Initial value problems for nonlinear dispersive equations at critical regularity

Abstract

Global regularity results for nonlinear dispersive equations hinge on a thorough understanding of the Cauchy problem in spaces of functions of low regularity. This is most challenging in scale invariant regimes as solutions interact strongly on multiple frequency-scales. Here, some recent progress on the critical well-posedness theory will be reviewed, with a focus on nonlinear Schrödinger and Dirac equations.