About this Mathematics article
Regularity criteria for the generalized viscous MHD equations by Yong Zhou is a Mathematics article available to read on EtoBox.
In this paper, we consider regularity criteria for solutions to the 3D generalized MHD equations with fractional dissipative term -(-) α u for the velocity field and -(-) β b for the magnetic field. For the case α = β, it is proved that if the velocity field belongs to L p,q with 2α/p + 3/q 2α -1 or the gradient of velocity field belongs to L p,q with 2α/p + 3/q 3α -1 on [0, T ], then the solution remains smooth on [0, T ]. The significance is that there is no restriction on the magnetic field. Moreover, the norms u L p,q and Λ α u L p,q are scaling dimension zero for 2α/p + 3/q = 2α -1 and 2α/p + 3/q = 3α -1 respectively. For 1 β α, we find that the minimum sum of α and β to guarantee the global existence of smooth solutions is 5/2. Furthermore, we show that the weak solution actually is strong if the corresponding vorticity field ω = ∇ × u satisfies certain condition in the high vorticity region.
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- Author
- Yong Zhou
- Publisher
- European Mathematical Society - EMS - Publishing House GmbH
- Published
- 2007
- Language
- EN
- Field
- Mathematics (Physical Sciences)