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

Abstract

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