Jacob Perry (UNC-CH), GMA Seminar

Title:  Localized energy for wave equations with degene= rate trapping

Abstract:  Local smoothing estimates for the Schr{\textasciidieresis}odinger equation are well established and show that locally in space and averaged in time, solutions gain one half of a derivative in regularity compared to the initial data.  Analogous estimates for solutions to the wave equation, so-called localized energy estimates, have also been studied, and provide a global integrability estimate (in both time and space).  When considering such estimates for equations on differentiable manifolds, in either case it is known that geodesic trapping requires a loss. For non-degenerate hyperbolic trapping, the loss is logarithmic.  For elliptic trapping, everything is lost except a logarithm.  Recently, Christianson and Wunsch demonstrated an algebraic loss for solutions to the Schro{\textasciidieresis}dinger equation on a surface of revolution with degenerate hyperbolic trapping. In this talk, we will review these prior results and consider the analogue for the wave equation on a warped product manifold with degenerate hyperbolic trapping, attaining an algebraic loss of derivative. We will then use a quasimode construction to show that our estimate is sharp.  This is a joint work with Robert Booth, Hans Christianson, and Jason Metcalfe