Scattering for defocusing cubic NLS under locally damped strong trapping

TL;DR

This paper investigates how localized damping in a trapping region affects the scattering behavior of the defocusing cubic 3D nonlinear Schrödinger equation, establishing global existence and scattering results despite trapping complexities.

Contribution

It introduces a novel approach of using localized damping to mitigate trapping effects in the cubic NLS, extending scattering results to more general trapping scenarios.

Findings

Global existence and scattering for initial data in H^s, 0 ≤ s < 1.

Damping can counteract trapping effects in Schrödinger equations.

Loss of smoothing due to trapping limits regularity to below H^1.

Abstract

We are interested in the scattering problem for the cubic 3D nonlinear defocusing Schr\"odinger equation with variable coefficients. Previous scattering results for such problems address only the cases with constant coefficients or assume strong variants of the non-trapping condition, stating that all the trajectories of the Hamiltonian flow associated with the operator are escaping to infinity. In contrast, we consider the most general setting, where strong trapping, such as stable closed geodesics, may occur, but we introduce a compactly supported damping term localized in the trapping region, to explore how damping can mitigate the effects of trapping. In addition to the challenges posed by the trapped trajectories, notably the loss of smoothing and of scale-invariant Strichartz estimates, difficulties arise from the damping itself, particularly since the energy is not, a priori, bounded. For $H^{1+\epsilon}$ initial data -- chosen because the local-in-time theory is a priori no better than for 3D unbounded manifolds, where local well-posedness of strong $H^1$ solutions is unavailable -- we establish global existence and scattering in $H^{s}$ for any $0 \leq s <1$ in positive times, the inability to reach $H^1$ being related to the loss of smoothing due to trapping.