On the generalized Magnetohydrodynamics-$$\alpha $$ equations with fractional dissipation in Lei–Lin and Lei–Lin–Gevrey spaces

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

“…where u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)) ∈ R 3 denotes the incompressible velocity field, and p(x, t) ∈ R the hydrostatic pressure; see [1,2,3,4,5,6,7,8,10,11,12,13,18,20,26,28,31] and their references. Here (−∆) α , with α ≥ 1/2, is the fractional Laplacian, see (2.1).…”

“…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).…”

“…Recently, some authors have published works that study the usual Navier-Stokes equations and their extensions in Lei-Lin and Lei-Lin-Gevrey spaces (see [1,4,7,9,18,24,28] and references therein). For example Melo, Souza and Santos [28], by studying the MHD-α equations, proved the existence of a unique global solution in C b ([0, ∞); X s a,σ (R 3 )). In addition, [28] presents analyticity and decay rates for global solutions in this same context (for more information, see [28] and references therein).…”

“…For example Melo, Souza and Santos [28], by studying the MHD-α equations, proved the existence of a unique global solution in C b ([0, ∞); X s a,σ (R 3 )). In addition, [28] presents analyticity and decay rates for global solutions in this same context (for more information, see [28] and references therein).…”