Sharp decay thresholds in weighted $L^\infty$ for wave kinetic equations with power-law dispersion

Xilu Zhu

REVIEW 3 major objections 5 minor 30 references

Sharp decay thresholds in weighted $L^\infty$ for wave kinetic equations with power-law dispersion

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves a sharp decay threshold for four-wave kinetic equations: local well-posedness in weighted L∞ holds exactly above s_c=4β+3−a/2 and fails below it.

desk verdict The threshold formula is likely right and the LWP proof is a real piece of work; the ill-posedness proof has one fragile geometric step that a referee should push on before the sharp dichotomy is accepted. read the letter →

arxiv 2607.14892 v1 pith:W2XGA63E submitted 2026-07-16 math.AP math-phmath.MP

Xilu Zhu This is my paper

classification math.APmath-phmath.MP

keywords wavekineticequationweightedL∞sharpthresholdlocalwell-posednessill-posednessresonancemanifoldpower-lawdispersiongain-losscancellation

verification ladder T0 review T1 audit T2 compute T3 formal

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 studies four-wave kinetic equations — effective equations for turbulent wave spectra — with power-law dispersion |p|^a and a collision kernel whose high-frequency strength is controlled by β. It identifies a single cutoff exponent, s_c = 4β + 3 − a/2, and proves that the Cauchy problem is locally well-posed in weighted L∞ spaces above this threshold and ill-posed below it. The threshold records a precise balance: stronger kernels (larger β) demand more decay, while the geometry of the resonance manifold (through a and the dimension of resonant integration) supplies the offsetting gain. If correct, this gives a complete answer to where a pointwise decay theory for these equations can and cannot exist, and it shows that cancellation between gain and loss terms is essential in the most delicate regimes.

What carries the argument

The key device is a parametrization of the four-wave resonance manifold, using the ratio c_* = (|p|^a+|p_1|^a)/|p+p_1|^a and a variable α that locates the point on the resonance surface. For the ill-posedness direction, the load-bearing object is the family of dyadic patches B_k with |p_1|≈|p_2|≈2^k, where the resonance surface is shown to have volume ≈2^{5k} and the derivative of the resonance function along the distinguished coordinate has size ≈2^{(a−1)k} with a definite sign. This volume-versus-Jacobian balance is what produces the term −a/2 in the threshold.

What would settle it

For a specific admissible exponent, say a=3 and β=0, compute the volume of B_k ∩ {Φ=0} and the derivative ∂_{z_∥}Φ on a grid of points: if the volume is significantly smaller than C·2^{5k} or the derivative fails to have uniform sign and size ≈2^{2k}, the sharpness claim would be falsified. A direct check of the claimed output growth, I_k[f_0](2^{2j}e_1) ≈ 2^{(4β−s+1−a)j}, would also settle the matter.

Watch

Extended reading notes

Core claim

The central result is a sharp well-posedness/ill-posedness dichotomy for the four-wave kinetic equation with dispersion ω(p)=|p|^a, 1≤a≤5, and kernel weight |p|^{2β}, 0≤β≤1, posed in weighted L∞ spaces. Under (H1) s>4β+3−a/2 with 0≤β≤(a−1)/4, or (H2) s=4β+3−a/2 with 0<β<(a−1)/4, the paper proves local well-posedness by establishing trilinear bounds for the full gain-loss collision operator. If neither condition holds — β>(a−1)/4, or s below the threshold, or the endpoint with β=(a−1)/4 — the paper proves ill-posedness by constructing initial data concentrated near a high-low-low-high resonant configuration. The formula s_c=4β+3−a/2 is therefore both necessary and sufficient for a contraction

Load-bearing premise

The load-bearing premise is that, for every exponent a in [1,5], the high-low-low-high resonance patch has surface measure ≈2^{5k} and the resonance function has a derivative of definite sign and size ≈2^{(a−1)k} uniformly on that patch; if this geometric control fails for some a, the ill-posedness construction collapses.

Editorial extensions

If this is right

For quadratic dispersion (a=2) with no kernel growth (β=0), the threshold reduces to s_c=2, matching the known sharp weighted L∞ threshold for that model.
The condition β≤(a−1)/4 appears as a necessary restriction on kernel strength for any local theory, generalizing the critical value β=1/4 for quadratic dispersion.
The threshold is a pure decay threshold: it controls the high-frequency tail of the spectrum pointwise and imposes no momentum regularity.
The proof shows that gain-loss cancellations are not optional in certain frequency regimes: separate bounds on T1, T2, T3 fail there, and cancellation must be extracted from the structure of the constrained resonance integral.
At the endpoint, a single high-low-low-high patch contributes only logarithmically; the ill-posedness construction must superpose many well-separated patches to obtain divergence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The volume-versus-Jacobian mechanism suggests a transferable heuristic: any collision kernel with high-frequency growth 4β, resonant integration dimension d, and Jacobian gain γ should yield a threshold 4β+d−γ; this could be tested on other collision kernels or higher dimensions.
The author leaves the inhomogeneous equation open, expecting the same threshold because the transport flow is an isometry; if the trilinear estimates extend to L∞_x L∞_{s,p}, the same dichotomy should follow.
The restriction 1≤a≤5 is tied to the proof; whether the formula s_c=4β+3−a/2 persists outside this range is a natural testable extension.
Since weighted L∞ controls spectra pointwise, the threshold could be probed numerically: simulate the collision integral on synthetic spectra with decay just above and just below s_c and check where the output tail loses polynomial decay.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: read this for the unified threshold s_c = 4β + 3 − a/2 and for the weighted-L∞ machinery; interrogate Section 5 before believing the sharpness claim in full. The paper deserves refereeing, but not unconditional acceptance as stated.

What is genuinely new and good: the formula organizes a two-parameter family of kinetic wave equations, recovers the Germain–Ionescu–Tran Schrödinger case a = 2, β = 0, and matches the Ampatzoglou–Léger quasilinear threshold β = 1/4. The upper-bound proof is substantial: resonance-manifold parametrization, cancellation via spherical averaging, and separate treatments for 2 ≤ a ≤ 5, 1 < a < 2, and a = 1. The spherical difference lemma is clean and likely useful beyond this paper. I also see no circularity: the parameters are in the model, and the lower bound is a genuine frequency-configuration argument.

Soft spots, in proportion. First, Theorem 1.1 overstates Theorem 5.1. The well-posedness assumptions (H1)/(H2) and the three ill-posedness scenarios do not cover all remaining parameter pairs; e.g. β = 0, a > 1 at s = s_c is neither covered by (H1)/(H2) nor by the listed ill-posedness scenarios. That is a statement gap, not a proof gap, but it should be fixed before the paper claims the full dichotomy.

Second, and more serious: the ill-posedness proof rests on the volume lower bound (5.3)–(5.6). The argument shows the derivative B_{z∥}Φ has the right sign and size only under a condition like k ≪ 2^j with a constant depending on a. The text then says “after shrinking ε ... C′R ≪ 2^k”, but R is not controlled uniformly in k, and the unique zero z∥^* is not proven to stay inside |z∥| ≤ 3·2^k for every k admitted in the sum (5.7), especially k near the upper end. If the root exits the patch, the volume bound fails and the logarithmic divergence in the endpoint case collapses. The stress-test note is correctly aimed. This is not a stylistic complaint; it is the only load-bearing step for the ill-posedness side. I would not desk-reject over it, but the author needs to close the k-uniformity gap.

Minor issues: a few typos and garbled passages (e.g. “s < 2β − 2” in the high-low case where context requires s < 2β + 2; Section 1.4 ends mid-sentence). Cosmetic.

Verdict: the central threshold is plausible and the upper bound is a real contribution. The lower-bound geometry is fragile enough that a serious referee should push on it. I would send it to review, and I would cite it for the unified formula and the parametrization method.

Referee Report

3 major / 5 minor

Summary. The paper studies the four-wave kinetic equation (1.2) in R^3 with power-law dispersion ω(p)=|p|^a, 1≤a≤5, and kernel weight |p|^{2β}, 0≤β≤1. Its central claim is that in the weighted spaces L^∞_s the sharp decay threshold is s_c = 4β+3−a/2: local well-posedness holds for s>s_c and 0≤β≤(a−1)/4, and at the endpoint s=s_c for 0<β<(a−1)/4; otherwise the problem is asserted to be ill-posed. The well-posedness proof decomposes the collision operator into low/high frequency regimes, uses a resonance-manifold parametrization, and exploits gain–loss cancellations in delicate regimes. The ill-posedness proof constructs data concentrated near high-low-low-high resonant configurations and, at the endpoint, superposes dyadic patches to produce a logarithmic divergence. The paper also discusses the physical meaning of the threshold and its consistency with known results for a=2, β=0.

Significance. If the results are correct, the paper gives a clean and attractive necessary-and-sufficient decay threshold for a model family of wave kinetic equations, unifying and extending the Germain–Ionescu–Tran theory and connecting with the Ampatzoglou–Léger threshold. The well-posedness side is impressively detailed: it isolates the exact cancellation structure, uses spherical-difference estimates, and appears internally coherent in most regimes. The paper also explicitly identifies the physical role of each term in s_c. However, the sharpness claim is currently not fully supported: the simplified theorem overstates the precise ill-posedness theorem, and the Section 5 construction contains unresolved technical points in the admissible dyadic set and in the uniformity of the geometric volume lower bound. These issues are load-bearing for the claimed optimality.

major comments (3)

[Theorem 1.1 vs Theorem 5.1; §5, preliminary comment (i)] Theorem 1.1 asserts ill-posedness whenever neither (H1) nor (H2) holds. But Theorem 5.1 proves ill-posedness only in the three listed scenarios: β>(a−1)/4, s<s_c, and s=s_c with β=(a−1)/4. The case β=0, 1<a≤5, s=s_c satisfies neither (H1) nor (H2) and is not covered by any scenario of Theorem 5.1. The paper itself acknowledges in the preliminary comment after Theorem 5.1 that when β=0 it is not possible to define the operators T_j for general inputs at this endpoint. Thus the claimed sharp threshold is not established on the entire endpoint line s=s_c. This is not a presentation issue: Theorem 1.1's 'otherwise' statement is one of the paper's headline claims. Please either restrict Theorem 1.1 to the cases proven in Theorem 5.1, or supply an argument for the missing endpoint cases.
[§5, Step 1: definition of A_i and the summation range in (5.7)] The admissible dyadic set A_i is not internally consistent as written. A_i is defined with `1≤2^k<ε2^{2i}` and `2^k/2^{2m} ∉ [1/4,4]`, but the exclusion condition appears to remove almost all positive integer k if it is meant for all m, and the parenthetical `for m=1` is unclear. Moreover, Case 3 states `#A_j ≈ 2^j`, which is incompatible with k∈N and 2^k<ε2^{2j}: there are only O(j) such integer exponents. A related issue is that (5.7) sums over `0≤k≪2^j` with `2^k∈A_j`; if k is the exponent, this allows scales 2^k far larger than |p^{(j)}|=2^{2j}, in which case p_3=p^{(j)}−z has |p_3|≈2^k and the asserted value f_0(p_3)=(2^{2j})^{−s} is false. The high-low-low-high construction only works if the low scales satisfy 2^k≪2^{2j}. This needs a precise definition of A_i and the admissible k-range, and the counting estimate in Case 3 must be corrected consistently. Without this, the lower bou
[§5, equations (5.3)–(5.6)] The geometric volume lower bound (5.3) is the load-bearing step of the ill-posedness proof. Its proof asserts that on B_k the resonance surface {Φ=0} has measure ≈2^{5k} and that the Jacobian is uniformly comparable to (2^{2j})^{a−1}. After (5.5), the text concludes that the unique zero z^*_∥ lies in [−C′R,C′R] and says 'After shrinking ε ... C′R≪2^k'. The quantity R is not defined or controlled. The displayed estimate only bounds Φ|_{z_∥=0} by (2^{2j})^{a−1}(2^{ak}/(2^{2j})^{a−1}+2^{2k}/2^{2j}); for k near the allowed upper end 2^k<ε2^{2j}, the first term can be large, and no proof is given that the root remains inside |z_∥|≤3·2^k for every k summed in (5.7). If the root exits the patch, or the derivative loses its sign/size on part of B_k, the measure lower bound fails. Since (5.3) is the only support for the lower bound (5.7), the ill-posedness side — especially the endpoint logarithm

minor comments (5)

[§3.2, high-low case] The line 'If s<2β−2' appears twice and seems to be a typo for 's<2β+2'. Please correct and check the surrounding estimates for signs.
[§1.4, last paragraph] The sentence 'For this reason, several geometric estimates' is incomplete. It needs to be finished or removed.
[§3.5 and Remark 1.6] Proposition 3.9 says 'Under the assumption (H1) and (H2)', but the intended meaning is 'under either (H1) or (H2)'. Please reword.
[§5, Step 1 notation] The notation `p_∥`, `p_⊥`, and `z_∥` is not defined with enough clarity in the extracted text; the distinction between the integer k and the dyadic scale 2^k is a recurring source of ambiguity. A table of notation or a clear sentence would help.
[§2.2, Proposition 2.2 Step 1] The estimates use `s>2.5`; for a=1 this is correct under (H1), but please make explicit that (H2) does not apply to a=1 and that the argument needs only H1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the threshold is derived from independent trilinear estimates and a separate ill-posedness construction; parameters are model data, not fitted to the conclusion.

full rationale

The derivation chain is self-contained. The threshold s_c=4β+3−a/2 is obtained by proving trilinear bounds for the collision operator under (H1)/(H2) in Sections 2–4, and separately by constructing initial data concentrated on high-low-low-high dyadic patches in Section 5 to prove failure below the threshold. Neither side assumes the threshold as an input: a and β are fixed model parameters, and the ill-posedness scaling is computed from the kernel weights, the Jacobian of the resonance constraint, and the volume of the chosen patch. The load-bearing geometric volume lower bound (5.3)–(5.6) is an asserted geometric fact whose uniformity could be questioned on correctness grounds, but it is not a circular reduction to the target formula. Citations to Germain–Ionescu–Tran and Ampatzoglou–Léger are used as comparison benchmarks and for a parametrization technique, not as self-citations or imported uniqueness theorems that force the result. The apparent mismatch between the simplified Theorem 1.1 and the precise Theorem 5.1 is a precision issue, not circularity. Overall, no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numbers are fitted to data, and a and β are model parameters rather than ad hoc constants. The proof relies on standard Fourier/spherical tools and on the formal kinetic model as a domain assumption. No new physical entities, forces, or conserved quantities are introduced.

assumptions (5)

standard math Funk–Hecke formula and Legendre/Jacobi polynomial bounds
Invoked in Lemmas 6.1–6.3 to control spherical averages; quoted from standard references [23] and [27].
standard math Coarea formula and delta-function integration on resonant manifolds
Used throughout Chapters 2–3 to convert collision integrals over resonance constraints into surface integrals.
domain assumption The homogeneous wave kinetic equation (1.2) with momentum and energy delta constraints is the object of study
The paper analyzes (1.2) directly; derivation of the kinetic equation from the quasilinear dispersive equation (1.1) is not part of the proof.
domain assumption Parameter ranges 1 ≤ a ≤ 5, 0 ≤ β ≤ 1, and the weighted L∞_s framework
These ranges define the model family and the function spaces in which the threshold is proved; the proof splits by values of a and β.
domain assumption Strong solutions are defined through the Duhamel integral formulation (4.1)
The local well-posedness and ill-posedness statements are interpreted with respect to this integral notion of solution.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Sharp decay thresholds in weighted $L^\infty$ for wave kinetic equations with power-law dispersion." pith.science (2026). https://pith.science/paper/W2XGA63E

@misc{pith2026260714892,
  author       = {Pith},
  title        = {Pith review of: Sharp decay thresholds in weighted $L^\infty$ for wave kinetic equations with power-law dispersion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W2XGA63E}},
  note         = {Machine review of arXiv:2607.14892}
}

abstract

We study four-wave kinetic equations in space dimension three with power-law dispersion $\omega(p)=|p|^a$ and collision kernels with high-frequency growth measured by $\beta$. In weighted $L^\infty$ spaces, we identify the sharp decay threshold $$ s_c=4\beta+3-\frac a2. $$ For $s>s_c$, we prove local well-posedness by establishing trilinear bounds for the full gain-loss collision operator. For $s<s_c$, we prove ill-posedness by constructing data concentrated near a high-low-low-high resonant configuration. This threshold captures the balance between the high-frequency strength of the kernel and the geometry of the resonant manifold. The proof also shows that gain-loss cancellations are essential in the most delicate regimes.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

30 extracted references · 4 linked inside Pith

[1]

Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting.American Journal of Mathematics, 147(4):1053–1158, 2025

Ioakeim Ampatzoglou, Charles Collot, and Pierre Germain. Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting.American Journal of Mathematics, 147(4):1053–1158, 2025

2025
[2]

On the ill-posedness of kinetic wave equations.Nonlinearity, 38:115004, 2025

Ioakeim Ampatzoglou and Tristan L´ eger. On the ill-posedness of kinetic wave equations.Nonlinearity, 38:115004, 2025

2025
[3]

On the optimal local well-posedness of the wave kinetic equation inL r

Ioakeim Ampatzoglou and Tristan L´ eger. On the optimal local well-posedness of the wave kinetic equation inL r. Journal of Differential Equations, 477:114531, 2026

2026
[4]

A collective description of electron interactions

David Bohm and David Pines. A collective description of electron interactions. i. magnetic interactions.Physical Review, 82(5):625–634, 1951

1951
[5]

Onset of the wave turbulence description of the longtime behavior of the nonlinear schr¨ odinger equation.Inventiones Mathematicae, 225(3):787–855, 2021

Tristan Buckmaster, Pierre Germain, Zaher Hani, and Jalal Shatah. Onset of the wave turbulence description of the longtime behavior of the nonlinear schr¨ odinger equation.Inventiones Mathematicae, 225(3):787–855, 2021

2021
[6]

On the derivation of the homogeneous kinetic wave equation.Communications on Pure and Applied Mathematics, 78(4):856–909, 2025

Charles Collot and Pierre Germain. On the derivation of the homogeneous kinetic wave equation.Communications on Pure and Applied Mathematics, 78(4):856–909, 2025

2025
[7]

Derivation of the homogeneous kinetic wave equation: Longer time scales

Charles Collot and Pierre Germain. Derivation of the homogeneous kinetic wave equation: Longer time scales. Journal of Functional Analysis, 290(1):111179, 2026

2026
[8]

On the derivation of the wave kinetic equation for NLS.Forum of Mathematics, Pi, 9:e6, 2021

Yu Deng and Zaher Hani. On the derivation of the wave kinetic equation for NLS.Forum of Mathematics, Pi, 9:e6, 2021

2021

Show all 30 references

[9]

Derivation of the wave kinetic equation: full range of scaling laws, 2023

Yu Deng and Zaher Hani. Derivation of the wave kinetic equation: full range of scaling laws, 2023

2023
[10]

Full derivation of the wave kinetic equation.Inventiones Mathematicae, 233(2):543–724, 2023

Yu Deng and Zaher Hani. Full derivation of the wave kinetic equation.Inventiones Mathematicae, 233(2):543–724, 2023

2023
[11]

Long time justification of wave turbulence theory

Yu Deng and Zaher Hani. Long time justification of wave turbulence theory. 2023

2023
[12]

Long time derivation of the boltzmann equation from hard sphere dynamics

Yu Deng, Zaher Hani, and Xiao Ma. Long time derivation of the boltzmann equation from hard sphere dynamics. Annals of Mathematics, 2024. To appear

2024
[13]

Miguel Escobedo and Juan J. L. Vel´ azquez.On the Theory of Weak Turbulence for the Nonlinear Schr¨ odinger Equation, volume 238 ofMemoirs of the American Mathematical Society. American Mathematical Society, 2015

2015
[14]

Linearized wave turbulence convergence results for three-wave systems.Communications in Mathe- matical Physics, 378:807–849, 2020

Erwan Faou. Linearized wave turbulence convergence results for three-wave systems.Communications in Mathe- matical Physics, 378:807–849, 2020

2020
[15]

Ionescu, and Minh-Binh Tran

Pierre Germain, Alexandru D. Ionescu, and Minh-Binh Tran. Optimal local well-posedness theory for the kinetic wave equation.Journal of Functional Analysis, 279(4):108570, 2020

2020
[16]

Stability of Rayleigh–Jeans equilibria in the kinetic FPU equation.arXiv preprint arXiv:2409.01507, 2024

Pierre Germain, Joonhyun La, and Angeliki Menegaki. Stability of Rayleigh–Jeans equilibria in the kinetic FPU equation.arXiv preprint arXiv:2409.01507, 2024

2024 arXiv
[17]

A review on the kinetic theory of oscillator chains.arXiv preprint arXiv:2606.01358, 2026

Pierre Germain, Joonhyun La, and Angeliki Menegaki. A review on the kinetic theory of oscillator chains.arXiv preprint arXiv:2606.01358, 2026

2026 arXiv
[18]

Local well-posedness for the kinetic MMT model

Pierre Germain, Joonhyun La, and Katherine Zhiyuan Zhang. Local well-posedness for the kinetic MMT model. Communications in Mathematical Physics, 406(1):18, 2025

2025
[19]

Inhomogeneous turbulence for the wick nonlinear schr¨ odinger equation

Zaher Hani, Jalal Shatah, and Hui Zhu. Inhomogeneous turbulence for the wick nonlinear schr¨ odinger equation. Communications on Pure and Applied Mathematics, 77(11):4100–4142, 2024

2024
[20]

On the wave turbulence theory for a stochastic KdV type equation – generalization for the inhomogeneous kinetic limit

Amirali Hannani, Matthew Rosenzweig, Gigliola Staffilani, and Minh-Binh Tran. On the wave turbulence theory for a stochastic KdV type equation – generalization for the inhomogeneous kinetic limit. 2022

2022
[21]

Oscar E. Lanford. Time evolution of large classical systems. InDynamical Systems, Theory and Applications, volume 38 ofLecture Notes in Physics, pages 1–111. Springer, 1975

1975
[22]

Almost sharp wave kinetic theory of multidimensional KdV type equations withdě3

Xiao Ma. Almost sharp wave kinetic theory of multidimensional KdV type equations withdě3. 2022

2022
[23]

Springer-Verlag, Berlin, 1966

Claus M¨ uller.Spherical Harmonics, volume 17 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1966

1966
[24]

Springer, Heidelberg, 2011

Sergey Nazarenko.Wave Turbulence, volume 825 ofLecture Notes in Physics. Springer, Heidelberg, 2011

2011
[25]

Local-in-time existence ofL 1 solutions to the gravity water wave kinetic equation

Yulin Pan and Xiaoxu Wu. Local-in-time existence ofL 1 solutions to the gravity water wave kinetic equation. 2026

2026
[26]

On the wave turbulence theory for a stochastic KdV type equation

Gigliola Staffilani and Minh-Binh Tran. On the wave turbulence theory for a stochastic KdV type equation. 2021

2021
[27]

American Mathematical Society, Providence, RI, 4th edition, 1975

G´ abor Szeg˝ o.Orthogonal Polynomials, volume 23 ofAmerican Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 4th edition, 1975. SHARP THRESHOLDS FOR W A VE KINETIC EQUATIONS 59

1975
[28]

Rigorous derivation of the wave kinetic equation for fullβ-FPUT system.arXiv preprint arXiv:2605.19308, 2026

Katja Vassilev and Boyang Wu. Rigorous derivation of the wave kinetic equation for fullβ-FPUT system.arXiv preprint arXiv:2605.19308, 2026

2026 arXiv
[29]

Vassilev

Katja D. Vassilev. One-dimensional wave kinetic theory.Communications in Mathematical Physics, 2025

2025
[30]

Rigorous derivation of the wave kinetic equation forβ-FPUT system.arXiv preprint arXiv:2506.02948, 2025

Boyang Wu. Rigorous derivation of the wave kinetic equation forβ-FPUT system.arXiv preprint arXiv:2506.02948, 2025. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA Email address:xiluzhu@umich.edu

2025 arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting.American Journal of Mathematics, 147(4):1053–1158, 2025

Ioakeim Ampatzoglou, Charles Collot, and Pierre Germain. Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting.American Journal of Mathematics, 147(4):1053–1158, 2025

2025

[2] [2]

On the ill-posedness of kinetic wave equations.Nonlinearity, 38:115004, 2025

Ioakeim Ampatzoglou and Tristan L´ eger. On the ill-posedness of kinetic wave equations.Nonlinearity, 38:115004, 2025

2025

[3] [3]

On the optimal local well-posedness of the wave kinetic equation inL r

Ioakeim Ampatzoglou and Tristan L´ eger. On the optimal local well-posedness of the wave kinetic equation inL r. Journal of Differential Equations, 477:114531, 2026

2026

[4] [4]

A collective description of electron interactions

David Bohm and David Pines. A collective description of electron interactions. i. magnetic interactions.Physical Review, 82(5):625–634, 1951

1951

[5] [5]

Onset of the wave turbulence description of the longtime behavior of the nonlinear schr¨ odinger equation.Inventiones Mathematicae, 225(3):787–855, 2021

Tristan Buckmaster, Pierre Germain, Zaher Hani, and Jalal Shatah. Onset of the wave turbulence description of the longtime behavior of the nonlinear schr¨ odinger equation.Inventiones Mathematicae, 225(3):787–855, 2021

2021

[6] [6]

On the derivation of the homogeneous kinetic wave equation.Communications on Pure and Applied Mathematics, 78(4):856–909, 2025

Charles Collot and Pierre Germain. On the derivation of the homogeneous kinetic wave equation.Communications on Pure and Applied Mathematics, 78(4):856–909, 2025

2025

[7] [7]

Derivation of the homogeneous kinetic wave equation: Longer time scales

Charles Collot and Pierre Germain. Derivation of the homogeneous kinetic wave equation: Longer time scales. Journal of Functional Analysis, 290(1):111179, 2026

2026

[8] [8]

On the derivation of the wave kinetic equation for NLS.Forum of Mathematics, Pi, 9:e6, 2021

Yu Deng and Zaher Hani. On the derivation of the wave kinetic equation for NLS.Forum of Mathematics, Pi, 9:e6, 2021

2021