Well-posedness and scattering for the mass-energy NLS on ${\mathbf{R}}^n\times {\mathcal{M}}^k$

Abstract

We study the nonlinear Schrödinger equation posed on product spaces ${\mathbf{R}}^n\times {\mathcal{M}}^k$, for $n\ge 1$ and $k\ge 1$, with ${\mathcal{M}}^k$ any $k$-dimensional compact Riemaniann manifold. The main results concern global well-posedness and scattering for small data solutions in non-isotropic Sobolev fractional spaces. In the particular case of $k=2$, $H^1$-scattering is also obtained.