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Results in Mathematics

, 73:103 | Cite as

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

  • Liangliang MaEmail author
Article

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.

Keywords

Magnetic Bénard system blow-up criterion a priori estimates multiplier space 

Mathematics Subject Classification

35Q35 35B65 35Q30 76D05 35Q30 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and ComputerPanzhihua UniversityPanzhihuaPeople’s Republic of China

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