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

Abstract

This study is devoted to investigating the blow-up criteria of strong solutions and regularity criterion of weak solutions for the magnetic Bénard system in $\mathbb {R}^3$ in a sense of scaling invariant by employing a different decomposition for nonlinear terms. Firstly, the strong solution $(u,b,\theta )$ of magnetic Bénard system is proved to be smooth on (0, T] provided the velocity field u satisfies

$$\begin{aligned} u\in {L}^{\frac{2}{1-r}}(0,T;\dot{\mathbb {X}}_r(\mathbb {R}^3))\quad ~~with\quad 0\le {r}<1, \end{aligned}$$

or the gradient field of velocity $\nabla {u}$ satisfies

$$\begin{aligned} \nabla {u}\in {L}^{\frac{2}{2-\gamma }}(0,T;\dot{\mathbb {X}}_\gamma (\mathbb {R}^3))\quad ~~with\quad 0\le {\gamma }\le {1}. \end{aligned}$$

Moreover, we prove that if the following conditions holds:

$$\begin{aligned} u\in {L}^\infty (0,T;\dot{\mathbb {X}}_1(\mathbb {R}^3))\quad and \quad \Vert u\Vert _{L^\infty (0,T;\dot{\mathbb {X}}_1(\mathbb {R}^3))}<\varepsilon , \end{aligned}$$

where $\varepsilon >0$ is a suitable small constant, then the strong solution $(u,b,\theta )$ of magnetic Bénard 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. $\tilde{\nabla }\tilde{u}$, $\tilde{\nabla }\tilde{b}$, $\tilde{\nabla }\theta$) 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 270:1–7, 2013) respectively on the blow-up criteria for the three-dimensional Boussinesq system and MHD system in the multiplier space.