TY - JOUR

T1 - Convergence rate estimates for the low Mach and Alfvén number three-scale singular limit of compressible ideal magnetohydrodynamics

AU - Cheng, Bin

AU - Ju, Qiangchang

AU - Schochet, Steve

N1 - Publisher Copyright:
© 2021 The authors. Published by EDP Sciences, SMAI.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Magnetohydrodynamics

KW - Rate of convergence

KW - Singular limit

UR - http://www.scopus.com/inward/record.url?scp=85101770826&partnerID=8YFLogxK

U2 - 10.1051/m2an/2020051

DO - 10.1051/m2an/2020051

M3 - מאמר

AN - SCOPUS:85101770826

VL - 55

SP - S733-S759

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

ER -