## Differential and Integral Equations

- Differential Integral Equations
- Volume 22, Number 3/4 (2009), 357-391.

### Sharp bilinear estimates and well-posedness for the 1-D Schrödinger-Debye system

Adán J. Corcho and Carlos Matheus

#### Abstract

We establish local and global well posedness for the initial-value problem associated to the one-dimensional Schröodinger-Debye (SD) system for data in 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 ($H^s\times H^s,\; s > 5/2$), Bidégaray [4] proved that there are one-parameter families of solutions of the SD system converging to certain solutions of the cubic nonlinear Schröodinger equation (NLS). Our results below $L^2\times L^2$ say that the SD system is not a good approach to 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.

#### Article information

**Source**

Differential Integral Equations, Volume 22, Number 3/4 (2009), 357-391.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019779

**Mathematical Reviews number (MathSciNet)**

MR2492826

**Zentralblatt MATH identifier**

1240.35511

**Subjects**

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35Q60: PDEs in connection with optics and electromagnetic theory

#### Citation

Corcho, Adán J.; Matheus, Carlos. Sharp bilinear estimates and well-posedness for the 1-D Schrödinger-Debye system. Differential Integral Equations 22 (2009), no. 3/4, 357--391. https://projecteuclid.org/euclid.die/1356019779