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A double-logarithmically improved regularity criterion of weak solutions for the 3D MHD equations

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Abstract

This paper devotes to establish an improved regularity criterion for weak solution of 3D incompressible MHD equations. We show that the weak solution is regular, provided that

$$\begin{aligned} \int _{0}^{T}{\frac{\Vert \pi (\cdot ,t)\Vert _{\dot{B}_{\infty ,\infty }^{-1}}^{2}}{\left( e+\ln \left( e+\Vert \pi (\cdot ,t)\Vert _{\dot{B} _{\infty ,\infty }^{-1}}\right) \right) \ln \left( e+\ln \left( e+\Vert \pi (\cdot ,t)\Vert _{\dot{B}_{\infty ,\infty }^{-1}}\right) \right) }}\ dt<\infty . \end{aligned}$$

As a consequence, this improves some previous works.

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Acknowledgements

The authors thank the referees for their careful reading and helpful suggestions, which improve the paper much. This research was supported by the Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University, Saudi Arabia, Grant No. (20-13-12-020) .

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Authors and Affiliations

  1. Department of Sciences Exactes, ENS of Mostaganem, University of Mostaganem, Box 227, Mostaganem, 27000, Algeria

    Sadek Gala

  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

  4. Department of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh, 11623, Saudi Arabia

    Ines Ben Omrane

Authors
  1. Ines Ben Omrane
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  2. Sadek Gala
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  3. Maria Alessandra Ragusa
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Correspondence to Sadek Gala.

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Omrane, I.B., Gala, S. & Ragusa, M.A. A double-logarithmically improved regularity criterion of weak solutions for the 3D MHD equations. Z. Angew. Math. Phys. 72, 114 (2021). https://doi.org/10.1007/s00033-021-01543-5

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  • DOI: https://doi.org/10.1007/s00033-021-01543-5

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