Local existence, uniqueness and lower bounds of solutions for the magnetohydrodynamics equations in Sobolev-Gevrey spaces

This paper proves the existence and uniqueness, and also establishes blow-up criteria, of solutions for the Magnetohydrodynamics equations in Sobolev-Gevrey spaces H-alpha,sigma(s) (R-3). More precisely, if it is assumed that the initial data (u(0), b(0)) belongs to H-alpha,sigma(s0)(R-3), with a > 0, sigma >= 1, s(0) > 1/2 and s(0) not equal 3/2, we demonstrate that there is a time T > 0 such that (u, b) is an element of C([0, T]; H-alpha,sigma(s)(R-3) for all s <= s(0). In addition, we show for instance that if T* < infinity is the first blow-up instant of the solution (u, b)(x, t); then, parallel to(u,b)(t)parallel to H-alpha,sigma(s)(R-3) >= C-1 parallel to(u,b)(t)parallel to 1-2(3)/3 L-2(R) exp {aC(2) parallel to(u,b)(t)parallel to-2/3 sigma -L-2(R) (T* - t) 1-/3 sigma } /(T* - t) r/3 for all t is an element of[0, T*), where s(0) > 3/2, 3/2 < s <= s(0), a > 0 and a >= 1. And also parallel to(u,b)(t)parallel to H-alpha,sigma(s)(R-3) >= a sigma 0+1/2 C(2)exp{aC(3) (T* - t) -1/3 sigma}/(T* - t) 2 where 1/2 < s(0) < 3/2, 1/2 < s <= s(0), a > 0 and sigma > 1. Here 2 sigma(0) is the integer part of 2 sigma. (C) 2019 Elsevier Inc. All rights reserved.