KdV and Almost Conservation Laws

Abstract

This short survey paper is concerned with a new method to prove global well-posedness results for dispersive equations below energy spaces, namely $H^{1}$ for the Schr\"odinger equation and $L^{2}$ for the KdV equation. The main ingredient of this method is the definition of a family of what we call almost conservation laws. In particular we analyze the Korteweg-de Vries initial value problem and we illustrate in general terms how the ``algorithm'' that we use to formally generate almost conservation laws can be used to recover the infinitely many conserved integrals that make the KdV an integrable system.