acad. Sci. Paris Sér. I Math. 325:1253-1256.
8. Hasegawa, A. (1985). Self-organization processes in continuous media. Adv. Phys. 34:1-42.
9. He, C., Xin, Z. (2005). On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differential Equations 213:235-254.
10. Kozono, H., Ogawa, T., Taniuchi, Y. (2002). The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z. 242:251-278.
11. Kozono, H., Taniuchi, Y. (2000). Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Comm. Math. Phys. 214:191-200.
12. N. Comm. Partial Differential Equations 26:2211-2226.
13. Politano, H., Pouquet, A., Sulem, P.-L. (1995). Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 2:2931-2939.
14. Prodi, G. (1959). Un teorema di unicità per le equazioni di Navier-Stokes [A uniqueness theorem for the Navier-Stokes equations]. Ann. Mat. Pura Appl. 48:173-182.
15. Serrin, J. (1963). The initial value problem for the Navier-Stokes equations. In: Langer, R. E., ed. Nonlinear Problems. Madison: Univ. of Wisconsin Press.
16. Sohr, H., von Wahl, W. (1984). On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations. Manuscripta Math. 49:27-59.
17. Wu, J. (1997). Viscous and inviscid magneto-hydrodynamics equations. J. d'Analyse Mathématique 73:251-265.
18. Wu, J. (2003). Generalized MHD equations. J. Differential Equations 195:284-312.
19. Wu, J. (2006). Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces. Comm. Math. Phys. 263:803-831.