Abstract
This study investigates the Cauchy problem of an incompressible magnetohydrodynamic system in the rotational framework. Under the assumption that the rotation speed is sufficiently large, the Cauchy problem is shown to be globally well-posed in \(H^{s}({{\mathbb {R}}}^3)\times (L^2\cap L^q) ({{\mathbb {R}}}^3)\) for \(\frac{1}{2}<s < \frac{3}{4}\) and \(3<q < \min \{\frac{6}{3-2s},\,\frac{27}{6 + 4s}\}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Babin, A. Mahalov, B. Nicolaenko, Regularity and integrability of 3D Euler and Navier–Stokes equations for rotating fluids, Asymptotic Analysis 15 (1997) 103–150.
A. Babin, A. Mahalov, B. Nicolaenko, Global regularity of 3D rotating Navier–Stokes equations for resonant domains, Indiana University Mathematics Journal 48 (1999) 1133–1176, https://doi.org/10.1512/iumj.1999.48.1856.
A. Babin, A. Mahalov, B. Nicolaenko, 3D Navier–Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana University Mathematics Journal 50 (2001) 1–35, https://doi.org/10.1512/iumj.2001.50.2155.
S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, Oxford University Press, Oxford, 1961.
J.Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Anisotropy and dispersion in rotating fluids, Studies in Applied Mathematics, vol. 31, North-Holland, Amsterdam, 2002, pp. 171–192.
J.Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Mathematical geophysics: An introduction to rotating fluids and the Navier–Stokes equations vol. 32, Oxford University Press, Oxford, 2006.
G. Duvaut, J.L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Archive for Rational Mechanics and Analysis 46 (1972) 241–279, https://doi.org/10.1007/bf00250512.
H. Fujita, T. Kato, On the Navier–Stokes initial value problem. \({\rm I}\), Archive for Rational Mechanics and Analysis 16 (1964) 269–315, https://doi.org/10.1007/bf00276188.
Y. Giga, K. Inui, A. Mahalov, S. Matsui, Uniform local solvability for the Navier–Stokes equations with the Coriolis force, Methods and Applications of Analysis 12 (2005) 381–394, https://doi.org/10.4310/maa.2005.v12.n4.a2.
Y. Giga, K. Inui, A. Mahalov, S. Matsui, Navier–Stokes equations in a rotating frame in \({{\mathbb{R}}}^3\) with initial data nondecreasing at infinity, Hokkaido Mathematical Journal 35 (2006) 321–364, https://doi.org/10.14492/hokmj/1285766360.
Y. Giga, K. Inui, A. Mahalov, J. Saal, Uniform global solvability of the rotating Navier–Stokes equations for nondecaying initial data, Indiana University Mathematics Journal 57 (2008) 2775–2791, https://doi.org/10.1512/iumj.2008.57.3795.
C. He, Y. Wang, On the regularity criteria for weak solutions to the magnetohydrodynamic equations, Journal of Differential Equations 238 (2007) 1–17, https://doi.org/10.1016/j.jde.2007.03.023.
C. He, Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, Journal of Differential Equations 213 (2005) 235–254, https://doi.org/10.1016/j.jde.2004.07.002.
M. Hieber, Y. Shibata, The Fujita–Kato approach to the Navier–Stokes equations in the rotational framework, Mathematische Zeitschrift 265 (2010) 481–491, https://doi.org/10.1007/s00209-009-0525-8.
T. Iwabuchi, R. Takada, Time periodic solutions to the Navier–Stokes equations in the rotational framework, Journal of Evolution Equations 12 (2012) 985–1000, https://doi.org/10.1007/s00028-012-0165-z.
T. Iwabuchi, R. Takada, Global solutions for the Navier–Stokes equations in the rotational framework, Mathematische Annalen 357 (2013) 727–741, https://doi.org/10.1007/s00208-013-0923-4.
T. Iwabuchi, R. Takada, Dispersive effect of the Coriolis force and the local well-posedness for the Navier–Stokes equations in the rotational framework, Funkcialaj Ekvacioj 58 (2015) 365–385, https://doi.org/10.1619/fesi.58.365.
T. Kato, Strong \(L^{p}\)-solutions of the Navier–Stokes equation in \(R^{m}\), with applications to weak solutions, Mathematische Zeitschrift 187 (1984) 471–480, https://doi.org/10.1007/bf01174182.
H. Kozono, T. Ogawa, Y. Taniuchi, Navier–Stokes equations in the Besov space near \(L^{\infty }\) and BMO, Kyushu Journal of Mathematics 57 (2003) 303–324, https://doi.org/10.2206/kyushujm.57.303.
B. Lehnert, Magnetohydrodynamic waves under the action of the Coriolis force, Astrophysical Journal 119 (1954) 647–654, https://doi.org/10.1086/145869.
A. Mahalov, B. Nicolaenko, T. Shilkin, \(L_{3, \infty }\)-solutions to the MHD equations, Journal of Mathematical Sciences 143 (2007) 2911–2923, https://doi.org/10.1007/s10958-007-0175-5.
V.-S. Ngo, A global existence result for the anisotropic rotating magnetohydrodynamical systems, Acta Applicandae Mathematicae 150 (2017) 1–42, https://doi.org/10.1007/s10440-016-0092-z.
M. Sermange, R. Temam, Some mathematical questions related to the MHD equations, Communications on Pure and Applied Mathematics 36 (1983) 635–664, https://doi.org/10.1002/cpa.3160360506.
J. Wu, Regularity results for weak solutions of the 3D MHD equations Discrete and Continuous Dynamical Systems 10 (2004) 543–556, https://doi.org/10.3934/dcds.2004.10.543.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions. J. Ahn’s work was partially supported by NRF Grant No. 2018R1D1A1B07047465, and J. Kim and J. Lee’s work was partially supported by NRF Grant No. 2016R1A2B3011647.
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.
Rights and permissions
About this article
Cite this article
Ahn, J., Kim, J. & Lee, J. Global solutions to 3D incompressible rotational MHD system. J. Evol. Equ. 21, 235–246 (2021). https://doi.org/10.1007/s00028-020-00576-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-020-00576-z