Sharp bilinear estimates and well-posedness for the 1-D Schr\"odinger-Debye system

Abstract

We establish local and global well-posedness for the initial value problem associated to the one-dimensional Schrodinger-Debye (SD) system for data in the Sobolev spaces with low regularity. To obtain local results we prove two new sharp bilinear estimates for the coupling terms of this system in the continuous and periodic cases. Concerning global results, in the continuous case, the system is shown to be globally well-posed in $H^s\times H^s, -3/14< s< 0$. For initial data in Sobolev spaces with high regularity ($H^s\times H^s, s>5/2$), Bid\'egaray \cite{Bidegaray} proved that there are one-parameter families of solutions of the SD system converging to certain solutions of the cubic \emph{nonlinear Schrodinger equation} (NLS). Our results bellow $L^2\times L^2$ say that the SD system is not a good approach of the cubic NLS in Sobolev spaces with low regularity, since the cubic NLS is known to be ill-posed below $L^2$. The proof of our global result uses the \textbf{I}-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.