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

Abstract

We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schrödinger 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 Schrödinger equation on domains, arxiv: math.AP/0512639, Bull. Soc. Math. France, submitted for publication] with a smoothing effect on exterior domains [N. Burq, P. Gérard, N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. I.H.P. (2004) 295–318].