On local existence, uniqueness and blow-up of solutions for the generalized MHD equations in Lei–Lin spaces

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

“…2. Theorem 1.1 establishes the case of Lei-Lin-Gevrey spaces for Theorem 1.1 obtained in [12] and, furthermore, proves that our solution (𝑢, 𝑏) belongs to 𝐿 2 𝑇 ( 𝑠+𝛼 𝑎,𝜎 (ℝ 3 )) × 𝐿 2 𝑇 ( 𝑠+𝛽 𝑎,𝜎 (ℝ 3 )). In addition, Corollary 1.3 proves that this same solution (𝑢, 𝑏) also belongs to…”

“…} , 𝜎 > 1, for all 𝑡 ∈ [0, 𝑇 * ). Compare all the blow-up criteria listed above with the results obtained in the papers [1,12] and notice that they are particular cases of our main results.…”

“…The MHD equations also present a high mathematical complexity and applications in other branches of science such as geophysics, astrophysics, and engineering. These are some of the reasons why these systems have attracted many researchers and, as a result, numerous papers have been published in the literature (see, for instance, [1,[6][7][8][10][11][12][13] and references therein).…”

“…For example, local well-posedness of the MHD equations was obtained by Sermange and Temam [14] (see also [2-5, 10, 13, 14, 18] and references included). Recently, by considering Lei-Lin spaces, Melo et al [12] proved the existence of a unique local solution for GMHD equations (1.1) as well as a blow-up criterion for this same solution (we refer also to [15][16][17]19] and references therein).…”

Well-posedness and blow-up of solutions for the 2D dissipative quasi-geostrophic equation in critical Fourier-Besov-Morrey spaces.

“…Also, the authors in [10] proved a blow-up criterion of the local-in-time solution of (1) in Lei-Lin space X 1−2 with subcritical dissipation, other related results can be found in [9,10,25,34]. We mention that the work [1] covers the critical dissipation (− ) 1∕2 in the context of the Boussinesq-Coriolis system in Fourier-Besov-Morrey spaces.…”