Kinetic decomposition for periodic homogenization problems

We develop an analytical tool which is adept for detecting shapes of oscillatory functions, is useful in decomposing homogenization problems into limit-problems for kinetic equations, and provides an efficient framework for the validation of multiscale asymptotic expansions. The main new result concerns a linear hyperbolic homogenizat ion problem which we transform to a hyperbolic limit problem for a kinetic equation. We establish conditions determining an effective equation and counterexamples for the case that such conditions fail. Second, we revisit some already known problems with our approach, applying in particular the kinetic decomposition to the problem of enhanced diffusion; it then leads to a diffusive limit problem for a kinetic equation that in turn yields the known effective equation of enhanced diffusion.