K-014

Nonlinear decay versus resonance expansions

Open Classical frontier Open in literature Mostly scoped Exterior Stability Physics-facing FV: medium

Near-Kerr (vacuum) Asymptotically flat Vacuum Spectral operator ExteriorNonlinearLinear

Linear ringdown expansions are useful, but the theorem-level nonlinear version is still missing.

Background / setting: Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.
Equation type: PDE level: spectral-operator.
Linearity: Includes or emphasizes nonlinear dynamics.
Regularity: Smooth / Sobolev classes as in the problem statement; tighten when citing a specific theorem.
Parameter regime: Subextremal Kerr moduli $|a|<M$ (or stated KN/KdS extension); smallness measured in the stability topology on Cauchy data.
Asymptotics: asymptotically-flat
Gauge / formulation: State gauge/fixing class compatible with cited stability or interior programs (e.g. generalized harmonic, double-null interior charts).

Entry imported without two independent bibliographic pointers in this repository; treat theorem-level claims as unverified here.

Relate late-time nonlinear Einstein dynamics near Kerr to a rigorous resonance/QNM plus tail expansion.

Nonlinear stability of vacuum Kerr is proved for sufficiently small $|a|/M$ (Klainerman–Szeftel).

Regime: Nonlinear Einstein vacuum, asymptotically flat, small angular momentum per unit mass.

Strongest unconditional nonlinear theorem toward the full subextremal conjecture.
Linearized Teukolsky/wave decay and mode stability on fixed subextremal Kerr are developed in depth (microlocal and physical-space methods).

Regime: Linearized gravity and scalar waves on exact Kerr.

Standard toolbox; not equivalent to nonlinear stability for all parameters.
Sharp Price-law exponents and nonlinear tail matching are understood in restricted settings (e.g. linearized models, Schwarzschild); sharp nonlinear Kerr curvature tails are not settled.

Regime: Late-time asymptotics; mixed linear vs nonlinear literature.

Locates what “sharp Price law” demands beyond integrated decay.

Progress summary: Partial progress exists in adjacent regimes;

A complete answer must state when resonance expansions are asymptotic or convergent and bound the nonlinear remainder in explicit norms.

Resonance theory; meromorphic resolvents; nonlinear normal forms; late-time asymptotics.

A complete answer must state when resonance expansions are asymptotic or convergent and bound the nonlinear remainder in explicit norms.

Would connect PDE stability theory directly to mathematical models of ringdown.

FV: medium

Some subquestions may formalize before the full statement.

See Formal verification for how this database uses these labels.

primary Nonlinear stability of Kerr for small angular momentum (program) — Klainerman, Szeftel (2021)
Theorem-level nonlinear stability of vacuum Kerr in a small $|a|/M$ regime; benchmark for full subextremal conjectures.
survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022)
Program overview, gauge structure, and relation between linear tools and nonlinear stability.

Heuristic matches on family, cluster, equation level, asymptotics, and relevance.

K-003 — Nonlinear asymptotic completeness near Kerr
K-005 — Sharp nonlinear Price law for curvature
K-006 — Kerr stability with BMS charges and nonlinear memory
K-659 — Prove sharp resolvent bounds near omega=0 for Kerr linearized Einstein operator, uniform in a/M.
K-001 — Full nonlinear stability of subextremal Kerr
K-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
K-009 — Einstein–massive Klein–Gordon near Kerr: classification of stable and unstable regimes
K-011 — Spin fields on dynamical near-Kerr backgrounds

Open: no complete theorem matching the statement is currently recorded on this site. : tighten if community consensus differs.

Last updated: 2026-04-05 · Last verified (editorial): 2026-04-06 (bulk-editorial-fixes) · Edit on GitHub →