Abstract
In this paper, we study global well-posedness of the three-dimensional MHD-Boussinesq equations. The global existence of axisymmetric strong solutions to the MHD-Boussinesq equations in the presence of magnetic diffusion is shown by providing some smallness conditions only on the swirl component of velocity. As a by-product, long-time asymptotic behaviors are also presented.
Similar content being viewed by others
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Abidi, H., Hmidi, T., Keraani, S.: On the global regularity of axisymmetric Navier–Stokes–Boussinesq system. Discrete Contin. Dyn. Syst. 29(3), 737–756 (2011)
Bian, D., Gui, G.: On 2-D Boussinesq equations for MHD convection with stratification effects. J. Differ. Equ. 261(3), 1669–1711 (2016)
Bian, D., Liu, J.: Initial-boundary value problem to 2D Boussinesq equations for MHD convection with stratification effects. J. Differ. Equ. 263(12), 8074–8101 (2017)
Bian, D., Pu, X.: Global smooth axisymmetric solutions of the Boussinesq equations for magnetohydrodynamics convection. J. Math. Fluid Mech. 22, Article No: 12 (2020)
Brandolese, L., Schonbek, M.: Large time decay and growth for solutions of a viscous Boussinesq system. Trans. Am. Math. Soc. 364(10), 5057–5090 (2012)
Cai, Y., Lei, Z.: Global well-posedness of the incompressible magnetohydrodynamics. Arch. Ration. Mech. Anal. 228(3), 969–993 (2018)
Chen, H., Fang, D., Zhang, T.: Regularity of 3D axisymmetric Navier–Stokes equations. Discrete Contin. Dyn. Syst. 37(4), 1923–1939 (2017a)
Chen, H., Fang, D., Zhang, T.: Global axisymmetric solutions of three dimensional inhomogeneous incompressible Navier–Stokes system with nonzero swirl. Arch. Ration. Mech. Anal. 223(2), 817–843 (2017b)
Duvaut, G., Lions, J.: Inéquations en thermoé lasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279 (1972)
Fang, D., Le, W., Zhang, T.: Global solutions of 3D axisymmetric Boussinesq equations with nonzero swirl. Nonlinear Anal. 166, 48–86 (2018)
Hmidi, T.: On a maximum principle and its application to the logarithmically critical Boussinesq system. Anal. PDE 4(2), 247–284 (2011)
Hmidi, T., Rousset, F.: Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(5), 1227–1246 (2010)
Hmidi, T., Keraani, S., Rousset, F.: Global well-posedness for a Boussinesq–Navier–Stokes system with critical dissipation. J. Differ. Equ. 249(9), 2147–2174 (2010)
Hmidi, T., Keraani, S., Rousset, F.: Global well-posedness for Euler–Boussinesq system with critical dissipation. Commun. Partial Differ. Equ. 36(3), 420–445 (2011)
Hou, T., Li, C.: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12(1), 1–12 (2005)
Jiu, Q., Liu, J.: Global regularity for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. Discrete Contin. Dyn. Syst. 35(1), 301–322 (2015)
Jiu, Q., Yu, H., Zheng, X.: Global well-posedness for axisymmetric MHD system with only vertical viscosity. J. Differ. Equ. 263(5), 2954–2990 (2017)
Larios, A., Pei, Y.: On the local well-posedness and a Prodi–Serrin-type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion. J. Differ. Equ. 263, 1419–1450 (2017)
Larios, A., Lunasin, E., Titi, E.: Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion. J. Differ. Equ. 255, 2636–2654 (2013)
Lei, Z.: On axially symmetric incompressible magnetohydrodynamics in three dimensions. J. Differ. Equ. 259, 3202–3215 (2015)
Lei, Z., Zhang, Q.: Criticality of the axially symmetric Navier–Stokes equations. Pac. J. Math. 289, 169–187 (2017)
Leonardi, S., Málek, J., Necas, J., Pokorny, M.: On axially symmetric flows in \(\mathbb{R} ^3\). Z. Anal. Anwend. 18, 639–649 (1999)
Li, Z.: Critical conditions on \(\omega ^{\theta }\) imply the regularity of axially symmetric MHD-Boussinesq. J. Math. Anal. Appl. 505, 125451 (2022)
Liu, Y.: Global well-posedness of 3D axisymmetric MHD system with pure swirl magnetic field. Acta Appl. Math. 155, 21–39 (2018)
Liu, Z., Han, P.: Decay for turbulent solutions of the magneto-hydrodynamic equations in an exterior domain. J. Math. Phys. 61, 091506, 20 pp (2020)
Liu, H., Bian, D., Pu, X.: Global well-posedness of the 3D Boussinesq-MHD system without heat diffusion. Z. Angew. Math. Phys. 70, Article No: 81 (2019)
Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow, Cambridge texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
Neustupa, J., Pokorny, M.: Axisymmetric flow of Navier–Stokes fluid in the whole space with non-zero angular velocity component, Proceedings of Partial Differential Equations and Applications (Olomouc, 1999). Math. Bohem. 126(2), 469–481 (2001)
Pan, X.: Global regularity of solutions for the 3D non-resistive and non-diffusive MHD-Boussinesq system with axisymmetric data, ar**v:1911.01550v2 (2020)
Pratt, J., Busse, A., Mueller, W.: Fluctuation dynamo amplified by intermittent shear bursts in convectively driven magnetohydrodynamic turbulence. Astron. Astrophys. 557(2), 906–908 (2013)
Schonbek, M.: \(L^2\) decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 88(3), 209–222 (1985)
Schrinner, M., Rädler, K., Schmitt, D., Rheinhardt, M., Christensen, U.: Mean-field view on rotating magnetoconvection and a geodynamo model. Astron. Nachr. AN. 326(3–4), 245–249 (2005)
Schrinner, M., Rädler, K., Schmitt, D., Rheinhardt, M., Christensen, U.: Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo. Fluid Dyn. 101, 81–116 (2007)
Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)
Wahl, W.: The equation \(u^{\prime }+A(t)u=f\) in a Hilbert space and \(L^{p}\)-estimates for parabolic equations. J. Lond. Math. Soc. 25(2), 483–497 (1982)
Wang, P., Guo, Z.: Global well-posedness for axisymmetric MHD equations with vertical dissipation and vertical magnetic diffusion. Nonlinearity 35, 2147–2174 (2022)
Zheng, S.: Nonlinear Evolution Equations. Monographs and Surveys in Pure and Applied Mathematics. Chapman and Hall/CRC, New York (2004)
Acknowledgements
The authors thank the reviewers for their helpful comments on the initial manuscript, which improved the paper significantly. Z. Guo was partially supported by Natural Science Foundation of Jiangsu Province (BK20201478) and Qing Lan Project of Jiangsu Universities. Z. Skalak was supported by the European Regional Development Fund, Project No. CZ.02.1.01/0.0/0.0/16_019/0000778.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Communicated by Anthony Bloch.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Guo, Z., Zhang, Z. & Skalák, Z. Global Well-Posedness and Asymptotic Behavior of the 3D MHD-Boussinesq Equations. J Nonlinear Sci 33, 61 (2023). https://doi.org/10.1007/s00332-023-09920-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00332-023-09920-2