Blow-Up Criteria and Regularity Criterion for the Three-Dimensional Magnetic Benard System in the Multiplier Space

This study is devoted to investigating the blow-up criteria of strong solutions and regularity criterion of weak solutions for the magnetic Benard system in R-3 in a sense of scaling invariant by employing a different decomposition for nonlinear terms. Firstly, the strong solution (u, b, theta) of magnetic Benard system is proved to be smooth on (0, T] provided the velocity field u satisfies u is an element of L2/1-r (0, T; (X) over dot(r) (R-3)) with 0 <= r < 1, or the gradient field of velocity del u satisfies del u is an element of L2/2-gamma (0, T; (X) over dot(gamma) (R-3)) with 0 <= gamma <= 1. Moreover, we prove that if the following conditions holds: u is an element of L-infinity (0, T; (X) over dot(1) (R-3)) and parallel to u parallel to(L infinity(0, T;(X) over dot1 (R3))) < is an element of, where is an element of > 0 is a suitable small constant, then the strong solution (u, b, theta) of magnetic Benard system can also be extended beyond t = T. Finally, we show that if some partial derivatives of the velocity components, magnetic components and temperature components (i.e.& nbsp;(del) over tilde(u) over tilde, (del) over tilde (b)over tilde>, (del) over tilde theta) belong to the multiplier space, the solution (u, b, theta) actually is smooth on (0, T). Our results extend and generalize the recent works (Qiu et al. in Commun Nonlinear Sci Numer Simul 16:1820-1824, 2011; Tian in J Funct Anal, 2017. https:// doi .org/10.1155/2017/3795172; Zhou and Gala in Z Angew Math Phys 61:193-199, 2010; Zhang et al. in Bound Value Probl 270:17, 2013) respectively on the blow-up criteria for the three-dimensional Boussinesq system and MHD system in the multiplier space.