Skip to main content
SpringerLink
Log in

Commutator estimates with fractional derivatives and local existence for the generalized MHD equations

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper considers the problem of the local existence for the generalized MHD equations with fractional dissipative terms \(\Lambda ^{2\alpha } u\) for the velocity field and \(\Lambda ^{2\beta } b\) for the magnetic field, respectively. Based on some new commutator estimates, local existence for the generalized MHD equations is established, which recovers and improves previous results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Brezis, H., Mironescu, P.: Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces. J. Evol. Equ. 4, 387–404 (2001)

    Article  MathSciNet  Google Scholar 

  2. Coifman, R., Meyer, Y.: Nonlinear harmonic analysis, operator theory and P.D.E. Beijing lectures in harmonic analysis, Beijing, (1984)

  3. Cao, C., Wu, J., Yuan, B.: The 2D incompressible magnetohydrodynamic equations with only magnetic diffusion. SIAM J. Math. Anal. 46, 588–602 (2014)

    Article  MathSciNet  Google Scholar 

  4. Chae, D.: On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces. Commun. Pure Appl. Math. 55, 654–678 (2002)

    Article  MathSciNet  Google Scholar 

  5. Chemin, J.Y.: Regularite de la trajectoire des particules d’un fluide parfait incompressible remplissant l’espace. J. Math. Pures Appl. 9(71), 407–417 (1992)

    MATH  Google Scholar 

  6. Chemin, J.Y.: Perfect incompressible fluids. In: Oxford Lecture Series in Mathematics and its Applications, vol. 14, The Clarendon Press, Oxford University Press, New York, (1998)

  7. Fan, J., Malaikah, H., Monaquel, S., Nakamura, G., Zhou, Y.: Global cauchy problem of 2D generalized MHD equations. Mon. Math. 175, 127–131 (2014)

    Article  MathSciNet  Google Scholar 

  8. Fan, J., Nakamura, G., Zhou, Y.: A regularity criterion for the 3D generalized MHD equations. Math. Phys. Anal. Geom. 17, 333–340 (2014)

    Article  MathSciNet  Google Scholar 

  9. Fefferman, C.-L., McCormick, D.-S., Robinson, J.-C., Rodrigo, J.-L.: Higher order commutator estimates and local existence for the non-resistive MHD equations and related models. J. Funct. Anal. 267, 1035–1056 (2014)

    Article  MathSciNet  Google Scholar 

  10. Jiang, Z., Wang, Y., Zhou, Y.: On regularity criteria for the 2D generalized MHD system. J. Math. Fluid Mech. 18, 331–341 (2016)

    Article  MathSciNet  Google Scholar 

  11. Jiu, Q., Zhao, J.: Global regularity of 2D generalized MHD equations with magnetic diffusion. Z. Angew. Math. Phys. 66, 677–687 (2015)

    Article  MathSciNet  Google Scholar 

  12. Kato, T.: Nonstationary flows of viscous and ideal fluids in \({\mathbb{R}}^3\). J. Funct. Anal. 9, 296–305 (1972)

    Article  Google Scholar 

  13. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)

    Article  MathSciNet  Google Scholar 

  14. Kenig, C., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Kortewegde-Vries equation. J. Am. Math. Soc. 4, 323–347 (1991)

    Article  Google Scholar 

  15. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  16. Mccormick, D., Robinson, J., Rodrigo, J.: Generalised Gagliardo-Nirenberg inequalities using weak lebesgue spaces and BMO. Milan J. Math. 81, 265–289 (2013)

    Article  MathSciNet  Google Scholar 

  17. Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13, 116–162 (1955)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Stein, E.M.: Interpolation of linear operators. Trans. Am. Math. Soc. 83, 482–492 (1956)

    Article  MathSciNet  Google Scholar 

  20. Tran, C.V., Yu, X.W., Zhai, Z.C.: On global regularity of 2D generalized magnetohydrodynamic equations. J. Differ. Equ. 254, 4194–4216 (2013)

    Article  MathSciNet  Google Scholar 

  21. Wu, J.: Generalized MHD equations. J. Differ. Equ. 195, 284–312 (2003)

    Article  MathSciNet  Google Scholar 

  22. Zhou, Y.: Local well-posedness for the incompressible Euler equations in the critical Besov spaces. Ann. Inst. Fourier (Grenoble) 54, 773–786 (2004)

    Article  MathSciNet  Google Scholar 

  23. Zhou, Y.: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. Henri Poincar Anal. Non Linear 24, 491–505 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the referees for helpful comments and suggestions on the manuscript. Z. Jiang is partially supported by NSFC (Grant No. 12071439) and ZJNSF (Grant No. LY19A010016).

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Zaihong Jiang & Yong Zhou

  2. Department of Mathematics, Tianshui Normal University, Tianshui, 741001, China

    Caochuan Ma

  3. School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, 519082, China

    Yong Zhou

Authors
  1. Zaihong Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Caochuan Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Zhou.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, Z., Ma, C. & Zhou, Y. Commutator estimates with fractional derivatives and local existence for the generalized MHD equations. Z. Angew. Math. Phys. 72, 111 (2021). https://doi.org/10.1007/s00033-021-01539-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01539-1

Keywords

Mathematics Subject Classification

Search

Navigation