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arxiv: 2604.15947 · v1 · submitted 2026-04-17 · 🧮 math.AP

On scattering and profile decomposition for critical nonlinear waves outside weakly trapping obstacles

David Lafontaine , Camille Laurent (CNRS , URCA) This is my paper

Pith reviewed 2026-05-10 08:20 UTC · model grok-4.3

classification 🧮 math.AP
keywords scatteringnonlinear wave equationprofile decompositionobstaclestrapped trajectoriesenergy-criticalDirichlet boundary conditionsexterior domains
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The pith

Scattering holds for large-data solutions of the defocusing energy-critical nonlinear wave equation outside two strictly convex obstacles in three dimensions

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes scattering for the defocusing energy-critical nonlinear wave equation subject to Dirichlet boundary conditions outside two strictly convex obstacles in three dimensions. This is achieved by first proving linear and nonlinear profile decompositions at infinite time under three linear hypotheses on the wave equation: global Strichartz estimates, weak non-trapping of the domain, and absence of trajectory reconcentration. These decompositions reduce the nonlinear problem to scattering for individual profiles once a rigidity condition excludes non-trivial solutions whose energy remains compactly supported for all time. A sympathetic reader would care because earlier scattering theorems for this equation required the absence of any trapped rays, whereas the new argument handles a concrete geometry that admits trapping and supplies a modular framework usable in other weakly trapping settings.

Core claim

Under the assumptions that the linear wave equation satisfies global Strichartz estimates, the domain is weakly non-trapping, and trajectories do not reconcentrate, linear and nonlinear profile decompositions in infinite time are established. These decompositions imply scattering for the defocusing energy-critical nonlinear wave equation with Dirichlet boundary conditions outside two strictly convex obstacles in dimension three, provided the only solution with compact flow is the trivial solution. The argument applies as a black box to other geometries satisfying the linear hypotheses.

What carries the argument

Infinite-time linear and nonlinear profile decompositions that break solutions into sums of rescaled profiles plus a remainder vanishing in the energy space and Strichartz norms

Load-bearing premise

The rigidity assumption that the only solution whose flow remains compact for all time is the zero solution, together with the linear wave equation satisfying global Strichartz estimates, weak non-trapping, and no trajectory reconcentration

What would settle it

A non-trivial finite-energy solution whose energy stays confined to a bounded region for all positive and negative times, or a concrete failure of global Strichartz estimates for the linear wave equation outside two strictly convex obstacles

Figures

Figures reproduced from arXiv: 2604.15947 by Camille Laurent (CNRS, David Lafontaine, URCA).

Figure 7.1
Figure 7.1. Figure 7.1: Illustration of the proof of Lemma 7.5 in the two dimensional case: t0 for a few points of ∂Θ1. Lemma 7.5. Let Θ1, Θ2 be two smooth, strictly convex subsets of R 3 with compact boundary verifying Assump￾tion 1.1. Let c1 > 0 and c := (c1, 0, 0). Denote χ(x) := |x − c| + |x + c|. Then, for any c1 > 0 fixed big enough, ∇χ(x) · (−n)(x) ≥ 0, ∀x ∈ ∂(Θ1 ∪ Θ2). Proof. We first do the proof in dimension 2, as it … view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: ∇χ and S(α) Collecting (7.10), (7.14) and (7.15) we get 1 T Z T 0 Z Ω∩B(0,A) |∇u(x, t)| 2 + |u(x, t)| 6 dxdt ≲ E T + E αT + ϵ(m(α)). We take (say) α := T −1/2 . Then all the right hand terms go to zero as T goes to infinity, and u scatters in H˙ 1 × L 2 by Lemma 7.4. □ 8. Geometric facts The purpose of this section is to verify that the exterior of two strictly convex obstacles satisfies the geomet￾rical… view at source ↗
read the original abstract

We prove scattering for the defocusing energy-critical non-linear wave equation with Dirichlet boundary conditions outside two strictly convex obstacles in dimension three. This is the first large data scattering result for such an equation in the presence of trapped trajectories. Our result is in fact more general and can be used as a black box in other geometries. More precisely, under the assumptions that the corresponding linear wave equation satisfies global Strichartz estimates, that the domain is weakly non-trapping and that trajectories do not reconcentrate, we show linear and nonlinear profile decompositions in infinite time. This implies scattering under the rigidity assumption that the only compact-flow solution is the trivial one.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves scattering for the defocusing energy-critical nonlinear wave equation with Dirichlet boundary conditions outside two strictly convex obstacles in three dimensions. This is achieved by establishing linear and nonlinear profile decompositions in infinite time under the black-box assumptions that the linear wave equation satisfies global Strichartz estimates, the domain is weakly non-trapping, and trajectories do not reconcentrate; scattering then follows from the rigidity hypothesis that the only compact-flow solution is the trivial one. The result is presented in a form that can be applied to other geometries satisfying the linear hypotheses.

Significance. If the result holds, it is a significant advance: it supplies the first large-data scattering theorem for this equation in the presence of trapped trajectories. The black-box isolation of the linear assumptions (global Strichartz, weak non-trapping, no reconcentration) and the adaptation of concentration-compactness techniques to the exterior domain are strengths that allow the argument to serve as a template for other settings. The manuscript thereby extends the reach of profile-decomposition methods beyond non-trapping domains while keeping the nonlinear analysis cleanly separated from the linear estimates.

minor comments (3)
  1. In the introduction, the relation between the specific geometry of two strictly convex obstacles and the general weak non-trapping assumption should be stated more explicitly, including a brief indication of how the two-obstacle case satisfies the linear hypotheses.
  2. The notation for the nonlinear profiles and the remainder term in the infinite-time decomposition (around the statement of the nonlinear profile decomposition) would benefit from a short table or diagram clarifying the parameters (time shifts, scaling, etc.) to improve readability.
  3. A few references to prior works on profile decompositions in exterior domains (e.g., the linear theory used for the Strichartz estimates) appear to be missing or cited only indirectly; adding them would strengthen the contextual placement of the black-box assumptions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment. We are pleased that the referee recognizes the significance of the first large-data scattering result in the presence of trapped rays and the value of the black-box formulation. Since the recommendation is for minor revision and no specific major comments are listed, we will prepare a revised version incorporating any minor editorial or presentational improvements.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under external linear assumptions

full rationale

The paper derives linear and nonlinear profile decompositions in infinite time from the black-box linear hypotheses (global Strichartz estimates, weak non-trapping, no trajectory reconcentration) via standard concentration-compactness techniques adapted to the exterior domain. Scattering then follows directly from the independent rigidity assumption that the only compact-flow solution is the zero solution. No step equates a derived quantity to its input by construction, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content reduces to the present result. The central claim is a logical implication from the stated external assumptions rather than an internal tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The central claim rests on four domain and dynamical assumptions listed in the abstract; no free parameters or new entities are introduced.

axioms (4)
  • domain assumption The corresponding linear wave equation satisfies global Strichartz estimates
    Invoked as a hypothesis for the profile decomposition
  • domain assumption The domain is weakly non-trapping
    Stated as an assumption enabling the result
  • domain assumption Trajectories do not reconcentrate
    Required for the nonlinear profile decomposition
  • domain assumption The only compact-flow solution is the trivial one
    Rigidity assumption used to conclude scattering

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