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A regularity criterion of the 3D MHD equations involving one velocity and one current density component in Lorentz space

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Abstract

In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution (u, b) becomes regular provided that one velocity and one current density component of the solution satisfy

$$\begin{aligned} u_{3}\in L^{\frac{30\alpha }{7\alpha -45}}\left( 0,T;L^{\alpha ,\infty }\left( {\mathbb {R}}^{3}\right) \right) \quad \text { with }\frac{45}{7} \le \alpha \le \infty , \end{aligned}$$
(0.1)

and

$$\begin{aligned} j_{3}\in L^{\frac{2\beta }{2\beta -3}}\left( 0,T;L^{\beta ,\infty }\left( {\mathbb {R}}^{3}\right) \right) \quad \text {with }\frac{3}{2}\le \beta \le \infty , \end{aligned}$$
(0.2)

which generalize some known results.

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Acknowledgements

This work was done while the second author was visiting the Catania University in Italy. He would like to thank the hospitality and support of the University, where this work was completed. This research is partially supported by P.R.I.N. 2019. The third author wish to thank the support of “RUDN University Program 5-100.”

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University-Kingsville, Kingsville, USA

    Ravi P. Agarwal

  2. Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6, 95125, Catania, Italy

    Sadek Gala & Maria Alessandra Ragusa

  3. RUDN University, 6 Miklukho - Maklay St, Moscow, Russia, 117198

    Maria Alessandra Ragusa

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  1. Ravi P. Agarwal
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  2. Sadek Gala
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  3. Maria Alessandra Ragusa
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Correspondence to Maria Alessandra Ragusa.

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Agarwal, R.P., Gala, S. & Ragusa, M.A. A regularity criterion of the 3D MHD equations involving one velocity and one current density component in Lorentz space. Z. Angew. Math. Phys. 71, 95 (2020). https://doi.org/10.1007/s00033-020-01318-4

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