The Korteweg--de Vries equation occurs as a model for unidirectional propagation of small amplitude long waves in numerous physical systems. The aim of this work is to propose a well-posed mixed initial--boundary value problem when the spacial domain is of finite extent. More precisely, we establish local existence of solutions for arbitrary initial data in the Sobolev space $H^1$ and global existence for small initial data in this space. In a second step we show global strong regularizing effects.
"An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval." Adv. Differential Equations 6 (12) 1463 - 1492, 2001.