Harmonic Analysis and Differential Equations Student Seminar: Local energy, resolvents, and wave decay in the asymptotically flat setting

Seminar | October 1 | 3:30-5 p.m. | 740 Evans Hall

 Katrina Morgan, MSRI

 Department of Mathematics

Asymptotically flat spacetimes (i.e. Lorentzian manifolds whose metric coefficients tend toward the flat metric as $|x| \to \infty $) arise in General Relativity, which has motivated many mathematical questions about wave behavior on such spacetimes. The dispersive estimate local energy decay has proven to be a powerful tool for studying these questions. It has been used to establish Strichartz estimates (global, mixed norm estimates often used in existence proofs) and pointwise estimates. The estimate holds if the underlying geometry allows waves to spread out enough to get decay of energy within compact sets. Local energy decay is connected to the presence of trapped geodesics and resolvent behavior. This talk will provide a brief overview of local energy estimates and the use of resolvents in studying wave behavior. The application of these tools in establishing the relationship between how quickly the background geometry tends toward flat and the pointwise decay rate of waves will be discussed. We find that a solution u to the wave equation on a spacetime which tends toward flat at a rate of $|x|^{-k}$ satisfies the pointwise bounds $|u|\le C_x t^{-k-2}$. This result extends the work of Tataru 2013 which proved a $t^{-3}$ pointwise decay rate for waves when the background geometry tends toward flat at a rate of $|x|^{-1}$.