Pré-Publication, Document De Travail Année : 2025

Scattering for defocusing cubic NLS under locally damped strong trapping

Résumé

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 $H^{\frac 12}$ in positive times, the loss of regularity in scattering being related to the loss of smoothing due to trapping.
Fichier principal
Vignette du fichier
2502.06306v1.pdf (637) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04939708 , version 1 (11-02-2025)

Identifiants

Citer

David Lafontaine, Boris Shakarov. Scattering for defocusing cubic NLS under locally damped strong trapping. 2025. ⟨hal-04939708⟩
0 Consultations
0 Téléchargements

Altmetric

Partager

More