KdV and Almost Conservation Laws

Details KdV and Almost Conservation Laws KdV and Almost Conservation Laws

2002-03-31

arxiv.org/abs/math/0204014v1

Mathematics

Analysis of PDEs

15 pages. This paper will appear in the AMS Proceedings of the Conference on Harmonic Analysis held at Mt. Holyoke College, Ju

Scientific paper

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.

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