Preprint A492/2006
Sharp bilinear estimates and well-posedness for the 1-D Schrödinger-Debye equation
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, the system is shown to be globally well-posed in $H^s\times H^s, -1/8< s< 0$. This is quite surprising in view of Bidegaray's theorem: in $H^s\times H^s, s>5/2$, there are one-parameter families of solutions of the SD system converging to certain solutions of the cubic NLS equation. In fact, since the cubic NLS is known to be ill-posed below $L^2$, the results of Bidegaray says that the existence of global solutions of SD system in $H^s\times H^s$ for negative Sobolev index $s$ is unexpected. The proof of our global result uses the \textbf{I}-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.