Colloquium: Local energy estimates for wave equations with degenerate trapping.

Event Type: 
Jacob Perry, University of North Carolina-Chapel Hill
Event Date: 
Thursday, September 28, 2017 -
3:30pm to 4:30pm
SMLC 356

Event Description: 

Title:  Local energy estimates for wave equations with degenerate trapping.


Local smoothing estimates for the Schrodinger 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 necessitates 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 Schrodinger 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.

Event Contact

Contact Name: Matthew Blair