Blow‐up of the maximal solution to 3D Boussinesq system in Lei‐Lin‐Gevrey spaces

Abstract

For arbitrary initial data in Lei‐Lin‐Gevrey spaces, we investigate the blow‐up phenomena in finite time to the local unique solution of the three‐dimensional Boussinesq system. We determine the blow‐up profile explicitly as a function of time, and we identify the low frequencies part as a solely responsible of this phenomena. Frequencies decomposition, functional spaces interpolation, and Leray theory are used.

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 −∆).…”

“…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].…”