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

Solutions for the Navier-Stokes equations with critical and subcritical fractional dissipation in Lei-Lin and Lei-Lin-Gevrey spaces

“…The fractional Laplacian (−∆) α has been studied in many works in the literature (see, for instance, [32,34] and references therein). To cite some models involving this kind of operator, we refer: Diffusion-reaction, Quasi-geostrophic, Cahn-Hilliard, Porous medium, Schrödinger, Ultrasound, Magnetohydrodynamics (MHD), Magnetohydrodynamics-α (MHD-α) and Navier-Stokes itself (see [1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,21,22,23,24,25,26,27,28,29,30,31,33,35] and references therein). It is important to recall that, by applying the Spectral Theorem, (−∆) α assumes the diagonal form in the Fourier variable, i.e., this is a Fourier multiplier operator with symbol |ξ| 2α (which extends Fourier multiplier property of −∆).…”

“…In fact, Wu [35] showed that the generalized Navier-Stokes equations (1.1) admit global classical solution provided that the initial data u 0 is smooth and α ≥ 5 4 . More precisely, [35] assumes that α ≥ 5/4 and u 0 ∈ H s (R 3 ), with s > 2α, to obtain a unique global classical solution for (1.1) (see also [1,4,5,7,8,9,15,16,17,18,27,28,29,31,35] and references therein).…”