Global existence and scattering for rough solutions of a nonlinear Schroedinger equation on R^3
arXiv:math/0301260
Abstract
We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger 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 $Ï$. This estimate bounds $\|Ï(x,t)\|_{L^4_{x,t}(\rr^3 \times \rr)}$, whereas the well-known Morawetz-type estimate of Lin-Strauss controls $\int_0^{\infty}\int_{\rr^3}\frac{(Ï(x,t))^4}{|x|} dx dt
Final version, to appear in Communications on Pure and Applied Mathematics: typos fixed, some expository remarks and references added, referee's suggestions incorporated