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arxiv: 2604.10366 · v1 · submitted 2026-04-11 · 🧮 math.AP · math-ph· math.CA· math.MP

Scattering for the Klein-Gordon-Schr\"odinger system in three dimensions with radial data

Vitor Borges , Tiklung Chan This is my paper

Pith reviewed 2026-05-10 15:13 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.CAmath.MP
keywords Klein-Gordon-Schrödinger systemglobal well-posednessscatteringradial dataStrichartz estimatesbilinear restriction estimatesthree dimensionsU^2 spaces
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The pith

Small radial initial data yield global well-posedness and scattering for the three-dimensional Klein-Gordon-Schrödinger system in the optimal known range.

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

The paper shows that the Klein-Gordon-Schrödinger system in three space dimensions admits global solutions that scatter to free waves whenever the initial data are small and radially symmetric and lie in the Sobolev spaces L² for the Schrödinger component and H to the power of minus one-half plus epsilon and H to the power of minus three-halves plus epsilon for the Klein-Gordon components. This range is the best currently known for global well-posedness even without the scattering conclusion. A reader would care because scattering means that after long times the nonlinear coupling becomes negligible and each field evolves essentially as a free wave. The argument proceeds by running a global iteration inside adapted function spaces that incorporate the radial symmetry to obtain the necessary space-time estimates.

Core claim

We prove global well-posedness and scattering for the 3D Klein-Gordon-Schrödinger system for small radial data in the best known global well-posedness range (u₀, n₀, n₁) ∈ L² × H^{-1/2 + ε} × H^{-3/2 + ε} for any ε > 0. The proof uses a global-in-time iteration scheme in the adapted function spaces U²_Φ, radial Strichartz estimates, and bilinear restriction estimates.

What carries the argument

A global iteration scheme inside the adapted function spaces U²_Φ that incorporates radial Strichartz estimates and bilinear restriction estimates to control the nonlinear interaction terms.

If this is right

  • Solutions exist for all positive and negative times and remain bounded in the given function spaces.
  • The nonlinear interaction terms become arbitrarily small in suitable norms for large times, so the solution scatters to a pair of free Klein-Gordon and Schrödinger waves.
  • The same iteration scheme yields uniform control on the space-time norms of the solution whenever the radial data are sufficiently small.
  • The result recovers the full range of local well-posedness data that were previously known only locally in time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If radial symmetry can be relaxed while retaining suitable angular regularity, the same iteration might extend to a larger class of data.
  • The technique of embedding radial Strichartz estimates into the U² framework may apply directly to other coupled dispersive systems that currently lack global results.
  • Removing the smallness assumption would require additional decay or conservation laws that are not used in the present argument.

Load-bearing premise

The initial data must be small and radially symmetric, because the estimates that close the iteration rely on radial symmetry and would not be available for general data.

What would settle it

An explicit small radial initial datum in the stated Sobolev spaces for which the corresponding solution develops a singularity in finite time or fails to approach a free solution as time tends to infinity.

Figures

Figures reproduced from arXiv: 2604.10366 by Tiklung Chan, Vitor Borges.

Figure 1
Figure 1. Figure 1: Strichartz admissible ranges for the radial wave and Schrödinger equa￾tions. Region (III) corresponds to σS < 0, regions (II) and (III) represent the increased radial Strichartz range for the wave, and regions (I)–(II) represent the in￾creased range for Schrödinger. The faintly colored endpoint L 2L 4 is included in the Schrödinger admissible range but not in the wave range. 8 [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 2
Figure 2. Figure 2: A visual breakdown of the cases in Propositions 4.3 and 4.4, respectively. In what follows we adopt the + sign convention on the Klein-Gordon flow, simply writing U 2 ⟨D⟩ , which does not alter the trilinear estimates. Similarly, we do not differentiate between N and N in the nonlinear analysis and always write N, as the difference will be irrelevant in our estimates. 4.3. Trilinear estimates – Schrödinger… view at source ↗
read the original abstract

We prove global well-posedness and scattering for the 3D Klein-Gordon-Schr\"odinger system for small radial data in the best known global well-posedness range $(u_0, n_0, n_1)\in L^2\times H^{ -\frac{1}{2} + \epsilon } \times H^{-\frac{3}{2} +\epsilon }$ for any $ \epsilon > 0 $. The proof uses a global-in-time iteration scheme in the adapted function spaces $U^2_\Phi$, radial Strichartz estimates, and bilinear restriction estimates.

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 global well-posedness and scattering for the three-dimensional Klein-Gordon-Schrödinger system with small radial initial data in the Sobolev range (u₀, n₀, n₁) ∈ L² × H^{-1/2+ε} × H^{-3/2+ε} for arbitrary ε > 0. The argument proceeds via a global-in-time contraction mapping in adapted U²_Φ spaces, relying on radial Strichartz estimates and bilinear restriction estimates to control the nonlinear interactions without derivative loss beyond the stated ε.

Significance. If the estimates close as claimed, the result advances the low-regularity theory for coupled dispersive PDEs by reaching the optimal known global well-posedness threshold in the radial setting. The use of U²_Φ spaces together with symmetry-adapted harmonic-analysis tools provides a clean, parameter-free closure for small data and yields scattering as a byproduct; this framework is reproducible and could extend to related systems.

minor comments (3)
  1. The abstract and introduction should explicitly display the system equations (including the precise coupling terms) for self-contained reading.
  2. Early in the paper, the precise definition and embedding properties of the adapted spaces U²_Φ relative to standard U² and V² spaces should be recalled, including any radial modifications.
  3. Figure captions (if any) and the statement of the main theorem should uniformly use the same notation for the initial-data spaces to avoid minor confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The report correctly summarizes the main result on global well-posedness and scattering for the 3D Klein-Gordon-Schrödinger system with small radial data in the indicated Sobolev spaces, and we appreciate the recognition of the significance of the U²_Φ framework combined with radial Strichartz and bilinear restriction estimates.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript presents a direct existence proof of global well-posedness and scattering via a standard contraction mapping argument in adapted U^2_Φ spaces. It relies on radial Strichartz estimates and bilinear restriction estimates that are invoked as known results under radial symmetry in 3D; these are external harmonic-analysis tools not derived or redefined within the paper. No self-definitional loops, fitted parameters renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the derivation chain. The small-data assumption closes the iteration without reducing the target statement to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard background results in dispersive PDE theory rather than new parameters or entities.

axioms (2)
  • domain assumption Radial Strichartz estimates hold for the Klein-Gordon and Schrödinger propagators in three dimensions
    Invoked to control the linear evolution inside the iteration scheme
  • domain assumption Bilinear restriction estimates are valid for the relevant frequency interactions
    Used to bound the nonlinear term in the coupled system

pith-pipeline@v0.9.0 · 5403 in / 1367 out tokens · 26058 ms · 2026-05-10T15:13:28.883897+00:00 · methodology

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Reference graph

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