Interactions of Fast and Slow Waves in Problems with Two Time Scales

This is the authors' abstract. We don't add key points for this paper.

We consider certain symmetric hyperbolic systems of nonlinear partial differential equations whose solutions vary on two time scales, a slow scale t and a fast scale t / ε. We show that if the initial data are not prepared correctly for the suppression of the fast scale motion, but contain errors of amplitude O(ε ^μ ), so that only μ time derivatives of the solution are bounded at $t = 0$, then fast waves of amplitude O(ε ^μ ) will be present in the solution, but the error introduced in the slow scale motion by nonlinear interactions of these waves will be of amplitude only O(ε 2μ ) + O(ε μ + 1 ). This holds for any μ > 0, and therefore extends the earlier results of Kreiss. In consequence, the effects of the fast waves can be controlled more easily than was previously thought, and their neglect in some physical situations is partially justified.