# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field. (English) Zbl 1185.35204
Summary: We consider regularity criterion for solutions to the 3D viscous incompressible MHD equations in Morrey-Campanato spaces. It is proved that if the vorticity field $\omega = \nabla \times u$ belongs to $\dot {\cal M}_{2, \frac{3}{r}}$ for $0 < r \leq 1$ on $[0,T]$, then the solution remains smooth on $[0,T]$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B65 Smoothness and regularity of solutions of PDE 76D03 Existence, uniqueness, and regularity theory 76W05 Magnetohydrodynamics and electrohydrodynamics
Full Text:
##### References:
 [1] Sermange, M.; Temam, R.: Some mathematical questions related to the MHD equations, Comm. pure appl. Math. 36, No. 5, 635-664 (1983) · Zbl 0524.76099 · doi:10.1002/cpa.3160360506 [2] He, C.; Xin, Z.: On the regularity of solutions to the magnetohydrodynamic equations, J. differential equations 213, No. 2, 235-254 (2005) · Zbl 1072.35154 · doi:10.1016/j.jde.2004.07.002 [3] Zhou, Y.: Remarks on regularities for the 3D MHD equations, Discrete contin. Dyn. syst. 12, 881-886 (2005) · Zbl 1068.35117 · doi:10.3934/dcds.2005.12.881 [4] Chen, Q.; Miao, C.; Zhang, Z.: On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. math. Phys. 284, No. 3, 919-930 (2008) · Zbl 1168.35035 · doi:10.1007/s00220-008-0545-y [5] Y. Zhou, S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys. (2009), in press (doi:10.1007/s00033-009-0023-1) · Zbl 1172.35063 [6] Wu, J.: Regularity criteria for the generalized MHD equations, Comm. partial differential equations 33, No. 1--3, 285-306 (2008) · Zbl 1134.76068 · doi:10.1080/03605300701382530 [7] Zhou, Y.: Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-linear mech. 41, No. 10, 1174-1180 (2006) · Zbl 1160.35506 · doi:10.1016/j.ijnonlinmec.2006.12.001 [8] Y. Zhou, J. Fan, On regularity criteria in terms of pressure for the 3D viscous MHD equations, Preprint (2008) [9] Zhou, Y.: Regularity criteria for the generalized viscous MHD equations, Ann. inst. H. poincar anal. Non linéaire 24, No. 3, 491-505 (2007) · Zbl 1130.35110 · doi:10.1016/j.anihpc.2006.03.014 · numdam:AIHPC_2007__24_3_491_0 [10] Kato, T.: Strong lp solutions of the Navier--Stokes equations in Morrey spaces, Bol. soc. Bras. mat. 22, No. 2, 127-155 (1992) · Zbl 0781.35052 [11] Taylor, M. E.: Analysis on Morrey spaces and applications to Navier--Stokes equations and other evolutions equations, Comm. partial differential equations 17, 1407-1456 (1992) · Zbl 0771.35047 · doi:10.1080/03605309208820892 [12] Lemarié-Rieusset, P. G.: The Navier--Stokes equations in the critical Morrey--campanato space, Rev. mat. Iberoam. 23, No. 3, 897-930 (2007) · Zbl 1227.35230 · euclid:rmi/1204128305 [13] Machihara, S.; Ozawa, T.: Interpolation inequalities in Besov spaces, Proc. amer. Math. soc. 131, 1553-1556 (2003) · Zbl 1022.46018 · doi:10.1090/S0002-9939-02-06715-1 [14] Wang, Y.: BMO and the regularity criterion for weak solutions to the magnetohydrodynamic equations, J. math. Anal. appl. 328, No. 2, 1082-1086 (2007) · Zbl 1107.35040 · doi:10.1016/j.jmaa.2006.05.054