Skip to main content
SpringerLink
Log in

Global Existence and Asymptotic Stability of 3D Generalized Magnetohydrodynamic Equations

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

In this paper, we study the global existence and asymptotic dynamics of generalized magnetohydrodynamic equations in \({\mathbb {R}}^3\), in which the dissipation terms are \(-\eta (-\Delta )^\alpha \) and \(-\mu (-\Delta )^\beta \), \(0<\alpha ,\,\beta <1\). With the help of combining the local existence and the a priori estimates, we establish the global existence and uniqueness of solution with small initial data. Moreover, we obtain the asymptotic decay rates of solutions by the method of energy estimates.

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. Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)

    ADS  MATH  Google Scholar 

  2. Biskamp, D.: Nonlinear Magnetohydrodynamics. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  3. Caflisch, R.E., Klapper, I., Steele, G.: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Commun. Math. Phys. 184, 443–455 (1997)

    ADS  MathSciNet  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  5. Cao, C., Wu, J.: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math. 226, 1803–1822 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Chae, D., Constantin, P., Wu, J.: Inviscid models generalizing the two dimensional Euler and the surface quasi-geostrophic equations. Arch. Ration. Mech. Anal. 202, 35–62 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Chae, D., Constantin, P., Córdoba, D., Gancedo, F., Wu, J.: Generalized surface quasi-geostrophic equations with singular velocities. Commun. Pure Appl. Math. 65, 1037–1066 (2012)

    MathSciNet  MATH  Google Scholar 

  8. 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)

    ADS  MathSciNet  MATH  Google Scholar 

  9. Constantin, P.: Geometric statistics in turbulence. SIAM Rev. 36, 73–98 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Constantin, P., Fefferman, C., Majda, A.: Geometric constraints on potentially singular solutions for the 3D Euler equations. Commun. Partial Differ. Equ. 21, 559–571 (1996)

    MATH  Google Scholar 

  11. Dong, H., Li, D.: On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces. J. Differ. Equ. 248, 2684–2702 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  12. Dong, B.Q., Jia, Y., Li, J., Wu, J.: Global regularity and time decay for the 2D magnetohydrodynamic equations with fractional dissipation and partial magnetic diffusion. J. Math. Fluid Mech. 20, 1541–1565 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  13. Duvant, G., Lions, J.L.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Rational Mech. Anal. 46, 241–279 (1972)

    ADS  MathSciNet  MATH  Google Scholar 

  14. Gibbon, J.D., Ohkitani, K.: Singularity formation in a class of stretched solutions of the equations for ideal magneto-hydrodynamics. Nonlinearity 14, 1239–1264 (2001)

    ADS  MathSciNet  MATH  Google Scholar 

  15. Granero-Belinchón, R.: Global solutions for a hyperbolic-parabolic system of chemotaxis. J. Math. Anal. Appl. 449, 872–883 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Guo, Y., Wang, Y.: Decay of dissipative equations and negative Sobolev spaces. Commun. Partial Differ. Equ. 37, 2165–2208 (2012)

    MathSciNet  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  18. He, C., Xin, Z.: Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations. J. Funct. Anal. 227, 113–152 (2005)

    MathSciNet  MATH  Google Scholar 

  19. Hmidi, T., Keraani, S., Rousset, F.: Global well-posedness for Euler–Boussinesq system with critical dissipation. Commun. Partial Differ. Equ. 36, 420–445 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Hmidi, T., Keraani, S., Rousset, F.: Global well-posedness for a Boussinesq–Navier–Stokes system with critical dissipation. J. Differ. Equ. 249, 2147–2174 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  21. Hmidi, T., Zerguine, M.: On the global well-posedness of the Euler–Boussinesq system with fractional dissipation. Physica D 239, 1387–1401 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  22. Ju, N.: Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Commun. Math. Phys. 251, 365–376 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. Lei, Z., Zhou, Y.: BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete Contin. Dyn. Syst. 25, 575–583 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Li, P., Zhai, Z.: Well-posedness and regularity of generalized Navier–Stokes equations in some critical Q-spaces. J. Funct. Anal. 259, 2457–2519 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, UK (2002)

    MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  28. Mohgooner, S.D., Sarayker, R.E.: \(L^2\) decay for solutions of the MHD equations. J. Math. Phys. Sci. 23, 35–53 (1989)

    MathSciNet  Google Scholar 

  29. Nirenberg, L.: On elliptic partial differential equations. Ann. Sci. Norm. Super. Pisa 13, 115–162 (1959)

    MathSciNet  MATH  Google Scholar 

  30. Núñez, M.: Estimates on hyperdiffusive magnetohydrodynamics. Physica D 183, 293–301 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  31. Priest, E., Forbes, T.: Magnetic Reconnection, MHD Theory and Applications. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  32. Schonbek, M.E., Schonbek, T.P., Siili, E.: Large time behaviour of solutions to the magnetohydrodynamics equations. Math. Ann. 304, 717–756 (1996)

    MathSciNet  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  34. Stein, E.: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton (1970) xiv+290 pp

  35. Tran, C., Yu, X., Zhai, Z.: Note on solution regularity of the generalized magnetohydrodynamic equations with partial dissipation. Nonlinear Anal. 85, 43–51 (2013)

    MathSciNet  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  37. Wang, Y.: Decay of the Navier–Stokes–Poisson equations. J. Differ. Equ. 253, 273–297 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  38. Wu, J.: Viscous and inviscid magnetohydrodynamics equations. J. Anal. Math. 73, 251–265 (1997)

    MathSciNet  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  42. Wu, J.: The generalized incompressible Navier–Stokes equations in Besov spaces, spaces. Dyn. Partial Differ. Equ. 1, 381–400 (2004)

    MathSciNet  MATH  Google Scholar 

  43. Wu, J.: Lower bounds for an integral involving fractional Laplacians and the generalized Navier–Stokes equations in Besov spaces. Commun. Math. Phys. 263, 803–831 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  46. Yamazaki, K.: Global regularity of logarithmically supercritical MHD system with zero diffusivity. Appl. Math. Lett. 29, 46–51 (2014)

    MathSciNet  MATH  Google Scholar 

  47. Yamazaki, K.: Stochastic Lagrangian formulations for dampled Navier–Stokes equations and Boussinesq system and their applications. Commun. Stoch. Anal. 12, 447–471 (2018)

    MathSciNet  Google Scholar 

  48. Yamazaki, K.: Global regularity of logarithmically supercritical MHD system with improved logarithmic powers. Dyn. Partial Differ. Equ. 15, 147–173 (2018)

    MathSciNet  MATH  Google Scholar 

  49. Yamazaki, K.: Remarks on the three and two and a half dimensional Hall-magnetohydrodynamics system: deterministic and stochastic cases. Complex Anal. Synerg. 5(9), 11 (2019)

    MathSciNet  Google Scholar 

  50. Zhu, S., Liu, Z., Zhou, L.: Global existence and asymptotic stability of the fractional chemotaxis-fluid system in \({\mathbb{R}}^3\). Nonlinear Anal. 183, 149–190 (2019)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work is partially supported by National Natural Science Foundation of China (11771380 and 11401515).

Author information

Authors and Affiliations

  1. School of Mathematical Science, Yangzhou University, Yangzhou, 225002, China

    Kerui Jiang, Zuhan Liu & Ling Zhou

Authors
  1. Kerui Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zuhan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ling Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kerui Jiang.

Ethics declarations

Conflict of interest

The authors declared that they have no conflicts of interest to this work.

Additional information

Communicated by G. P. Galdi

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, K., Liu, Z. & Zhou, L. Global Existence and Asymptotic Stability of 3D Generalized Magnetohydrodynamic Equations. J. Math. Fluid Mech. 22, 9 (2020). https://doi.org/10.1007/s00021-019-0475-9

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00021-019-0475-9

Keywords

Mathematics Subject Classification

Search

Navigation