Scattering for defocusing cubic NLS under locally damped strong trapping
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916646270009344 |
|---|---|
| author | Lafontaine, David Shakarov, Boris |
| author_facet | Lafontaine, David Shakarov, Boris |
| contents | We are interested in the scattering problem for the cubic 3D nonlinear defocusing Schrödinger 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 $H^{1+ε}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06306 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scattering for defocusing cubic NLS under locally damped strong trapping Lafontaine, David Shakarov, Boris Analysis of PDEs 35Q55, 35B40 We are interested in the scattering problem for the cubic 3D nonlinear defocusing Schrödinger 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 $H^{1+ε}$ 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. |
| title | Scattering for defocusing cubic NLS under locally damped strong trapping |
| topic | Analysis of PDEs 35Q55, 35B40 |
| url | https://arxiv.org/abs/2502.06306 |