Skip to main content
SpringerLink
Log in

A Regularity Criterion for the 3D Generalized MHD Equations

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

This paper proves a new regularity criterion for the 3D generalized MHD system with fractional diffusion terms ( − Δ)αu and ( − Δ)βb with \(0<\alpha <\frac 54\leqslant \beta \).

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. Wu, J.: Generalized MHD equations. J. Diff. Equat. 195, 284–312 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Wu, J.: Regularity criteria for the generalized MHD equations. Comm. Partial Differ. Equ. 33, 285–306 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fan, J., Gao, H., Nakamura, G.: Regularity criteria for the generalized magnetohydrodynamic equations and the quasi-geostrophic equations. Taiwan. J. Math. 15(3), 1059–1073 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Zhou, Y.: Regularity criteria for the generalized viscous MHD equations. Ann. I. H. Poincare-AN. 24, 491–505 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Wu, J.: Global regularity for a class of generalized magnetohydrodynamic equations. J. Math. Fluid Mech. 13, 295–305 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Ji, E.: On two-dimensional magnetohydrodynamic equations with fractional diffusion. Nonlinear Anal. 80, 55–65 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Tran, C.V., Yu, X., Zhai, Z.: On global regularity of 2D generalized magnetohydrodynamic equations. J. Diff. Equat. 254(10), 4194–4216 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Fan, J., Malaikah, H., Monaquel, S., Nakamura, G., Zhou, Y.: Global Cauchy problem of 2D generalized MHD equations. To appear in Monatsh. Math (2014)

  9. Wang, Y., Zhou, Y., Alsaedi, A., Hayat, T., Jiang, Z.: Global regularity for the incompressible 2D generalized liquid crystal model with fractional diffusions. Appl. Math. Lett. 35, 18–23 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhu, M., Jin, Y.: Global regularity for the incompressible 2D liquid crystal model with fractional diffusion. Preprint (2014)

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037, People’s Republic of China

    Jishan Fan

  2. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Ahmed Alsaedi, Tasawar Hayat & Yong Zhou

  3. Department of Mathematics, Quaid-I-Azam University 45320, Islamabad, 44000, Pakistan

    Tasawar Hayat

  4. Department of Mathematics, Inha University, Incheon, 402-751, Korea

    Gen Nakamura

  5. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, People’s Republic of China

    Yong Zhou

Authors
  1. Jishan Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmed Alsaedi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tasawar Hayat
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gen Nakamura
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Zhou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fan, J., Alsaedi, A., Hayat, T. et al. A Regularity Criterion for the 3D Generalized MHD Equations. Math Phys Anal Geom 17, 333–340 (2014). https://doi.org/10.1007/s11040-014-9159-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11040-014-9159-0

Keywords

Search

Navigation