• DOI:10.1002/CPA.20029
• Corpus ID: 16423475

Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3

@article{Colliander2003GlobalEA,
  title={Global existence and scattering for rough solutions of a nonlinear Schr{\"o}dinger equation on ℝ3},
  author={James E. Colliander and Markus Keel and Gigliola Staffilani and Hideo Takaoka and Terence Tao},
  journal={Communications on Pure and Applied Mathematics},
  year={2003},
  volume={57}
}

• J. Colliander, M. Keel, +2 authors T. Tao
• Published 23 January 2003
• Mathematics
• Communications on Pure and Applied Mathematics

We prove global existence and scattering for the defocusing, cubic, nonlinear Schrödinger equation in $H^{\scriptscriptstyle S}$ (ℝ3) for s > ${4 \over 5}$. The main new estimate in the argument is a Morawetz‐type inequality for the solution ϕ. This estimate bounds \documentclass{article}\usepackage{amsfonts}\pagestyle{empty}\begin{document}\begin{displaymath}{\|\phi \left( x, t \right) \|}_{\textstyle{L^{4}}_{\scriptstyle{x,t}} \bigl( \mathbb{R} \times \mathbb{R} \bigr)} \, ,\end{displaymath}\end{document} whereas the well‐known Morawetz‐type estimate of Lin‐Strauss controls \documentclass{article}\usepackage[intlimits]{amsmath}\usepackage{amsfonts}\pagestyle{empty}\begin{document}\begin{displaymath}\int_{0}^{\infty} \int_{\textstyle{\mathbb{R}^3}} {(\phi\,(x,\,t))^{4} \over |\,x\,|} \>dx \>dt \,. \end{displaymath}\end{document} © 2004 Wiley Periodicals, Inc.