Advertisement

A Regularity Criterion in Terms of Pressure for the 3D Viscous MHD Equations

  • Sadek Gala
  • Maria Alessandra Ragusa
  • Zujin Zhang
Article

Abstract

In this note, we are concerned with the regularity of solutions of the MHD equation in terms of the pressure. More precisely, it is proved that if the pressure satisfies the critical growth condition
$$\begin{aligned} \pi (x,t)\in L^{\frac{2}{2+r}}\left( 0,T,\overset{.}{B}_{\infty ,\infty }^{r}(\mathbb {R}^{3})\right) \end{aligned}$$
for \(-1\le r\le 1\), then the solution remains smooth on (0, T]. The finding is mainly based on the innovative function decomposition methods together with Besov space techniques. Here \(\overset{.}{B}_{\infty ,\infty }^{r}\) denotes the homogeneous Besov space.

Keywords

MHD equations Regularity criteria Besov spaces 

Mathematics Subject Classification

35Q35 35B65 76D05 

Notes

Acknowledgments

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. The authors are indebted to Professor Yong Zhou who kindly sent us the preprint [29]. The authors want to express their sincere thanks to the editor and the referee for their invaluable comments and suggestions.

References

  1. 1.
    Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Berlin (1976)CrossRefMATHGoogle Scholar
  2. 2.
    Cabannes, H.: Theoretical Magneto Fluid dynamics. Academic Press, New York (1970)Google Scholar
  3. 3.
    Chemin, J.Y.: Perfect Incompressible Fluids. Oxford lecture series in mathematics and its applications, vol. 14. The Clarendon Press, New York (1998)MATHGoogle Scholar
  4. 4.
    Chen, Q., Miao, C., Zhang, Z.: The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations. Commun. Math. Phys. 275, 861–872 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chen, Q., Miao, C., Zhang, Z.: On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations. Commun. Math. Phys. 284, 919–930 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, Q., Miao, C., Zhang, Z.: On the uniqueness of weak solutions for the 3D Navier-Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 2165–2180 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, Q., Miao, C.: Existence theorem and blow-up criterion of the strong solutions to the two-fluid MHD equation in \(\mathbb{R}^{3}\). J. Differ. Equ. 239, 251–271 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Duan, H.L.: On regularity criteria in terms of pressure for the 3D viscous MHD equations. Appl. Anal. 91, 947–952 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Duvaut, G., Lions, J.L.: Inéquations en thermoé lasticité et magnéto-hydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279 (1972)CrossRefMATHGoogle Scholar
  10. 10.
    Gagliardo, E.: Proprieta di alcune classi di funzioni in pia variabili. Richerche Mat. 7, 102–137 (1958)MathSciNetMATHGoogle Scholar
  11. 11.
    Gagliardo, E.: Proprieta di alcune classi di funzioni in pia variabili. Richerche Mat. 9, 24–51 (1959)MATHGoogle Scholar
  12. 12.
    Gala, S.: Extension criterion on regularity for weak solutions to the 3D MHD equations. Math. Meth. Appl. Sci 33, 1496–1503 (2010)MathSciNetMATHGoogle Scholar
  13. 13.
    He, C., Wang, Y.: Remark on the regularity for weak solutions to the magnetohydrodynamic equations. Math. Methods Appl. Sci. 31, 1667–1684 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    He, C., Xin, Z.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 213, 235–254 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jia, Y., Gui X., Dong, B.: A new pressure regularity criterion of three-dimensional magnetohydrodynamic equations in Besov Spaces. (preprint 2014)Google Scholar
  16. 16.
    Jia, X.J., Zhou, Y.: A new regularity criterion for the 3D incompressible MHD equations in terms of one component of the gradient of pressure. J. Math. Anal. Appl. 396, 345–350 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P.: Electrodynamics of Continuous Media, 2nd edn. Butterworth-Heinemann, Oxford (1999)Google Scholar
  18. 18.
    Li, T., Qin, T.: Physics and Partial Differential Equations, vol. I, 2nd edn. Higher Education Press, Beijing (2005)Google Scholar
  19. 19.
    Meyer, Y., Gerard P., Oru, F.: Inégalités de Sobolev précisées, Séminaire Équations aux dérivées partielles (Polytechnique) (1996–1997), Exp. No 4, p. 8Google Scholar
  20. 20.
    Moreau, R.: Magnetohydrodynamics. Kluwer Academic Publishers, Dordrecht (1990)CrossRefMATHGoogle Scholar
  21. 21.
    Nirenberg, L.: On elliptic partial differential equations. Ann. Sc. Norm. Sup. Pisa, Ser. III 13, 115–162 (1959)MathSciNetMATHGoogle Scholar
  22. 22.
    Polovin, R.V., Demutskii, V.P.: Fundamentals of Magnetohydrodynamics. Consultants Bureau, New York (1990)Google Scholar
  23. 23.
    Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36(5), 635–664 (1983)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Triebel, H.: Theory of Function Spaces. II., Monographs in Mathematics, vol. 84. Birkhäuser, Basel (1992)Google Scholar
  25. 25.
    Wang, B., Chen, M.: An improved pressure regularity criterion of magnetohydrodynamic equations in critical Besov spaces. Bound. Value Probl. 66, pp. 10 (2015)Google Scholar
  26. 26.
    Wu, J.: Regularity results for weak solutions of the 3D MHD equations. Discret. Contin. Dyn. Syst. 10, 543–556 (2004)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Zhang, X., Jia, Y., Dong, B.: On the pressure regularity criterion of the 3D Navier-Stokes equations. J. Math. Anal. Appl. 393, 413–420 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Zhang, Z.J., Li, P., Yu, G.H.: Regularity criteria for the 3D MHD equations via one directional derivative of the pressure. J. Math. Anal. Appl. 401, 66–71 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zhou, Y., Fan, J.: On regularity criteria in terms of pressure for the 3D viscous MHD equations. (preprint 2009)Google Scholar
  30. 30.
    Zhou, Y.: Remarks on regularities for the 3D MHD equations. Discret. Contin. Dyn. Syst. 12, 881–886 (2005)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Zhou, Y.: Regularity criteria for the 3D MHD equations in terms of the pressure. Int. J. Non-linear Mech. 41, 1174–1180 (2006)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Zhou, Y.: On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \(\mathbb{R}^{3}\). Z. Angew. Math. Phys. 57, 384–392 (2006)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Zhou, Y.: On regularity criteria in terms of pressure for the Navier-Stokes equations in \(\mathbb{R}^{3}\). Proc. Am. Math. Soc. 134, 149–156 (2006)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    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)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Zhou, Y., Gala, S.: Regularity criteria in terms of the pressure for the Navier-Stokes Equations in the critical Morrey-Campanato space. Z. Anal. Anwend. 30, 83–93 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Authors and Affiliations

  • Sadek Gala
    • 1
    • 2
  • Maria Alessandra Ragusa
    • 2
  • Zujin Zhang
    • 3
  1. 1.Department of MathematicsUniversity of MostaganemMostaganemAlgeria
  2. 2.Dipartimento di Matematica e InformaticaUniversità di CataniaCataniaItaly
  3. 3.School of Mathematics and Computer ScienceGannan Normal UniversityGanzhouChina

Personalised recommendations