Develop a mathematically rigorous definition of nonlinear QNMs as poles of a suitable nonlinear response functional.
Summary
Develop a mathematically rigorous definition of nonlinear QNMs as poles of a suitable nonlinear response functional.
Why this matters
Novel conceptual foundation for nonlinear ringdown.
Exact scope
- Background / setting
- asymptotically flat general relativity context; see family and coupling tags for matter model.
- Equation type
- PDE level: spectral-operator.
- Linearity
- both linearized and fully nonlinear aspects
- Regularity
- Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.
- Parameter regime
- Subextremal Kerr (or Kerr–de Sitter where tagged); spectral parameters $(l,m)$ and frequency $ω$ regimes as in cited microlocal frameworks.
- Asymptotics
- asymptotically flat
- Gauge / formulation
- State gauge/fixing class compatible with cited stability or interior programs (e.g. generalized harmonic, double-null interior charts).
Status explanation
Theorem status follows literature as summarized in known results and references (not upgraded without verified solution pointers).
Problem statement
Develop a mathematically rigorous definition of nonlinear QNMs as poles of a suitable nonlinear response functional.
What is already known
- Microlocal/resolvent frameworks yield decay and mode stability for waves on exact Kerr and Kerr–de Sitter under stated spectral assumptions.Regime: Linear waves; fixed background.Standard input for QNM expansions and superradiance discussions.
- Nonlinear Kerr stability is proved in a small-$|a|/M$ vacuum window (Klainerman–Szeftel).Regime: Nonlinear vacuum, restricted parameters.Closest nonlinear analogue for exterior stability conjectures.
- Complete QNM expansion as a spectral representation (including branch cuts) for Kerr remains an open mathematical framework problem.Regime: Spectral theory on Kerr.Distinguishes partial mode stability from full expansion/completeness.
Progress summary: Context: Novel conceptual foundation for nonlinear ringdown.
What remains open
Develop a mathematically rigorous definition of nonlinear QNMs as poles of a suitable nonlinear response functional.
Mathematical prerequisites
Match hypotheses to primary sources cited on this page; state minimal regularity, gauge class, and parameter windows in any claimed theorem.
Completion criteria
Prove a theorem or give a rigorous counterexample that matches the scoped statement under explicitly listed hypotheses.
Implications if solved
Impact depends on the solved formulation; sharpen once the statement is pinned to a literature-compatible theorem.
Formal verification suitability
FV: low
Speculative or open-ended; formalization boundary unclear.
See Formal verification for how this database uses these labels.
References
- primary Global analysis of linear waves on Kerr–de Sitter space — Hintz, Vasy (2016) Linear wave decay and spectral gap on Kerr–de Sitter; standard microlocal input for $Lambda>0$ decay.
- survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022) Program overview, gauge structure, and relation between linear tools and nonlinear stability.
Related problems
Related by shared tags
- K-506 — High-frequency Kerr quasinormal-mode laws with explicit remainder bounds
- K-624 — Prove a sharp characterization of near-extremal QNM clustering with explicit remainders.
- K-625 — Prove completeness/expansion of solutions in Kerr via QNMs plus branch-cut contributions (mathematical ringdown expansion).
- K-627 — Establish pseudospectrum bounds for Kerr wave operators and relate to transient growth near superradiance.
- K-636 — Prove sharp semiclassical quantization of Kerr QNMs with explicit high-frequency error bounds.
- K-661 — Determine whether QNM expansions are stable under small nonlinearities (nonlinear resonance theory).
- K-682 — Prove that Kerr QNMs are stable under small perturbations of the metric in a topology relevant to stability proofs.
Editorial / maintainer notes
Source manifest: N-081 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.