On well-posedness of the Cauchy problem for 3D MHD system in critical Sobolev–Gevrey space

Abstract

This paper is devoted to the study of the three-dimensional periodic MHD system. We prove the local in time well-posedness for arbitrary large in H 1=2 a;r ðT 3 Þ initial data as well as global in time well-posedness when initial data satisfies a smallness condition. We also provide an unusual global in time existence criteria. It is based on breaking up the Fourier sums P k2Z 3 jûðt; kÞj, P k2Z 3 jbðt; kÞj of the solution (u, b) into low frequency modes up to m and high frequency modes down to m. We prove that there exists a wave-number m such that if the map t7 !F m ðtÞ ¼ P jkj m jûðt; kÞj þ P jkj m jbðt; kÞj À P jkj [ m jûðt; kÞj À P jkj [ m jbðt; kÞj keeps its sign constant on the time interval of existence [0, T), then the global in time existence of the solution follows.

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 −∆).…”