Moderately fast three-scale singular limits

A uniform existence theorem is proven for quasilinear symmetric hyperbolic systems containing two small parameters tending to zero at different rates for more general initial data than required in a recent paper of Cheng, Ju, and Schochet. An iterated filtering scheme is developed, for which filtered spatially periodic solutions converge to a limit profile as the two parameters tend to zero. Necessary conditions are given for the occurrence of resonance, in which the fast part of the limit influences the slow part. The small Mach and small Alfven number limit of the ideal compressible MHD equations is shown to be nonresonant, and an example where resonance does occur is presented.