Global existence for defocusing cubic NLS and Gross-Pitaevskii equations in three dimensional exterior domains

We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schrodinger and Gross-Pitaevskii equations on the exterior of a nontrapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate [R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds. and nonlinear Schrodinger equation on domains, arxiv:math.AP/0512639, Bull. Soc. Math. France, submitted for publication] with a smoothing effect on exterior domains [N. Burq, P. Gerard, N. Tzvetkov, On nonlinear Schrodinger equations in exterior domains, Ann. I.H.P. (2004) 295-3181]. (c) 2007 Elsevier Masson SAS. All rights reserved.