Propagation of singularities for rough metrics

Smith, Hart

doi:10.2140/apde.2014.7.1137

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Propagation of singularities for rough metrics

Hart F. Smith

Vol. 7 (2014), No. 5, 1137–1178

DOI: 10.2140/apde.2014.7.1137

Abstract

We use a wave packet transform and weighted norm estimates in phase space to establish propagation of singularities for solutions to time-dependent scalar hyperbolic equations that have coefficients of limited regularity. It is assumed that the second order derivatives of the principal coefficients belong to $L_{t}^{1} L_{x}^{\infty}$ , and that $u$ is a solution to the homogeneous equation of global Sobolev regularity $s_{0} = 0$ or 1. It is then proven that the $H^{s_{0} + 1}$ wavefront set of $u$ is a union of maximally extended null bicharacteristic curves.

Keywords

wavefront set, wave equation, propagation of singularities

Mathematical Subject Classification 2010

Primary: 35A21, 35L10

Milestones

Received: 18 February 2014

Accepted: 11 April 2014

Published: 27 September 2014

Authors

Hart F. Smith
Department of Mathematics University of Washington Box 354350 Seattle, WA 98195-4350 United States

Vol. 7, No. 5, 2014