On the Blow-up Criterion of 3D-NSE in Sobolev–Gevrey Spaces

Abstract

In [5], Benameur proved a blow-up result of the non regular solution of (N SE) in the Sobolev-Gevrey spaces. In this paper we improve this result, precisely we give an exponential type explosion in Sobolev-Gevrey spaces with less regularity on the initial condition. Fourier analysis is used.

The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions

“…Also, we prove the blowup inequality for 1 2 < s < 3 2 whereas only the value s = 1 was considerded in [4]. For further blow-up results for the Navier-Stokes and MHD systems we refer to [1,2,4,6,7,8,11,12,13,14,15] and references therein.…”

“…Remark 1. It is important to point out that the existence result obtained for the spaceḢ 1 a,σ (R 3 ) by J. Benameur and L. Jlali [4] is a particular case of Theorem 1.1. In fact, it is enough to take s = 1 and b = 0 in this last statement.…”

“…In fact, it is enough to take s = 1 and b = 0 in this last statement. Furthermore, Theorem 1.1 generalizes [4] from the Navier-Stokes equations to MHD system (1).…”

“…In a recent paper, J. Benameur and L. Jlali [4] proved blow-up criteria for the Navier-Stokes equations in Sobolev-Gevrey spaces. Our current paper extends the results of [4] from the Navier-Stokes to the MHD system.…”

“…In a recent paper, J. Benameur and L. Jlali [4] proved blow-up criteria for the Navier-Stokes equations in Sobolev-Gevrey spaces. Our current paper extends the results of [4] from the Navier-Stokes to the MHD system. Also, we prove the blowup inequality for 1 2 < s < 3 2 whereas only the value s = 1 was considerded in [4].…”