Generalised Fourier integral operator methods for hyperbolic equations with singularities

Abstract

This article addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalised functions. We employ the recently developed theory of generalised Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalised solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalised Fourier integral operators are obtained.