Global existence and scattering for rough solutions of a nonlinear Schroedinger equation on R^3

Details Global existence and scattering for rough solutions of a nonlinear Schroedinger equation on R^3 Global existence and scattering for rough solutions of a nonlinear Schroedinger equation on R^3

2003-01-23

arxiv.org/abs/math/0301260v2

Mathematics

Analysis of PDEs

Final version, to appear in Communications on Pure and Applied Mathematics: typos fixed, some expository remarks and reference

Scientific paper

We prove global existence and scattering for the defocusing, cubic nonlinear Schr\"odinger equation in $H^s(\rr^3)$ for $s > {4/5}$. The main new estimate in the argument is a Morawetz-type inequality for the solution $\phi$. This estimate bounds $\|\phi(x,t)\|_{L^4_{x,t}(\rr^3 \times \rr)}$, whereas the well-known Morawetz-type estimate of Lin-Strauss controls $\int_0^{\infty}\int_{\rr^3}\frac{(\phi(x,t))^4}{|x|} dx dt

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