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

TL;DR

This paper investigates the global behavior of solutions to the nonlinear Schrödinger equation on product spaces combining Euclidean and compact manifold components, establishing well-posedness and scattering results for small initial data.

Contribution

It extends the analysis of NLS to product spaces with compact manifolds, providing new well-posedness and scattering results, including $H^1$-scattering for the case $k=2$.

Findings

Global well-posedness for small data in non-isotropic Sobolev spaces.

Scattering results for small solutions in the considered setting.

$H^1$-scattering established when $k=2$.

Abstract

We study the nonlinear Schr\"odinger equation posed on product spaces $\mathbf R^n\times \mathcal M^k$, for $n\geq 1$ and $k\geq1$, 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.