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March/April 2009 Sharp bilinear estimates and well-posedness for the 1-D Schrödinger-Debye system
Adán J. Corcho, Carlos Matheus
Differential Integral Equations 22(3/4): 357-391 (March/April 2009).

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.

Citation

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Adán J. Corcho. Carlos Matheus. "Sharp bilinear estimates and well-posedness for the 1-D Schrödinger-Debye system." Differential Integral Equations 22 (3/4) 357 - 391, March/April 2009.

Information

Published: March/April 2009
First available in Project Euclid: 20 December 2012

zbMATH: 1240.35511
MathSciNet: MR2492826

Subjects:
Primary: 35Q55, 35Q60

Rights: Copyright © 2009 Khayyam Publishing, Inc.

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Vol.22 • No. 3/4 • March/April 2009
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