New blow‐up criteria for local solutions of the 3D generalized MHD equations in Lei–Lin–Gevrey spaces

Abstract

This paper determines the existence of a unique local solution for the 3D generalized magnetohydrodynamics equations. In order to be more precise, our solution is obtained by involving Lei–Lin–Gevrey spaces as well as Lei–Lin spaces. Furthermore, we present five new blow‐up criteria for this same system when the maximal time of existence is finite. It is important to point out that one of these criteria is obtained by assuming fractional Laplacians with equal powers.

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

“…Another motivating work was written by Melo and Rocha [24,30]. This article proves the local existence, as well as blow-up criteria, for solutions of the generalized Magnetohydrodynamics equations in…”

“…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 −∆).…”

“…where the fractional dissipations α and β belong to the interval (1/2, 1], and s ∈ (max{1 [24,30] for more information on the blow-up criteria proved in these works).…”

“…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].…”

“…. See [30] for a study of the generalized Magnetohydrodynamics equations. In this case, one infers that u ∈ L p T (X…”