21st Century COE Program : Mathematics of Nonlinear Structure via Singularity

Fabrice Planchon : Nonlinear Dispersive Equations, Expository Lectures

Schrödinger equations with variable coefficients

Abstract

There are essentially two kinds of estimates for Schrödinger equations: so-called local smoothing estimates, and dispersive estimates (dispersion, Strichartz...). In the case of variable coefficients, and unlike for the wave equation, at lot less is known about dispersive estimates. We will first consider a 1D model and obtain results for very rough coefficients (allowing jumps). Then we will apply these results to deal with Benjamin-Ono equations. In higher dimensions, we will review results with rough potentials and if time permits, variable metrics.

21st Century COE Program : Mathematics of Nonlinear Structure via Singularity