Volume 55, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Page(s)||S733 - S759|
|Published online||26 February 2021|
Convergence rate estimates for the low Mach and Alfvén number three-scale singular limit of compressible ideal magnetohydrodynamics
Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK
2 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China
3 School of Mathematical Sciences, Tel-Aviv University, Tel Aviv 69978, Israel
* Corresponding author: firstname.lastname@example.org
Accepted: 23 July 2020
Convergence rate estimates are obtained for singular limits of the compressible ideal magnetohydrodynamics equations, in which the Mach and Alfvén numbers tend to zero at different rates. The proofs use a detailed analysis of exact and approximate fast, intermediate, and slow modes together with improved estimates for the solutions and their time derivatives, and the time-integration method. When the small parameters are related by a power law the convergence rates are positive powers of the Mach number, with the power varying depending on the component and the norm. Exceptionally, the convergence rate for two components involve the ratio of the two parameters, and that rate is proven to be sharp via corrector terms. Moreover, the convergence rates for the case of a power-law relation between the small parameters tend to the two-scale convergence rate as the power tends to one. These results demonstrate that the issue of convergence rates for three-scale singular limits, which was not addressed in the authors’ previous paper, is much more complicated than for the classical two-scale singular limits.
Mathematics Subject Classification: 35B25 / 35L45 / 35Q35 / 76W05
Key words: Rate of convergence / singular limit / magnetohydrodynamics
© The authors. Published by EDP Sciences, SMAI 2021
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