Hyperbolic systems with discontinuous coefficients: Generalized wavefront sets

Abstract

sWe study linear hyperbolic systems of pseudo-differential equations with nonsmooth, possibly discontinuous symbols and distributional data. This note initiates an approach whereby solutions are constructed in the dual of the Colombeau algebra of generalized functions. We compute the generalized wave front set of the solution to a transport equation with discontinuous propagation speed and delta functions as initial data. The generalized wave front set turns out to have a more refined and informative structure than the wavefront set of the corresponding distributional limit.