Convergence Rate Estimates for the Low Mach and Alfv\'en Number Three-Scale Singular Limit of Compressible Ideal Magnetohydrodynamics

TL;DR

This paper establishes convergence rate estimates for the three-scale singular limit of compressible ideal magnetohydrodynamics equations, considering different rates at which Mach and Alfvén numbers tend to zero, revealing complex dependencies and sharp rates.

Contribution

It provides the first detailed analysis of convergence rates in the three-scale singular limit, including sharp estimates and the effect of power-law relations between small parameters.

Findings

Convergence rates depend on the component and the norm used.

Rates are positive powers of the Mach number, varying with the parameters.

For certain components, the rate involves the ratio of the two small parameters, and is proven sharp.

Abstract

Convergence rate estimates are obtained for singular limits of the compressible ideal magnetohydrodynamics equations, in which the Mach and Alfv\'en 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.