Propagation of singularities for the wave equation on manifolds with corners

TL;DR

This paper studies how singularities in solutions to the wave equation propagate on manifolds with corners, extending previous results to smooth and Sobolev singularities using microlocal analysis techniques.

Contribution

It provides a new proof for the propagation of singularities at hyperbolic points on manifolds with corners, generalizing Lebeau's results to smooth settings.

Findings

Wave front set propagates along maximally extended generalized broken bicharacteristics.

New microlocal positive commutator estimates are developed.

Results apply to manifolds with corners, including smooth boundaries.

Abstract

In this paper we describe the propagation of smooth (C^\infty) and Sobolev singularities for the wave equation on smooth manifolds with corners M equipped with a Riemannian metric g. That is, for X=MxR, P=D_t^2-\Delta_M, and u locally in H^1 solving Pu=0 with homogeneous Dirichlet or Neumann boundary conditions, we show that the wave front set of u is a union of maximally extended generalized broken bicharacteristics. This result is a smooth counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary. Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners).