Dispersive Evolution Equations (and why Newton is still relevant)

Abstract: "Dispersive equations" form a fourth category of famous partial differential equations of mathematical physics (the first three are the elliptic, hyperbolic and parabolic equations). They include the Schrodinger equation, the Korteweg-deVries equation (and its linear version, the Airy equation). Recently, some higher-order equations have been considered in this category, modelling various wave phenomena. The talk (intended for general audience) will describe two major methods in the study of global spacetime properties of solutions to dispersive equations:

1) The "Limiting Absorption Principle" (LAP) for partial differential operators states that the resolvents of certain classes of partial differential operators (including all elliptic operators but also important higher-order dispersive operators) can be extended continuously to the (continuous) spectrum, in suitable operator topologies. We illustrate this classical theory (initially due to S. Agmon) by deriving (in a very elementary way) various global space-time estimates, for the Schrodinger equation as well as the wave equation. The connection to "trace theorems" on curved manifolds is the key to such estimates, and also the key for the second method:

2) "Oscillatory Integrals and Strichartz Estimates", which leads to global L^p (spacetime) estimates. One aspect is the study of polynomial phase functions (in Fourier integrals), with certain classifications going back to Newton.

Refreshments will be served at 3:45 p.m. (Math Lounge, MATX 1115).