Smoothing estimates for the periodic KdV equation

Abstract

In this thesis we aim at understanding an article of M. B. Erdo gan and N. Tzirakis on the famous KdV (Korteweg-de Vries) equation, entitled \Global smoothing for the periodic KdV", which appeared in International Mathematics Research Notices in 2012. The article establishes smoothing estimates in the case of the periodic KdV equation. Roughly speaking, smoothing estimates indicate that the solutions to the equation turn out to be smoother than the initial data, and constitute a subject closely related to well-posedness problems. This smoothing e ect of a dispersive partial di erential equation (PDE) on its solutions has been studied extensively, but the global smoothing e ect in the periodic case was inaccessible prior to the paper of Erdo gan and Tzirakis. The most important tools they have used are the so-called Bourgain spaces, introduced by J. Bourgain, that are de ned speci cally for each equation and re ect the dispersion relation of the equation, and the so-called di erentiation by parts method.