Well-Posedness, Blow-up Criteria and Stability for Solutions of the Generalized MHD 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

“…where f ∈ L 1 T (X s a,σ (R 3 )) (provided that T > 0 is arbitrary). It is worth to point out that the main ideas that will be presented below were firstly established by Orf [31] and generalized a few years later by Guterres, Melo, Rocha, and Santos [17], in the case of Sobolev-Gevrey spaces. See [36,Lemma 2.6] for examples of particular cases.…”

“…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).…”

“…Motivated by these works, we present global and local solutions for the Navier-Stokes equations (1.1), with fractional dissipation of order α ≥ 1/2, in Lei-Lin and Lei-Lin-Gevrey spaces (we refer to [8,28,30,33] and papers therein). Moreover, it is worth to point out that we have adapted some of the ideas applied in the paper [17].…”