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On the regularity criterion of weak solutions for the 3D MHD equations

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Abstract

The paper deals with the 3D incompressible MHD equations and aims at improving a regularity criterion in terms of the horizontal gradient of velocity and magnetic field. It is proved that the weak solution (u, b) becomes regular provided that

$$\begin{aligned} \left( \nabla _{h}u,\nabla _{h}b\right) \in L^{\frac{8}{3}}\left( 0,T;\overset{\cdot }{B}_{\infty ,\infty }^{-1}\left( \mathbb {R}^{3}\right) \right) . \end{aligned}$$

The result is an extension of regularity criterion for 3D Navier–Stokes equations in Besov space due to Fang and Qian (Commun Pure Appl Anal 13:585–603, 2014) [see also (Ni et al. in J Math Anal Appl 396:108–118, 2012)].

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References

  1. Cao, C., Wu, J.: Two regularity criteria for the 3D MHD equations. J. Differential Equations 248, 2263–2274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Duvaut, G., Lions, J.L.: Inéquations en thermoé lasticité et magnéto-hydrodynamique. Arch. Rational Mech. Anal. 46, 241–279 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fang, D.Y., Qian, C.Y.: Regularity criteria for 3D Navier-Stokes equations in Besov space. Commun. Pure Appl. Anal. 13, 585–603 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. He, C., **n, Z.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differential Equations 213, 235–254 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jia, X., Zhou, Y.: Regularity criteria for the 3D MHD equations involving partial components. Nonlinear Anal. Real World Appl. 13, 410–418 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jia, X., Zhou, Y.: Regularity criteria for the 3D MHD equations via partial derivatives. Kinet. Relat. Models 5, 505–516 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Mahalov, A., Nicolaenko, B., Shilkin, T.: \(L^{3,\infty }\) solutions to the MHD equations. J. Math. Sci. 143, 2911–2923 (2007)

    Article  MathSciNet  Google Scholar 

  8. Meyer, Y., Gerard, P., Oru, F.: Inégalités de Sobolev précisées, Séminaire Équations aux dérivées partielles (Polytechnique) (1996–1997), Exp. \(\text{N}^{\circ }\)4, p. 8

  9. Ni, L., Guo, Z., Zhou, Y.: Some new regularity criteria for the 3D MHD equations. J. Math. Anal. Appl. 396, 108–118 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)

    Book  MATH  Google Scholar 

  12. Yuan, B., Li, X.: Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete Contin. Dyn. Syst. 9, 2167–2179 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhou, Y.: Remarks on regularities for the 3D MHD equations. Discrete Continuous Dyn. Syst. 12, 881–886 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhou, Y., Gala, S.: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys. 61, 193–199 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhou, Y., Gala, S.: A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field. Nonlinear Anal. 72, 3643–3648 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The part of the work was carried out, while the first author was long-term visitor at University of Catania. The hospitality and support of Catania University are graciously acknowledged. Maria Alessandra Ragusa is supported by the Ministry of Education and Science of the Russian Federation (Agreement No. 02. a 03.21.0008). The authors would like to thank the referees for valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Mostaganem, Box 227, 27000, Mostaganem, Algeria

    Sadek Gala

  2. Dipartimento di Matematica e Informatica, Universit à 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. Sadek Gala
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  2. Maria Alessandra Ragusa
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Correspondence to Sadek Gala.

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Gala, S., Ragusa, M.A. On the regularity criterion of weak solutions for the 3D MHD equations. Z. Angew. Math. Phys. 68, 140 (2017). https://doi.org/10.1007/s00033-017-0890-9

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  • DOI: https://doi.org/10.1007/s00033-017-0890-9

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