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A nonlinear wave-Dirac system on near-Minkowski spacetimes has global small-data solutions.

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Small-data solutions of a nonlinear tensorial wave-Dirac system on non-trapping asymptotically flat spacetimes are shown to exist globally with quantitative weighted-energy decay.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection Real structural insight—Clifford cancellation in a first-order spinor bootstrap—but the Kerr-type framing oversells the metric class and several load-bearing proofs are only sketched. the 3 major comments →

arxiv 2607.13775 v2 pith:EEDXFC65 submitted 2026-07-15 math.AP math-phmath.MP

A spinor-adapted geometric approach for nonlinear Dirac systems and its application to a tensorial wave-Dirac system near Minkowski spacetime

Seokchang Hong This is my paper
classification math.AP math-phmath.MP MSC 35Q7583C6035B4035L05
keywords nonlinear Dirac equationwave-Dirac systemglobal existencenull decompositionClifford algebraweighted energy estimatesasymptotically flat spacetimesmall-data stability
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper proves global existence and quantitative decay for small smooth solutions of a tensorial wave-Dirac system on (1+3)-dimensional asymptotically flat spacetimes close to Minkowski, under a non-trapping assumption. The system couples a Dirac spinor to a closed antisymmetric 2-form through the same first-order, Clifford-multiplication nonlinearity that appears in Maxwell-Dirac and Einstein-Dirac models, but without gauge or derivative-loss complications. The proof keeps the Dirac equation first-order throughout, using the Dirac current for the basic energy and only squaring the Dirac operator to borrow wave-type spacetime estimates. A structural point is that the Clifford algebra kills the most singular null interactions between the spinor and the extreme tensor components, so weak decay like t^{-1/2-δ} suffices to close the bootstrap. If correct, this gives a template for treating nonlinear Dirac systems, and specifically the spinor part of Einstein-Dirac, by geometric energy methods.

Core claim

The paper establishes that for N ≥ 11 and small initial data satisfying X^D_N[ψ]²(0) + X^T_N[F]²(0) ≤ ε0², the tensorial wave-Dirac system admits a global solution with uniform high-order energy bounds, bounded weighted spinor energy, at most logarithmically growing weighted wave energy for the extreme tensor components α and α̲, and explicit improved decay estimates, e.g. (1)E^D_{≤N−4}[ψ]²(τ) ≲ τ^{-1+δ}. The mechanism is a coupled energy method: commute by modified vector fields adapted to the spinor bundle, use the causal Dirac current for L² control of ψ, use the squared Dirac equation for spacetime and weighted-energy estimates, and exploit the null decomposition of both the spinor and t

What carries the argument

The central objects are the Clifford-compatible null decomposition of the Dirac spinor into components ψ± carried along null directions, and the null decomposition of the antisymmetric 2-form into components (α, α̲, ρ, σ). The carrying mechanism is the commutator identity for the spinorial Lie derivative together with an r^p-weighted energy hierarchy obtained after squaring the Dirac operator; the Clifford identity (γL)² = 0 excludes the worst null-null interactions. The analysis is organized on a double null foliation, with spinor energy defined by the Dirac current and spacetime control supplied by the wave equation for the spinor field.

Load-bearing premise

The load-bearing premise is that the non-spherically symmetric part of the metric perturbation decays no slower than r^{-2} and that the background has no trapped null geodesics; if only r^{-1} angular decay is available, the commutator gain that closes the r^p hierarchy is lost.

What would settle it

Take a background perturbation whose angular part behaves like h_ang ≍ r^{-1} sinθ, the long-range rotating term present in stationary black-hole metrics, and compute the rotation-commutator estimate in Proposition 4.3, Case 3: the gain becomes r^{-1-η} with η = 0 instead of η = 1, so the weighted hierarchy loses its r-gain and the claimed bootstrap closure fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Small-data global existence holds with quantitative energy and decay rates, so the continuation criterion shows uniform energy bounds prevent blow-up.
  • Nonlinear Dirac systems need not be fully reduced to wave equations: the first-order Dirac current energy is indispensable, and wave reduction alone is not sufficient to close the argument.
  • Because the Clifford algebra restricts which nonlinear interactions occur, weak decay such as t^{-1/2-δ} is enough, so bootstrap schemes of this type can close with relatively little dispersion.
  • The coupled energy hierarchy yields bounded weighted spinor energies and only mild logarithmic growth for the extreme wave components, giving a concrete template for analyzing the spinor part of Einstein-Dirac systems.

Where Pith is reading between the lines

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

  • The r^{-2} angular decay assumption on the background metric is likely load-bearing: for generic stationary rotating metrics, where the off-diagonal angular component decays only like r^{-1}, the commutator gain in the rotation-vector-field estimates disappears and the hierarchy may fail to close.
  • The regularity threshold N ≥ 11 and the eight-derivative gap are probably not optimal; the same null structure may allow lower regularity with sharper Sobolev or Strichartz estimates.
  • Since the null decomposition used for the tensor field is the Maxwell decomposition, adapting this method to Maxwell-Dirac would mainly require treating the gauge structure separately.
  • For Einstein-Dirac, the spinorial nonlinearities appear manageable once the quasilinear derivative loss of the Einstein equations is handled, suggesting that the derivative loss is the primary obstruction rather than the Dirac coupling.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a spinor-adapted geometric energy method for a semilinear tensorial wave-Dirac system, iγ^μ∇_μψ = iF^{μν}γ_μγ_νψ, ∇_μF^{μν} = ⟨ψ,γ^νψ⟩, dF=0, on (1+3)-dimensional asymptotically flat Lorentzian manifolds. The background metric is assumed to satisfy the decomposition h = h_rad + h_ang with decay rates (1.13), together with a non-trapping condition. The method combines the Dirac-current energy, null decomposition of the Dirac equation, Kosmann–Lie commutation, Lichnerowicz squaring to a wave-type equation, and r^p-weighted energy hierarchies. The main result, Theorem 5.3, asserts global existence and quantitative energy/decay bounds for small data with N ≥ 11 and 0 < δ < 1/20, under a bootstrap argument whose closure occupies Sections 6–12. The paper also claims that the Clifford algebra null structure excludes the most singular nonlinear interactions and that only weak decay t^{-1/2-δ} is needed for stability.

Significance. If the proof is completed and the hypotheses are precisely as stated, this would be a significant contribution to the geometric analysis of nonlinear Dirac systems. The combination of the first-order Dirac current with wave-type estimates from squaring the Dirac operator, the null decomposition of the spinor, and the r^p hierarchy is a natural and potentially powerful framework for future work on Maxwell-Dirac and Einstein-Dirac systems. The paper is careful about the Clifford-algebra structure and the Kosmann Lie derivative, and the explicit null-structure identities in Sections 3 and 6 are useful. However, the current manuscript has not yet delivered a complete, verifiable proof of the main theorem, and the claimed 'Kerr-type' scope is not supported by the stated angular decay hypothesis. The value of the paper therefore depends on whether these gaps can be closed or the claims appropriately narrowed.

major comments (3)
  1. [§1.3, (1.13); §4.2, (4.39)-(4.52), Proposition 4.3] The angular decay assumption h_ang = O(r^{-2}) (η=1) is load-bearing: Proposition 4.3, Case 3 uses the rotation-commutator gain r^{-1-η} to obtain the weight gain (p−η) in (4.41), and the paper then fixes η=1. This is exactly the r^{-2} decay of h_ang. However, the paper motivates this class as having 'the same long-range radial structure as the Kerr metric' (§1.3). Standard Kerr-type stationary perturbations have angular modes decaying like r^{-1} (e.g., g_{tφ} ~ r^{-1} sin^2θ). For η=0, the hierarchy (4.41) gives no weight gain, and the nonlinear closure in Sections 7–10, which repeatedly exploits the extra r^{-1} from rotation commutators, does not follow as written. The theorem is internally consistent under (1.13), but the stated Kerr motivation is not; the strengthened decay is doing essential work and must either be proved for Kerr-type metrics or explicitly removed from the claim
  2. [§1.3, §2.1, Theorem 5.3] The non-trapping condition is assumed but never quantified or proven for the class of metrics satisfying (1.13). This is not a minor caveat: Schwarzschild and Kerr spacetimes contain trapped null geodesics at the photon sphere, so the family of metrics with 'the same long-range radial structure as the Kerr metric' is not contained in the non-trapping class. If the intended application is genuinely 'near Minkowski spacetime', the manuscript should either prove that sufficiently small perturbations satisfying (1.13) are non-trapping, or state the non-trapping condition as an explicit, separate hypothesis and discuss what class of metrics satisfies it. As written, Theorem 5.3 is conditional on an unverified geometric assumption that also excludes the motivating black-hole-type examples.
  3. [§7–§10, §12] The manuscript does not contain a complete proof of the nonlinear bootstrap closure. Key estimates are repeatedly deferred with phrases such as 'the remaining task is obvious' (§7.2.3), 'we omit the repetitive details' (Proposition 9.2), and 'the proof is somewhat schematic' (§9.1). In particular, the top-order 'derivative loss' integrals, where ∇_L∇_Lψ is replaced via the squared Dirac equation and then integrated by parts in the angular variables, are central to the stated mechanism and are only sketched. Section 12 describes an iteration whose 'sufficiently many iterations' is never quantified, and the final passage from improved estimates to the theorem's bounds is asserted rather than demonstrated. For a theorem of this scope, these omitted details are load-bearing; the reader cannot currently verify that the bootstrap closes.
minor comments (5)
  1. [Throughout] There are numerous typographical errors that obscure the mathematics: 'yileds' (p. 32), 'merelt' (Prop. 6.3), 'Furthremore' (§8.1), 'improdved' (Cor. 8.2), 'compltes' (p. 40). The paper would benefit from a careful proofreading pass.
  2. [§1.5.2 vs §3.2] The notation L_Z is used both for the ordinary Lie derivative on tensors and for the Kosmann–Lie derivative on spinors. The paper acknowledges this abuse, but it creates confusion in estimates where both objects appear. A distinct notation, e.g., L_Z^S for the spinorial derivative, would improve readability.
  3. [Theorem 5.3] The statement 'there exists a small 0 ≤ ε < ε_0' is unusual: ε is used as a bootstrap smallness parameter but the strict inequality and the dependence on C in later estimates are not fully specified. Please clarify the ordering of quantifiers and the role of ε versus ε_0.
  4. [§4.2, (4.37)-(4.42)] In the statement of Proposition 4.3, the right-hand sides of (4.40)–(4.42) use the notation (p−1)E^D and (p−η)E^D but the summation over Z∈{L,L,Ω} is not fully aligned with the derivative order of the fields appearing in the left-hand sides. A more explicit index convention would help.
  5. [Appendix B] The local well-posedness argument is sketched via a Picard iteration, but the contraction estimate (B.23) appears to lose a power of Y(0) (the factor C'τY(0) should likely be C'τ Y(0)^{1/2} or similar). Please check the exponents in this estimate.

Circularity Check

0 steps flagged

No significant circularity: the proof is a self-contained bootstrap under explicit assumptions; no fitted input is relabeled as a prediction and no load-bearing claim reduces to a self-citation.

full rationale

The derivation chain is not circular. The main theorem is proved under explicit hypotheses (1.12)-(1.13), including the angular decay h_ang = O(r^-2). Proposition 4.3 uses exactly this assumed decay, with η = 1 fixed in §4.2, to obtain the rotation-commutator gain and hence the r^p hierarchy. This is an assumption on the admissible background class, not a quantity defined in terms of the theorem's conclusion. The nonlinear argument in §5.1-§12 is a standard bootstrap: energy bounds are assumed and then improved using the equations, and the continuity argument extends the solution. That is a closed-loop estimate, not circular reasoning. Local well-posedness in Appendix B is a separate Picard-iteration argument. No parameter is fitted to a subset of the data and then renamed as a prediction, no invariant is defined in terms of the desired result, no load-bearing uniqueness theorem is imported from the authors' prior work, and no central premise is justified solely by a self-citation. The potential caveats noted in the paper—that the r^-2 angular decay may not be met by generic Kerr-type metrics and that 'after sufficiently many iterations' in §12.3 is sketched rather than fully written out—are concerns about correctness, completeness, or scope of the stated assumptions, not circularity.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 1 invented entities

The central claim rests on (i) standard spin geometry (Clifford algebra, spin connection, Lichnerowicz squaring); (ii) three geometric domain assumptions about the background — non-trapping, |∂^α h_rad| ≲ r^{−1−|α|}, |∂^α h_ang| ≲ r^{−2−|α|}, and the hybrid time-slice/null foliation; and (iii) the linear r^p hierarchy (Props 4.2-4.4) whose proof is partially omitted. There are no fitted data parameters: the smallness exponents (δ < 1/20, derivative gap 8, N ≥ 11) are proof-design choices. The only invented object is the toy model itself, which is explicitly labeled as such.

free parameters (4)
  • δ (bootstrap smallness exponent) = 0 < δ < 1/20 (chosen, not fitted)
    Chosen small enough that all exponent inequalities close (e.g., −1/4 + 5δ/2 + ε < 0, −1/2 + 3δ < 0, 3δ growth tolerable); a proof-design parameter, not a datum.
  • Derivative gap (N ≥ 11) = gap = 8 derivatives
    Creates the high-low separation so every quadratic interaction contains a lower-order factor; regularity is chosen by hand to make the Sobolev/energy bookkeeping work.
  • Bootstrap constants ε, C = 0 ≤ ε < ε0; C ≥ 1
    Standard bootstrap constants: bounds are of the form C²ε²τ^{kδ}; chosen to make the assumed estimates improve, not fitted to data.
  • r^p / r^q hierarchy exponents = p ∈ [0, 2−δ], q ∈ [2, 3+δ]
    Weights follow the Dafermos-Rodnianski template; the specific shifts by δ are adjusted algebraically so the coupled hierarchy closes.
axioms (5)
  • standard math Spin structure and Clifford representation exist on (M,g) with γ(μ)γ(ν)+γ(ν)γ(μ) = −2g_{μν}I, and the Lichnerowicz identity γ^μ∇_μ(γ^ν∇_νψ) = g^{μν}∇_μ∇_νψ + (1/4)Rψ (Prop 4.1).
    Background from Lawson-Michelsohn [53], Wernli [67], LeFloch-Ma-Zhang [52]; the squaring formula is checked in the text and is standard.
  • domain assumption The spacetime is non-trapping and asymptotically flat with h = h_rad + h_ang, |∂^α h_rad| ≤ c_α r^{−1−|α|}, |∂^α h_ang| ≤ c'_α r^{−2−|α|}, constants small.
    Eqs. (1.12)-(1.13), (2.4). Non-trapping is assumed, not proved; the r^{−2} angular decay (η = 1) is required for the rotation commutation gain in Prop 4.3.
  • domain assumption Global null foliation: Σ_τ = {t=τ} for r ≤ R and Σ_τ = {u=τ−R, v≥τ+R} for r ≥ R, with J⁺(Σ_0) = ∪_τ Σ_τ.
    Section 2.1, eq. (2.5); defines the flux/energy domains used in every divergence-theorem argument; existence of such a foliation for general small perturbations is assumed.
  • domain assumption Gauge choice Ω = 1 on C_u, C_v so that ∇_L L = 0 and ∇_{L̲} L̲ = 0.
    Section 2.3; simplifies connection coefficients (ξ = ξ̲ = ω = ω̲ = 0); standard in the null-frame formalism.
  • standard math Linear r^p hierarchy and commutator bounds (Propositions 4.2-4.4).
    Stated and partially proved in §4; the proof of Prop 4.3 Case 2 is omitted ('We omit the further details'); these estimates underwrite every nonlinear estimate, so their correctness is load-bearing.
invented entities (1)
  • Tensorial wave-Dirac system (1.1): iγ^μ∇_μψ = iF^{μν}γ_μγ_νψ, ∇_μF^{μν} = ⟨ψ,γ^νψ⟩, dF = 0 no independent evidence
    purpose: Toy model isolating the null interaction between antisymmetric 2-form fields and spinors, without Einstein derivative loss or Maxwell gauge structure.
    A deliberately simplified model introduced to develop the spinor-adapted method; not a physical theory, so no external falsifiable handle; its validation is internal (the theorem).

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of A spinor-adapted geometric approach for nonlinear Dirac systems and its application to a tensorial wave-Dirac system near Minkowski spacetime." pith.science (2026). https://pith.science/paper/EEDXFC65

@misc{pith2026260713775,
  author       = {Pith},
  title        = {Pith review of: A spinor-adapted geometric approach for nonlinear Dirac systems and its application to a tensorial wave-Dirac system near Minkowski spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EEDXFC65}},
  note         = {Machine review of arXiv:2607.13775}
}
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abstract

We study a nonlinear tensorial wave-Dirac system on $(1+3)$-dimensional asymptotically flat spacetimes as a semilinear model motivated by the Maxwell-Dirac and Einstein-Dirac systems. The purpose of this model is to isolate the interaction between the null geometry of antisymmetric tensor fields and the intrinsic first-order geometry of the Dirac equation while avoiding the derivative loss mechanism of the Einstein equations and the gauge structure of the Maxwell equations. Our analysis preserves the first-order nature of the Dirac equation throughout the nonlinear argument. The Dirac current provides the fundamental energy identity, while quantitative spacetime estimates are obtained from the wave equation arising from the squared Dirac operator. Combining these ingredients with integrated local energy decay estimates and $r^p$-weighted energy hierarchies, we establish a coupled energy method for the tensorial and spinorial components. A key observation is that the Clifford algebra is compatible with the null decomposition of antisymmetric tensor fields and excludes the most singular nonlinear interactions. As a consequence, we establish the global existence of small-data solutions together with quantitative weighted energy and decay estimates. Remarkably, combining the null structure with dyadic argument, weak decay such as $t^{-\frac12-\delta}$ is sufficient to obtain nonlinear stability of the system, in the spirit of \cite{DHRT}. We expect that the geometric ideas developed in this paper provide a useful starting point for the study of more general nonlinear Dirac systems, including the Maxwell-Dirac and Einstein-Dirac equations.

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Reference graph

Works this paper leans on

69 extracted references · 2 canonical work pages

  1. [1]

    Alcubierre,The Dirac equation in general relativity and3 + 1formalism, available in arXiv:2503.03918

    M. Alcubierre,The Dirac equation in general relativity and3 + 1formalism, available in arXiv:2503.03918. alinhac

    Pith/arXiv arXiv

  2. [2]

    Alinhac,Geometric Analysis of Hyperbolic Differential Equations: An Introduction, London Mathematical Society Lecture Note Series: 374, (2010)

    S. Alinhac,Geometric Analysis of Hyperbolic Differential Equations: An Introduction, London Mathematical Society Lecture Note Series: 374, (2010). artzcaccia

    2010

  3. [3]

    Ben-Artzi, F

    J. Ben-Artzi, F. Cacciafesta, A.S. de Suzzoni, J. Zhang,Global Strichartz estimates for the Dirac equation on symmetric spaces, Forum of Mathematics, Sigma, 10, E25, (2022). batic

    2022

  4. [4]

    Batic,Scattering for massive Dirac fields on the Kerr metric, J

    D. Batic,Scattering for massive Dirac fields on the Kerr metric, J. Math. Phys.48, 022502, (2007). bar

    2007

  5. [5]

    B¨ ar,The Dirac operator on hyperbolic manifolds of finite volume, J

    C. B¨ ar,The Dirac operator on hyperbolic manifolds of finite volume, J. Diff. Geom.53, (1999), 439–488. behe

    1999

  6. [6]

    Bejenaru and S

    I. Bejenaru and S. Herr,The cubic Dirac equation: small initial data inH 1(R3), Comm. Math. Phys.335(1), (2015), 43–83. behe1

    2015

  7. [7]

    Bejenaru and S

    I. Bejenaru and S. Herr,The cubic Dirac equation: small initial data inH 1/2(R2), Comm. Math. Phys.343(2), (2016), 515–562. beherr

    2016

  8. [8]

    Bejenaru and S

    I. Bejenaru and S. Herr,On global well-posedness and scattering for the massive Dirac-Klein-Gordon system, J. Eur. Math. Soc. (JEMS),19:8, (2017), 2445–2467. 102 S. HONG boucan

    2017

  9. [9]

    Bournaveas and T

    N. Bournaveas and T. Candy,Global well-posedness for the massless cubic Dirac equation, Int. Math. Res. Notices. No. 22, (2016), 6735–6828. boucanma

    2016

  10. [10]

    Bournaveas, T

    N. Bournaveas, T. Candy, S. Machihara.A note on the Chern-Simons-Dirac equations in the Coulomb gauge, Discrete and Continuous Dynamical Systems,34No. 7, (2014), 2693–2701. brand

    2014

  11. [11]

    Brandling, K

    V. Brandling, K. Kr¨ oncke,Global existence of Dirac-wave maps with curvature term on expanding spacetimes, Calc. Var. 57, No. 119, (2018) cacciasu1

    2018

  12. [12]

    Cacciafesta and A

    F. Cacciafesta and A. S. de Suzzoni,Weak dispersion for the Dirac equation on asymptotically flat and warped products spaces, Discrete Contin. Dyn. Syst.,39(8), (2019), 4359–4398. cacciasu

    2019

  13. [13]

    Cacciafesta and A

    F. Cacciafesta and A. S. de Suzzoni,Local in time Strichartz estimates for the Dirac equation on spherically symmetric spaces, Int. Math. Res. Not., Vol. 2022, Issue 4, (2022), 2729–2771. caccia1

    2022

  14. [14]

    Cacciafesta, A.S

    F. Cacciafesta, A.S. de Suzzoni, and L. Meng,Strichartz estimates for the Dirac equations on asymptotically flat manifolds, available in doi.org/10.2422/2036-2145.202203-026. caccia2

    arXiv 2036

  15. [15]

    Cacciafesta, E

    F. Cacciafesta, E. Danesi, L. Meng,Strichartz estimates for the half-wave/Klein-Gordon and Dirac equations on compact manifolds without boundary, Math. Ann., doi.org/10.1007/s00208-023-02716-5. canhe2

    work page doi:10.1007/s00208-023-02716-5

  16. [16]

    Candy, S

    T. Candy, S. Herr,Transference of bilinear restriction estimates to quadratic variation norms and the Dirac-Klein-Gordon system, Anal. PDE,11, No. 5, (2018), 1171–1240. canhe1

    2018

  17. [17]

    Candy, S

    T. Candy, S. Herr,Conditional large initial data scattering results for the Dirac-Klein-Gordon system, Forum of Mathe- matics, Sigma,6, (2018). canhe

    2018

  18. [18]

    Candy, S

    T. Candy, S. Herr,On the Majorana condition for nonlinear Dirac systems, Ann. I. H. Poincar´ e, Vol. 35, (2018), 1707–1717. chagla

    2018

  19. [19]

    J. M. Chadam, R. T. Glassey,On certain global solutions of the Cauchy problem for the (classical) coupled Klein-Gordon- Dirac equations in one and three space dimensions, Arch. Ration. Mech. Anal.,54, (1974), 223–237. chagla1

    1974

  20. [20]

    J. M. Chadam, R. T. Glassey,On the Maxwell-Dirac equations with zero magnetic field and their solution in two space dimensions, J. Math. Anal. Appl.,53(1976), 495–597. chen

    1976

  21. [21]

    Chen,Global stability of Minkowski spacetime for a spin-1/2 field, Advances in Theoretical and Mathematical Physics, 29, No

    X. Chen,Global stability of Minkowski spacetime for a spin-1/2 field, Advances in Theoretical and Mathematical Physics, 29, No. 2, (2025), 485–556. chohonglee

    2025

  22. [22]

    Y. Cho, S. Hong, K. Lee,Scattering and nonscattering of the Hartree-type nonlinear Dirac system at critical regularity, SIAM J. Math. Anal.55, No. 4, (2023). chohongoz

    2023

  23. [23]

    Y. Cho, S. Hong, T. Ozawa,Charge conjugation approach to scattering for the Hartree type Dirac equations with chirality, J. Math. Phys.64, 021508, (2023). chokwonleeyang

    2023

  24. [24]

    Y. Cho, S. Kwon, K. Lee, C. Yang,The modified scattering for Dirac equations of scattering-critical nonlinearity, Adv. Diff. Equ., 29(3/4), (2024), 179–222. choklee

    2024

  25. [25]

    Y. Cho, K. Lee,The global dynamics for the Maxwell-Dirac system, available in arXiv:2406.18887. choleeoz

    Pith/arXiv arXiv

  26. [26]

    Y. Cho, K. Lee, T. Ozawa,Small data scattering of 2D Hartree type Dirac equations, J. Math. Anal. Appl., 506, 125549, (2022). christoklai

    2022

  27. [27]

    Christodoulou, S

    D. Christodoulou, S. Klainerman,The Global Nonlinear Stability of the Minkowski SpacePrinceton University Press, Princeton, (1993). cloos

    1993

  28. [28]

    Cloos,On the long-time behavior of the three-dimensional dirac-maxwell equation with zero magnetic field, (2020)

    C.C. Cloos,On the long-time behavior of the three-dimensional dirac-maxwell equation with zero magnetic field, (2020). daferrod1

    2020

  29. [29]

    Dafermos, I

    M. Dafermos, I. Rodnianski,The redshift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math., 52, (2009), 859–919. daferrod

    2009

  30. [30]

    Dafermos, I

    M. Dafermos, I. Rodnianski,A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds, Invent. Math.,185, (2011), 467–559. dahorod

    2011

  31. [31]

    Dafermos, G

    M. Dafermos, G. Holzegel, I. Rodnianski,Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case|a| ≪m, Annals of PDE,5, No. 1, (2019). dahorod1

    2019

  32. [32]

    Dafermos, G

    M. Dafermos, G. Holzegel, I. Rodnianski,The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Math.222, No. 1, (2019), 1–214. DHRT

    2019

  33. [33]

    Dafermos, G

    M. Dafermos, G. Holzegel, I. Rodnianski, M. Taylor,Quasilinear wave equations on asymptotically flat spacetimes with applications to Kerr black holes, Annal. & PDE., Vol. 19 (2026), No. 5, 909–1028. DR

    2026

  34. [34]

    Dafermos, I

    M. Dafermos, I. Rodnianski,A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, XVIth International Congress on Mathematical Physics, pp. 421-432 (2010). dirac1928

    2010

  35. [35]

    P. A. M. Dirac,The quantum theory of the electron, Proc. Roy. Soc. London A,117(1928), 610–624. GEOMETRIC ANALYSIS OF NONLINEAR DIRAC EQUATIONS 103 donglefloch

    1928

  36. [36]

    S. Dong, P. G. LeFloch, Z. Wyatt,Global evolution ofU(1)Higgs Boson: nonlinear stability and uniform energy bounds, Ann. Henri Poincar´ e, 22, (2021), 677–713. escobedovega

    2021

  37. [37]

    Escobedo, L

    M. Escobedo, L. Vega,A semilinear Dirac equation inH s(R3)fors >1, SIAM J. Math. Anal. Vol. 28, No. 2, (1997), 338–362. finster

    1997

  38. [38]

    Finster, N

    F. Finster, N. Kamran, J. Smoller, S.-T. Yau,Decay rates and probability estimates for massive Dirac particles in the Kerr-Newman black hole Geometry, Commun. Math. Phys.230, (2002), 201–244 galyag

    2002

  39. [39]

    Galstian, K

    A. Galstian, K. Yagdjian,The self-interacting Dirac fields in FLR W spacetime, Nonlinear Diff. Equ. Appl. (2022) 29:62. gavrusoh

    2022

  40. [40]

    Gavrus, S.-J

    C. Gavrus, S.-J. Oh,Global well-posedness of high dimensional Maxwell-Dirac for small critical data, Memoirs. Amer. Math. Soc.,264, No. 1279, (2020). geosha

    2020

  41. [41]

    Georgiev, B

    V. Georgiev, B. Shakarov,Global large data solutions for 2D Dirac equation with Hartree type interaction, IMRN, Vol. 2022, Issue 17, (2022), 12803–12820. geogiev

    2022

  42. [42]

    Georgiev,Small amplitude solutions of the Maxwell-Dirac equations, Indiana University Mathematics Journal, Vol

    V. Georgiev,Small amplitude solutions of the Maxwell-Dirac equations, Indiana University Mathematics Journal, Vol. 40, no. 3, (1991), 845–883. ginoux

    1991

  43. [43]

    Ginoux, O

    N. Ginoux, O. M¨ uller,Global solvability of massless Dirac-Maxwell systems, Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, Vol. 35, No. 6, (2018), 1645–1654. hanicolas

    2018

  44. [44]

    H¨ afner, J.-P

    D. H¨ afner, J.-P. Nicolas.Scattering of massless Dirac fields by a Kerr black hole, Rev. Math. Phys.16. No. 1, (2004), 29–123. hafnernicolas

    2004

  45. [45]

    H¨ afner, J.-P

    D. H¨ afner, J.-P. Nicolas,The characteristic Cauchy problem for Dirac fields on curved backgrounds, Journal of Hyperbolic Diff. Equ. Vol. 8, No. 3, (2011), 437–483. herrh

    2011

  46. [46]

    S. Herr, S. Hong,Strichartz estimates for the half Klein-Gordon equation on asymptotically flat backgrounds and applica- tions to cubic Dirac equations, available in arXiv:2502.13670. herrifrimspitz

    Pith/arXiv arXiv

  47. [47]

    S. Herr, M. Ifrim, M. Spitz,Modified scattering for the three dimensional Maxwell-Dirac system, Ann. PDE 12, 14 (2026). herrmaul

    2026

  48. [48]

    S. Herr, C. Maul´ en, C. Mu˜ noz,Decay of solutions of nonlinear Dirac equations, Commun. Math. Phys. 407, 94 (2026). holzegel10

    2026

  49. [49]

    Holzegel,Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem, available on arXiv:1010.3216

    G. Holzegel,Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem, available on arXiv:1010.3216. klai

    Pith/arXiv arXiv

  50. [50]

    Klainerman,The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, (1986), 293–326

    S. Klainerman,The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, (1986), 293–326. kwonleeyang

    1986

  51. [51]

    S. Kwon, K. Lee, C. Yang,The modified scattering of two dimensional semi-relativistic Hartree equations, J. Evol. Equ. 24, No. 64, (2024). leflochmazhang

    2024

  52. [52]

    LeFloch, Y

    P.G. LeFloch, Y. Ma, W. Zhang,The Global nonlinear stability of Minkowski spacetime with self-gravitating massive Dirac fields, available on arXiv:2510.20626. lawson

    arXiv

  53. [53]

    H. B. Lawson, M.-L. Michelsohn,Spin geometry, Princeton, NJ: Princeton University Press, (1989) klee

    1989

  54. [54]

    Lee,Scattering results for the(1+4)dimensional massive Maxwell-Dirac system under Lorenz gauge condition, available in arXiv:2312.13621

    K. Lee,Scattering results for the(1+4)dimensional massive Maxwell-Dirac system under Lorenz gauge condition, available in arXiv:2312.13621. lizhang

    Pith/arXiv arXiv

  55. [55]

    J. Li, Y. Zhang,A vector field method for some nonlinear Dirac models in Minkowski spacetime, Journal of Differential Equations,273, (2021): 58–82. mazhang

    2021

  56. [56]

    S. Ma, L. Zhang,Sharp decay estimates for massless Dirac fields on a Schwarzschild background, J. Func. Anal.282, (2022), 109375. machihara

    2022

  57. [57]

    Machihara, M

    S. Machihara, M. Nakamura, K. Nakanishi, and T. Ozawa,Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equations, J. Funct. Anal.219(1), (2005), 1–20. parker

    2005

  58. [59]

    Pasqualotto,The Spin±1Teukolsky Equations and the Maxwell System on Schwarzschild, Ann

    F. Pasqualotto,The Spin±1Teukolsky Equations and the Maxwell System on Schwarzschild, Ann. Henri Poincar´ e,20, (2019), 1263–1323. psarelli1

    2019

  59. [60]

    Psarelli,Maxwell-Dirac equations in four-dimensional Minkowski space, Comm

    M. Psarelli,Maxwell-Dirac equations in four-dimensional Minkowski space, Comm. PDE. (2005), 30(1–2), 97–119. strichartz

    2005

  60. [61]

    R. S. Strichartz,Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J.44(3), (1977), 705–714. tes

    1977

  61. [62]

    Tesfahun,Long-time behavior of solutions to cubic Dirac equation with Hartree type nonlinearity inR 1+2, Int

    A. Tesfahun,Long-time behavior of solutions to cubic Dirac equation with Hartree type nonlinearity inR 1+2, Int. Math. Res. Not., 2020, (2020), 6489–6538. tes1

    2020

  62. [63]

    Tesfahun,Small data scattering for cubic dirac equation with hartree type nonlinearity inR 1+3, SIAM J

    A. Tesfahun,Small data scattering for cubic dirac equation with hartree type nonlinearity inR 1+3, SIAM J. Math. Anal., 52 (2020), 2969–3003. thirring

    2020

  63. [64]

    Thirring,A soluble relativistic field theory, Annals of Physics,3, (1958), 91–112

    W. Thirring,A soluble relativistic field theory, Annals of Physics,3, (1958), 91–112. 104 S. HONG treude

    1958

  64. [65]

    J.-H. Treude,Decay in outgoing null directions of solutions of the massive Dirac equation in certain asymptotically flat, static spacetimes, (Dissertation), available in 10.5283/epub.32344 wang

    work page doi:10.5283/epub.32344

  65. [66]

    Wang,On global existence of 3D charge critical Dirac-Klein-Gordon system, Int

    X. Wang,On global existence of 3D charge critical Dirac-Klein-Gordon system, Int. Math. Res. Notices,2015, (2015), 10801–10846. wernli

    2015

  66. [67]

    Wernli,Lecture notes on spin geometry, available on arXiv:1911.09766

    K. Wernli,Lecture notes on spin geometry, available on arXiv:1911.09766. yag

    Pith/arXiv arXiv 1911

  67. [68]

    Yagdjian,Global in time self-interacting Dirac fields in the de Sitter space, J

    K. Yagdjian,Global in time self-interacting Dirac fields in the de Sitter space, J. Evol. Equ. (2022) 22:22. cyang

    2022

  68. [69]

    Yang,Scattering results for Dirac Hartree-type equations with small initial data, Commun

    C. Yang,Scattering results for Dirac Hartree-type equations with small initial data, Commun. Pure Appl. Anal.,18, (2019), 1711–1734. zhaowu

    2019

  69. [70]

    P. Zhao, X. Wu,On the local existence for the characteristic initial value problem for the Einstein-Dirac system, available in arXiv:2509.04167 Universit¨at M¨unster, Mathematisches Institut, Einsteinstrasse 62 48149 M¨unster, Germany Email address:seokchangh.11@uni-muenster.de

    Pith/arXiv arXiv

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.