## 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.