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This paper proves the existence of a unique global in time solution for the generalized Magnetohydrodynamics- α  equations in Sobolev–Gevrey (and Sobolev) spaces (including critical cases) if it is assumed that the initial data is small enough. In addition, the authors show that this same solution is analytic and study its long time behavior.

Time decay rates for the generalized MHD- α equations in Sobolev–Gevrey spaces