Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3
Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3
复制标题
DOI：
10.1002/cpa.20029
复制
发表时间：
2003-01
影响因子：
3
通讯作者：
J. Colliander;M. Keel;G. Staffilani;H. Takaoka;H. Takaoka;T. Tao
中科院分区：
文献类型：
--
作者：
J. Colliander;M. Keel;G. Staffilani;H. Takaoka;H. Takaoka;T. Tao
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.
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.