Abstract
In this paper, we consider the 3D density-dependent magnetohydrodynamic equations with vacuum in the whole space \(\mathbb {R}^{3}\), and provide a regularity criterion involving the velocity field in BMO space norm. This work generalizes the regularity criterion of the constant density MHD equations to the density-dependent one.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, Q., Tan, Z., Wang, Y.: Strong solutions to the incompressible magnetohydrodynamic equations. Math. Methods Appl. Sci. 34, 94–107 (2011)
Coifman, R., Lions, P.L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)
Fan, J., Li, F., Nakamura, G., Tan, Z.: Regularity criteria for the three-dimensional magnetohydrodynamic equations. J. Differ. Equ. 256, 2858–2875 (2014)
Fan, J., Zhou, Y.: Uniform local well-posedness for the density-dependent magnetohydrodynamic equations. Appl. Math. Lett. 24, 1945–1949 (2011)
Fan, J., Samet, B., Zhou, Y.: Global strong solutions of the density dependent incompressible MHD system with zero resistivity in a bounded domain. Math. Model. Anal. 24, 95–104 (2019)
Fan, J., Samet, B., Zhou, Y.: A regularity criterion for a density-dependent incompressible liquid crystals model with vacuum. Hiroshima Math. J. 49, 129–138 (2019)
Gala, S.: A note on Div-Curl lemma. Serdica Math. J. 33, 339–350 (2007)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Springer Tracts Natur. Philos., vol. 38. Springer-Verlag, NewYork (1994)
Gerbeau, J.F., Le Bris, C., Claude, L.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Numer. Math. Sci. Comput. Oxford University Press, Oxford (2006)
Landau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media, 2nd edn. Pergamon, NewYork (1984)
Triebel, H.: Theory of Function Spaces, Monograph in Mathematics. Birkhauser Verlag, Basel (1983)
Wu, H.: Strong solution to the incompressible MHD equations with vacuum. Comput. Math. Appl. 61, 2742–2753 (2011)
Zhou, Y., Fan, J.: A regularity criterion for the density-dependent magnetohydrodynamic equations. Math. Meth. Appl. Sci. 33, 1350–1355 (2010)
Zhou, Y., Gala, S.: Logarithmically improved regularity criteria for the Navier–Stokes equations in multiplier spaces. J. Math. Anal. Appl. 356, 498–501 (2009)
Zhou, Y., Gala, S.: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys. 61, 193–199 (2010)
Acknowledgements
This work was done while the second author was visiting the Catania University in Italy. He would like to thank the hospitality and support of the University, where this work was completed. This research is partially supported by Piano della Ricerca 2016–2018-Linea di intervento 2: “Metodi variazionali ed equazioni differenziali”. The third author wish to thank the support of “RUDN University Program 5–100”. The authors would like to thank the anonymous referee for his/her helpful suggestions and comments which led to improvement of the presentation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Alghamdi, A.M., Gala, S., Ragusa, M.A. et al. A Regularity Criterion for the 3D Density-Dependent MHD Equations. Bull Braz Math Soc, New Series 52, 241–251 (2021). https://doi.org/10.1007/s00574-020-00199-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-020-00199-5