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Scattering for defocusing cubic NLS under locally damped strong trapping

David Lafontaine, Boris Shakarov

TL;DR

This work analyzes the cubic defocusing NLS with a variable-coefficient Laplacian Δ_G on ℝ^3 in the presence of a compactly supported damping a that acts where trapping occurs. Despite potentially strong trapping, the authors prove global existence and scattering for initial data u_0 ∈ H^{1+ε} and any 0 ≤ s < 1, under the exterior control condition supp(G−I) ⊂ supp(a). The approach combines Strichartz estimates with a derivative loss due to trapping, a perturbative Morawetz argument to obtain uniform energy bounds and local energy decay, and a bilinear Morawetz analysis to control nonlinear interactions, followed by a cautious decomposition of the solution to connect damped dynamics to free-space scattering. The results show that localized damping can mitigate trapping effects sufficiently to yield global well-posedness and scattering, albeit with a loss of half a derivative relative to the undamped, non-trapped case, and they lay groundwork for exterior control and stabilization questions in dispersive equations with variable geometry.

Abstract

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.

Scattering for defocusing cubic NLS under locally damped strong trapping

TL;DR

This work analyzes the cubic defocusing NLS with a variable-coefficient Laplacian Δ_G on ℝ^3 in the presence of a compactly supported damping a that acts where trapping occurs. Despite potentially strong trapping, the authors prove global existence and scattering for initial data u_0 ∈ H^{1+ε} and any 0 ≤ s < 1, under the exterior control condition supp(G−I) ⊂ supp(a). The approach combines Strichartz estimates with a derivative loss due to trapping, a perturbative Morawetz argument to obtain uniform energy bounds and local energy decay, and a bilinear Morawetz analysis to control nonlinear interactions, followed by a cautious decomposition of the solution to connect damped dynamics to free-space scattering. The results show that localized damping can mitigate trapping effects sufficiently to yield global well-posedness and scattering, albeit with a loss of half a derivative relative to the undamped, non-trapped case, and they lay groundwork for exterior control and stabilization questions in dispersive equations with variable geometry.

Abstract

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 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 solutions is unavailable -- we establish global existence and scattering in for any in positive times, the inability to reach being related to the loss of smoothing due to trapping.

Paper Structure

This paper contains 13 sections, 17 theorems, 178 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strichartz Estimates for the undamped linear Schrödinger flow
  4. Local, and conditionally global, well-posedness theory
  5. Uniform bound on the energy and local energy decay
  6. Mass and energy laws
  7. A perturbative Morawetz identity
  8. Proof of Proposition \ref{['prpUnifEst']}
  9. Local energy decay
  10. Bilinear estimates
  11. End of the proof
  12. The exterior of an obstacle
  13. Mildly trapping case

Key Result

Theorem 1.1

For any $\epsilon>0$, equation (eq:DNLS) is locally well-posed in $H^{1+\epsilon}(\mathbb R^3)$. In addition, under the control condition (eq:control), for any $u_0 \in H^{1+\epsilon}(\mathbb R^3)$, the associated forward maximal solution is global in positive times, verifies $\Vert u \Vert_{L^\inft

Figures (2)

  • Figure 1.1: Schematic representation of the control condition (\ref{['eq:control']}) and of typical Hamilton trajectories.
  • Figure A.1: Schematic representation of the control condition in the case of an obstacle. Only the non star-shaped portion of the boundary needs to be controlled.

Theorems & Definitions (38)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2: Global Strichartz estimates with dramatic loss for the damped flow
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Claim 2.6
  • ...and 28 more