Smoothing estimates for the periodic KdV equation

In this thesis we aim at understanding an article of M. B. Erdoğan 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 effect of a dispersive PDE (partial differential equation) on its solutions has been studied extensively, but the global smoothing effect in the periodic case was inaccessible prior to the paper of Erdoğan and Tzirakis. The most important tools they have used are the Bourgain spaces, introduced by J. Bourgain, that are defined specifically for each equation and reflect the dispersion relation of the equation, and also the differentiation by parts method was employed.