Skip to main content
SpringerLink
Log in

Analyticity of Mild Solution for the 3D Incompressible Magneto-hydrodynamics Equations in Critical Spaces

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

The Cauchy problem for the 3D incompressible magneto-hydrodynamics equations in critcal spaces is considered. We first prove the global well-posedness of mild solution for the system in some time dependent spaces. Furthermore, we obtain analyticity of the solution.

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. Bae, H.: Existence and analyticity of Lei–Lin solution to the Navier–Stokes equations. Proc. Amer. Math. Soc., 143, 2887–2892 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benameur, J.: Long time decay to the Lei–Lin solution of 3D Navier–Stokes equations. J. Math. Anal. Appl., 422, 424–434 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cannone, M., Chen, Q., Miao, C.: A losing estimate for the Ideal MHD equations with application to blow-up criterion. SIAM J. Math. Anal., 38, 1847–1859 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Chi, M., Xu, F., Wu, Y.: A logarithmic improvement of regularity criterion for the MHD equations in terms of the pressure. Appl. Math. Comput., 327, 46–54 (2018)

    MathSciNet  Google Scholar 

  6. Chen, Q., Miao, C., Zhang, Z.: The Beale–Kato–Majda criterion for the 3D magneto-hydrodynamics equations. Commun. Math. Phys., 275, 861–872 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Q., Miao, C., Zhang, Z.: On the regularity criterion of weak solution for the 3D viscous magnetohydrodynamics equations. Comm. Math. Phys., 284, 919–930 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Q., Miao, C., Zhang, Z.: Existence theorem and blow-up criterion of strong solutions to the two-fluid MHD equation in R3. J. Diff. Equations, 239, 251–271 (2007)

    Article  MATH  Google Scholar 

  9. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal., 87, 359–369 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. He, C., Xin, Z.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Diff. Equations, 213, 235–254 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lei, Z., Lin, F.: Global mild solutions of Navier–Stokes equations. Comm. Pure Appl. Math., 64, 1297–1304 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lemari-Rieusset, P. G.: On the analyticity of mild solutions for the Navier–Stokes equations. C. R. Acad. Sci. Paris, Ser. I, 330, 183–186 (2000)

    Article  Google Scholar 

  13. Lemari-Rieusset, P. G.: Recent Developments in the Navier–Stokes Problem, Research Notes in Mathematics, Chapman Hall/CRC, Boca Raton, 2002

    Google Scholar 

  14. Miao, C., Yuan, B.: On the wellposedness of the Cauchy problem for an MHD system in Besov spaces. Math. Methods. Appl. Sci., 32, 53–76 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Miao, C., Yuan, B., Zhang, B.: Well-posedness for the incompressible magneto-hydrodynamics system. Math. Methods. Appl. Sci., 30, 961–976 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, Y., Wang, K.: Global well-posedness of the three dimensional magnetohydrodynamics equations. Nonl. Anal. RWA, 17, 245–251 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, Y.: Asymptotic decay of solutions to the 3D MHD equations. Nonl. Anal., 132, 115–125 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wu, J.: Bounds and new approaches for the 3D MHD equations. J. Nonlinear Sci., 12, 395–413 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wu, J.: Regularity results for weak solutions of the 3D MHD equations. Discrete. Contin. Dynam. Systems, 10, 543–556 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wu, J.: Regularity criteria for the generalized MHD equations. Commun. Partial Diff. Equations, 33, 285–306 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xu, F.: A regularity criterion for the 3D incompressible magneto-hydrodynamics equations. J. Math. Anal. Appl., 460, 634–644 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Xu, F., Li, X., Cui, Y., et al.: A scaling invariant regularity criterion for the 3D incompressible magnetohydrodynamics equations. Z. Angew. Math. Phy., 68, 1–8 (2017)

    Article  Google Scholar 

  24. Xu, F., Zhang, X., Wu, Y., et al.: Global existence and temporal decay for the 3D compressible Hallmagnetohydrodynamic system. J. Math. Anal. Appl., 438, 285–310 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the referees for their time and comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University, Weihai, 264200, P. R. China

    Sheng Wang

  2. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255049, P. R. China

    Yu Bing Ren & Fu Yi Xu

Authors
  1. Sheng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Bing Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fu Yi Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fu Yi Xu.

Additional information

Supported by Research supported by the National Natural Science Foundation of China (Grant Nos. 11501332, 11771043, 11371221), the Natural Science Foundation of Shandong Province (Grant No. ZR2015AL007), China Postdoctoral Science Foundation funded project (Grant No. 2014M561893), Postdoctoral innovation fund of Shandong Province (Grant No. 201502015), the Open Fund of State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research Fund (Grant No. IWHR-SKL-201407), and the Specialized Research Foundation for the Doctoral Program of Higher Education of China (Grant No. 20123705110001), and Young Scholars Research Fund of Shandong University of Technology

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S., Ren, Y.B. & Xu, F.Y. Analyticity of Mild Solution for the 3D Incompressible Magneto-hydrodynamics Equations in Critical Spaces. Acta. Math. Sin.-English Ser. 34, 1731–1741 (2018). https://doi.org/10.1007/s10114-018-8043-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-018-8043-4

Keywords

MR(2010) Subject Classification

Search

Navigation