On global regular solutions to magnetohydrodynamics in axi-symmetric domains

Open Access
Article

Abstract

We consider mhd equations in three-dimensional axially symmetric domains under the Navier boundary conditions for both velocity and magnetic fields. We prove the existence of global, regular axi-symmetric solutions and examine their stability in the class of general solutions to the mhd system. As a consequence, we show the existence of global, regular solutions to the mhd system which are close in suitable norms to axi-symmetric solutions.

Keywords

Magnetohydrodynamics Stability of axially symmetric solutions Global existence of regular solutions 

Mathematics Subject Classification

35Q35 76D03 76W05 

References

  1. 1.
    Nowakowski, B., Zajączkowski, W.M.: Stability of two-dimensional magnetohydrodynamic motions in the periodic case. Math. Methods Appl. Sci. 39(1), 44–61 (2016). doi:10.1002/mma.3459 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Zadrzyńska, E., Zajączkowski, W.M.: Stability of two-dimensional Navier–Stokes motions in the periodic case. J. Math. Anal. Appl. 423(2), 956–974 (2015). doi:10.1016/j.jmaa.2014.10.026 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ströhmer, G.: About an initial-boundary value problem from magnetohydrodynamics. Math. Z. 209(3), 345–362 (1992). doi:10.1007/BF02570840 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ströhmer, G.: About the equations of non-stationary magneto-hydrodynamics with non-conducting boundaries. Nonlinear Anal. Theory Methods Appl. 39(5), 629–647 (2000). doi:10.1016/S0362-546X(98)00226-0 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Duvaut, G., Lions, J.-L.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279 (1972)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36(5), 635–664 (1983). doi:10.1002/cpa.3160360506 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cao, C., Wu, J.: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math. 226(2), 1803–1822 (2011). doi:10.1016/j.aim.2010.08.017 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cao, C., Regmi, D., Wu, J.: The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J. Differ. Equations 254(7), 2661–2681 (2013). doi:10.1016/j.jde.2013.01.002 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fan, J., Malaikah, H., Monaquel, S., Nakamura, G., Zhou, Y.: Global Cauchy problem of 2D generalized MHD equations. Monatsh. Math. 175(1), 127–131 (2014). doi:10.1007/s00605-014-0652-0 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Huang, X., Wang, Y.: Global strong solution to the 2D nonhomogeneous incompressible MHD system. J. Differ. Equations 254(2), 511–527 (2013). doi:10.1016/j.jde.2012.08.029 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jiu, Q., Zhao, J.: Global regularity of 2D generalized MHD equations with magnetic diffusion. Z. Angew. Math. Phys. 66(3), 677–687 (2015). doi:10.1007/s00033-014-0415-8 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kang, E., Lee, J.: Regularizing model for the 2D MHD equations with zero viscosity. Abstr. Appl. Anal. 2012, 212786-1–212786-7 (2012)Google Scholar
  13. 13.
    Wei, Z., Zhu, W.: Global well-posedness of 2D generalized MHD equations with fractional diffusion. J. Inequal. Appl. 2013, 489 (2013). doi:10.1186/1029-242X-2013-489 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Zhou, Y., Fan, J.: Global Cauchy problem for a 2D Leray-\(\alpha \)-MHD model with zero viscosity. Nonlinear Anal. 74(4), 1331–1335 (2011). doi:10.1016/j.na.2010.10.005 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Solonnikov, V.A.: Estimates for solutions of a non-stationary linearized system of Navier–Stokes equations. Trudy Mat. Inst. Steklov. 70, 213–317 (1964)MathSciNetGoogle Scholar
  16. 16.
    Mucha, P.: Stability of nontrivial solutions of the Navier–Stokes system on the three-dimensional torus. J. Differ. Equations 172(2), 359–375 (2001). doi:10.1006/jdeq.2000.3863 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zajączkowski, W.: On global regular solutions to the Navier–Stokes equations in cylindrical domains. Topol. Methods Nonlinear Anal. 37(1), 55–85 (2011)MathSciNetMATHGoogle Scholar
  18. 18.
    Nowakowski, B.: Global existence of strong solutions to micropolar equations in cylindrical domains. Math. Methods Appl. Sci. 38(2), 311–329 (2015). doi:10.1002/mma.3070 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Solonnikov, V.: Overdetermined elliptic boundary-value problems. J. Math. Sci. 1(4), 477–512 (1973) [translation from: Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 153—231 (Russian)]Google Scholar
  20. 20.
    Nowakowski, B., Ströhmer, G., Zajączkowski, W.M.: Large time existence of special strong solutions to MHD equations in cylindrical domains. J. Math. Fluid Mech. (2015) (submitted) Google Scholar
  21. 21.
    Cholewa, J., Dłotko, T.: Global Attractors in Abstract Parabolic Problems. Cambridge University Press, Cambridge (2000)CrossRefMATHGoogle Scholar
  22. 22.
    Zajączkowski, W.: Global special regular solutions to the Navier–Stokes equations in a cylindrical domain without the axis of symmetry. Topol. Methods Nonlinear Anal. 24(1), 69–105 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ukhovskii, M. R., Iudovich, V. I.: Axially symmetric flows of idealand viscous fluids filling the whole space. J. Appl. Math. Mech. 32, 52–61 (1968) [translation from: Prikl. Mat. Meh. 32 (1968), 59–69 (Russian)]Google Scholar
  24. 24.
    Nowakowski, B.: Large time existence of strong solutions to micropolar equations in cylindrical domains. Nonlinear Anal. Real World Appl. 14(1), 635–660 (2013). doi:10.1016/j.nonrwa.2012.07.023 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Bernard Nowakowski
    • 1
  • Wojciech M. Zajączkowski
    • 1
    • 2
  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  2. 2.Cybernetics Faculty, Institute of Mathematics and CryptologyMilitary University of TechnologyWarsawPoland

Personalised recommendations