Parametrized invariant manifolds: a recipe for multiscale modeling?

Given a fine-scale physical theory characterized by an evolutionary system of equations and a set of quantities, defined from the variables of the fine theory, that serve as a coarse representation of the fine scale phenomena, a method for developing an evolutionary set of equations for the coarse quantities is presented. This set may be considered as governing equations for a coarse model corresponding to the fine one since its solutions can be formally shown to generate exact/approximate solutions to the fine theory in a well-defined sense. The method for generating the coarse representation is illustrated on model problems.