On the ill-posedness of kinetic wave equations

Ioakeim Ampatzoglou; Tristan L\'eger

REVIEW 2 major objections 3 minor 29 references

On the ill-posedness of kinetic wave equations

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper establishes a sharp well-posedness threshold at β=1/4 for 4-wave kinetic equations derived from quasilinear Schrödinger systems, with ill-posedness for β>1/4.

desk verdict Sharp threshold claim is credible; ill-posedness side is solid and self-contained, well-posedness side rests on an unproved self-cited lemma. read the letter →

arxiv 2411.12868 v2 pith:WGAKKSP2 submitted 2024-11-19 math.AP math-phmath.MP

Ioakeim Ampatzoglou , Tristan Léger This is my paper · ORCID

classification math.APmath-phmath.MP MSC 35Q5535R2535B3035Q35

keywords kineticwaveequationquasilinearSchrödingerwell-posednessill-posednessweightedL∞spacescollisionalaveragingturbulence4-waveresonance

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 tries to establish exactly when a family of kinetic wave equations derived from quasilinear Schrödinger systems is solvable. It claims a sharp threshold at $\beta = 1/4$ in the derivative-loss parameter: for $0 \le \beta \le 1/4$ the equation is locally well-posed in polynomially weighted $L^\infty$ spaces, while for $\beta > 1/4$ both the full equation and its gain-only part are ill-posed, meaning strong solutions cannot be constructed even from small data. This matters because existing derivations of kinetic equations from nonlinear wave equations only hold as long as the kinetic equation has a smooth solution, so knowing the exact solvability threshold tells where that derivation program can succeed. The paper also gives physical interpretations of the number $1/4$, connecting it to Sobolev control of potential energy and to the finite/infinite capacity of wave-action cascades.

What carries the argument

The central object on the well-posedness side is the collisional averaging estimate of Lemma 2.1: for $k_1,k_2 \in \mathbb{R}^3$, the angular average $F(k_1,k_2)=\int_{S^2} \langle k_1^*\rangle^{-3} d\sigma$ is bounded by $(1+|k_1|^2+|k_2|^2)^{-1}$, where $k_1^* = (k_1+k_2)/2 + |k_1-k_2|\sigma/2$. This estimate converts the singular cross-section into a bound that makes the collision operator contractive in weighted $L^\infty$ for $\beta \le 1/4$, via the trilinear bounds of Lemma 2.3. On the ill-posedness side, the load-bearing machinery is the isotropic reduction of the collision operator and the study of the second Picard iterate: for radial data the operator splits according to which of $\omega_1,\omega_2,\omega_3$ is smallest, and the dominant contribution obeys $C^{234}[n_1^0,n_1^0,n_1^0] \approx \omega_1^{2\beta - 1/2 - M/2}$, which becomes unbounded relative to the weighted norm exactly when $\beta > 1/4$.

What would settle it

Compute the angular average $F(k_1,k_2)=\int_{S^2} \langle k_1^*\rangle^{-3} d\sigma$ for large $|k_1|=|k_2|$; Lemma 2.1 predicts decay like $(1+|k_1|^2+|k_2|^2)^{-1}$, so a direct numerical evaluation that violates this bound would invalidate the well-posedness proof. Alternatively, evaluate the second Picard iterate for the gain-only equation with $f_0=\langle k\rangle^{-M}$ at $\beta>1/4$: the paper predicts the dominant term grows like $\omega_1^{2\beta-1/2-M/2}$, so checking that integral for fixed $\beta,M$ would settle the ill-posedness claim.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 1.1: for the 4-wave kinetic equation (1.3) derived from the quasilinear Schrödinger model (1.1), local well-posedness in weighted $L^\infty$ holds precisely when $0 \le \beta \le 1/4$, and fails for $\beta > 1/4$ for both the full collision operator and its gain-only counterpart. Ill-posedness is shown through instantaneous loss of smoothness: for an explicit initial datum the second Picard iterate is already unbounded, so no strong solution in the sense of Definition 2.5 can exist. A further point of the argument is that the gain-only equation becomes ill-posed for the simple radial datum $\langle k_1\rangle^{-M}$, whereas the full equation needs an oscillatory radial datum of the form $A + \cos(N|k_1|^2)$ over $\langle k_1\rangle^{M}$, because cancellations in the collision operator damp low frequencies.

Load-bearing premise

The positive half of the theorem relies on a collisional averaging estimate (Lemma 2.1) quoted without proof from the authors' earlier preprint; if that estimate fails, the well-posedness side for $0 \le \beta \le 1/4$ is not established.

Editorial extensions

If this is right

For $0 \le \beta \le 1/4$ and $M>6$, the collision operator is a contraction on a ball in $\langle k\rangle^{-M}L^\infty$, producing a unique local strong solution for each initial datum in that space.
For $\beta > 1/4$, no weighted-$L^\infty$ strong solution can be constructed for the full or gain-only equation, and this remains true for arbitrarily small initial data.
Because both the full and gain-only equations share the same threshold, solving the gain-only equation is a legitimate path to solving the full 4-wave kinetic equation.
The inhomogeneous version $\partial_t f + v \cdot \nabla_x f = C[f]$ inherits the same well-posedness result on the same time scale, since free transport is an isometry on the weighted spaces used here.

Reading between the lines

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

Editorial extension: the threshold $\beta \le 1/4$ matches the range where Sobolev embedding gives $\| |\nabla|^\beta u \|_{L^4} \lesssim \|u\|_{\dot H^1}$, suggesting that kinetic-equation well-posedness tracks the ability of kinetic energy to control potential energy; the paper notes this heuristic but does not derive the threshold from it.
Editorial extension: the same second-Picard-iterate strategy might yield explicit thresholds for other quasilinear wave kinetic models once an analogue of the collisional averaging estimate is known.
Editorial extension: a natural test is whether the threshold is independent of the choice of weighted $L^\infty$ spaces; in $L^p$ or exponentially weighted spaces the exponent $1/4$ could move, because the angular-averaging bound is space-dependent.

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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

Bottom line: this paper probably fixes the local well-posedness threshold at beta = 1/4 for these kinetic wave equations. The ill-posedness proofs are the real contribution, and they look right. The well-posedness half is not self-contained: it inherits the key averaging lemma from the authors' own unpublished preprint, so the headline result is conditional on material outside the paper.

What is new: the sharp threshold; the proof that gain-only and full equations share it; and the mechanism for full-equation ill-posedness—oscillatory initial data that cancel the usual gain-loss compensation. The gain-only section is very clean, with a two-sided estimate (4.7) that pins down the exact size. The isotropic reduction and the decomposition of the collision operator are done carefully.

Soft spots, in descending order of importance. (1) Lemma 2.1 is the load-bearing wall for the whole well-posedness section. It is quoted from [4] with no proof. The lemma is plausible—even the extreme cases k1 ≈ -k2 or k1 >> k2 give the claimed decay—but for a paper advertised as sharp, an unpublished self-cited lemma is not enough. The referee should demand a proof or a public version of [4]. (2) In Section 5, the inequality 2β + 5/2 - M/2 < -1 is asserted for M > 10, β in [0,1]. That's false (e.g., β=1, M=11 gives -1 exactly; M=10.5 gives -0.75). The proof only needs the exponent to be negative, and M > 10 ensures that (since 4β+5 ≤ 9), so this is a repairable slip, not a hole. (3) The simplified theorem and abstract say 'weighted L∞ spaces' without the M > 6 / M > 10 qualifications. That's a presentational issue; the precise statements inside are fine.

The physical discussion (Remarks 1.3–1.5) is explicitly heuristic and clearly marked; I wouldn't hold that against the paper. Citation pattern looks fair—prior work on β=0 and soft potentials is cited, and the self-citation is of a directly relevant lemma, not padding.

Who this is for: anyone working on the derivation program for wave kinetic equations or on well-posedness of kinetic equations. It deserves a serious referee. I'd send it, with instructions to fix the exponent, add a proof or reference for Lemma 2.1, and align the abstract with the precise M conditions.

Referee Report

2 major / 3 minor

Summary. The paper studies the four-wave kinetic wave equation (1.3) derived from the quasilinear Schrödinger model (1.1), and claims a sharp local well-posedness/ill-posedness threshold at β = 1/4. The well-posedness half (Theorem 2.6) covers 0 ≤ β ≤ 1/4 in the spaces ⟨k⟩^{-M}L∞ with M > 6, via a fixed-point argument whose key input is a collisional averaging lemma (Lemma 2.1) quoted from the authors' earlier preprint [4]. The ill-posedness half treats β > 1/4: Theorem 4.2 handles the gain-only equation for M > 6, and Theorem 5.1 handles the full equation for M > 10, both by exhibiting isotropic initial data whose evolution immediately leaves the weighted space. The gain-only proof uses monotonicity and a two-sided asymptotic (4.7) for the second Picard iterate; the full-equation proof uses an oscillatory radial datum and a lower bound (5.2) isolating the main contribution.

Significance. If the result is correct, it gives a rare exact well-posedness threshold for a kinetic wave equation and connects derivative loss in the underlying quasilinear NLS model to solvability of the kinetic equation. The gain-only ill-posedness half is essentially self-contained: the two-sided bound (4.7) and the monotonicity argument are clean and convincing, and the construction of the oscillatory datum for the full equation in Section 5 is inventive. However, the positive half rests entirely on an unproved, self-cited averaging lemma, and the final step of the full-equation proof contains a false exponent inequality, so the paper as written is conditional rather than definitive.

major comments (2)

[§2.1, Lemma 2.1 and Lemma 2.3] Lemma 2.1, quoted from the unpublished preprint [4] without proof, is the sole mechanism that makes the trilinear bounds (2.4)–(2.6) close for β ≤ 1/4. In the proof of Lemma 2.3, the angular average of ⟨k*⟩^{-3} is replaced by 1/(1+|k1|^2+|k2|^2), and this is exactly what turns the prefactor |k1−k2||k1|^{2β}|k*_1|^{2β} into an O(1) quantity. Since [4] is not available to the reader, the well-posedness half of Theorem 1.1 is not established by the present manuscript. I request that a proof of Lemma 2.1 be included, or that the theorem be stated with this dependence made fully explicit.
[§5.6, after (5.21)] The assertion “using that since M > 10 and β ∈ [0,1] we have 2β + 5/2 − M/2 < −1” is false: for M = 10.1 and β = 1 the exponent is −0.55. The subsequent choice ω1 = 2πB/N with N/(2πB)(A3C4 + A3C5 + A2C6) < A2C1/10 controls terms of size ω1^{-1}, but it does not control A3C5ω1^{2β+5/2−M/2} when that exponent lies in (−1,0). The gap is repairable, because M > 10 implies 2β + 5/2 − M/2 < −1/2, so taking ω1 large independently of the displayed constraint makes this error term small; the proof should be corrected accordingly.

minor comments (3)

[§4.3] After defining f01(k1) = (1+|k1|^4)^{-M/4}, the line “n01(ω1) := ⟨ω1⟩^{-M}” is inconsistent with n0(ω1) = f0(k1); it should read ⟨ω1⟩^{-M/2}. All subsequent estimates in Section 4 use the M/2 convention, as does Section 5, so this is a typo with serious potential to mislead.
[Theorem 1.1 and abstract] The simplified statements should make the M-dependence explicit: the well-posedness theorem requires M > 6, the gain-only ill-posedness theorem requires M > 6, and the full-equation ill-posedness theorem requires M > 10. As written, “weighted L∞ spaces” hides this difference.
[§5.1 and Remark 4.3] In the displayed definition of I5 there is a stray comma in “C124, [n01,n01,n01]”; moreover, Remark 4.3 says “initial datum leading to uniqueness” where “ill-posedness” is meant.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the ill-posedness half is self-contained, and the well-posedness half depends on a stated, parameter-free averaging lemma from the authors' preprint rather than on a fitted input or definitional equivalence.

full rationale

The ill-posedness direction (Theorems 4.2 and 5.1) is self-contained: it exhibits radial initial data, derives the isotropic collision operator from the parametrized resonant manifold, and proves unboundedness of the second Picard iterate by explicit lower bounds (e.g., (4.2): C234[n01,n01,n01] is comparable to omega_1^{2*beta-1/2-M/2}) and non-resonant integration by parts for the oscillating datum. No parameter is fitted and no conclusion is imported from a benchmark. The well-posedness direction (Theorem 2.6) is a Banach fixed-point argument whose trilinear estimates (2.4)-(2.6) are consequences of Lemma 2.1, quoted without proof from the authors' earlier preprint [4]. This is a load-bearing self-citation: if the angular average in (2.1) were weaker, the bound |k1-k2||k1|^{2*beta}|k1*|^{2*beta}/(1+|k1|^2+|k2|^2) <= 1 would fail and the beta <= 1/4 contraction would not close. However, Lemma 2.1 is a concrete parameter-free estimate with no fitted constants, it is stated in full in the manuscript, and it is not defined in terms of Theorem 1.1; the present paper simply does not reproduce its proof. Thus the issue is missing proof supply in this manuscript, not a circular reduction of the theorem to its own conclusion. I also flag a separate correctness gap in Section 5.6: the text asserts 'since M > 10 and beta in [0,1] we have 2*beta + 5/2 - M/2 < -1', which is not implied because the exponent can be greater than -1 for M just above 10; this does not affect the circularity assessment. Overall, no equation is equivalent to its input by construction, so the paper receives a low circularity score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper contributes a sharp threshold but imports three nontrivial tools from prior literature: the collisional averaging estimate from the authors' own preprint [4], the parametrization of the collision operator from [4], and the angular integration formula from [29]. None of these is proved in the present paper, and none is machine-checked or supported by code. The central claim does not appear to be a definitional tautology.

assumptions (3)

domain assumption Collisional averaging estimate (Lemma 2.1, Eq. 2.1): ∫_{S^2} ⟨k1*⟩^{-3} dσ ≲ (1+|k1|^2+|k2|^2)^{-1}.
Quoted from the authors' earlier preprint [4]; the entire well-posedness proof (Section 2) and the trilinear bounds depend on it.
domain assumption Angular integration formula P = 32π^3 (ω1ω2ω3ω4)^{-1/2} min{√ω1,√ω2,√ω3,√ω4} for the resonant angular average (Lemma 3.1).
Used in Lemma 3.1 to derive the isotropic collision operator (3.2); cited from [29, Appendix A] as a known formula.
domain assumption Parametrization of G1, G2, L1, L2 by k1*, k2* (Eqs. 1.14-1.17), solving the momentum-energy system (1.20)-(1.21).
The isotropic reduction and all later estimates use this parametrization, quoted from [4].

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Pith. "Pith review of On the ill-posedness of kinetic wave equations." pith.science (2026). https://pith.science/paper/WGAKKSP2

@misc{pith2026241112868,
  author       = {Pith},
  title        = {Pith review of: On the ill-posedness of kinetic wave equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WGAKKSP2}},
  note         = {Machine review of arXiv:2411.12868}
}

read the original abstract

In this article we identify a sharp ill-posedness/well-posedness threshold for kinetic wave equations (KWE) derived from quasilinear Schr\"{o}dinger models. We show well-posedness using a collisional averaging estimate proved in our earlier work \cite{AmLe}. Ill-posedness manifests as instantaneous loss of smoothness for well-chosen initial data. We also prove that both the gain-only and full equation share the same well-posedness threhold, thus legitimizing a gain-only approach to solving 4-wave kinetic equations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 18 canonical work pages

[4]

Ampatzoglou, T

I. Ampatzoglou, T. L´ eger,Scattering theory for the inhomogeneous kinetic wave equat ion, Preprint (2024)

work page 2024
[1]

Ampatzoglou, Global well-posedness of the inhomogeneous kinetic wave eq uation near vacuum , Kinet

I. Ampatzoglou, Global well-posedness of the inhomogeneous kinetic wave eq uation near vacuum , Kinet. Relat. Models, Volume 17, Issue 6: 838-854, (2024)

work page 2024
[2]

Ampatzoglou, C

I. Ampatzoglou, C. Collot, P. Germain, Derivation of the kinetic wave equation for quadratic dispe rsive problems in the inhomogeneous setting , to appear in Amer. J. Math. (2024)

work page 2024
[3]

Ampatzoglou, J.K

I. Ampatzoglou, J.K. Miller, N. Pavlovi´ c, M. Taskovi´ c , Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted L∞ spaces, arXiv:2405.03984 (2024)

arXiv 2024
[5]

B´ enyi, T

A. B´ enyi, T. Oh, The Sobolev inequality on the torus revisited , Publ. Math. Debrecen 83 (2013), no. 3, 359–374

work page 2013
[6]

Buckmaster, P

T. Buckmaster, P. Germain, Z. Hani, J. Shatah, Onset of the wave turbulence description of the longtime behavior of the nonlinear Schr¨ odinger equation, Invent. Math. 225 (2021), no. 3, 787–855

work page 2021
[7]

X. Chen, J. Holmer, Well/ill-posedness bifurcation for the Boltzmann equatio n with constant collision kernel, to appear in Ann. PDE

work page
[8]

Collot, H

C. Collot, H. Dietert, P. Germain, Stability and cascades for the Kolmogorov-Zakharov spectru m of wave turbulence, Arch. Ration. Mech. Anal. 248 (2024), no. 1 , Paper No. 7, 31 pp

work page 2024

Show all 29 references

[9]

Collot, P

C. Collot, P. Germain, On the derivation of the homogeneous kinetic wave equation , to appear in Comm. Pure Appl. Math., arXiv:1912.10368

1912 arXiv
[10]

Collot, P

C. Collot, P. Germain, Derivation of the homogeneous kinetic wave equation: longe r time scales , arXiv:2007.03508

2007 arXiv
[11]

Y. Deng, Z. Hani, On the derivation of the wave kinetic equation for NLS , Forum Math. Pi 9 (2021), Paper No. e6, 37 pp

2021
[12]

Y. Deng, Z. Hani, Full derivation of the wave kinetic equation , Invent. math. 233 (2023), no. 2, 543–724

2023
[13]

Y. Deng, Z. Hani, Long time justiﬁcation of wave turbulence theory , arXiv:2311.10082

arXiv
[14]

Y. Deng, Z. Hani, X. Ma Long time derivation of Boltzmann equation from hard sphere dynamics, Preprint arXiv:2408.07818 (2024)

2024 arXiv
[15]

Escobedo, A

M. Escobedo, A. Menegaki, Instability of singular equilibria of a wave kinetic equati on, arXiv:2406.05280

arXiv
[16]

Escobedo, J.J.L

M. Escobedo, J.J.L. Vel´ azquez,On the theory of weak turbulence for the nonlinear Schr¨ odinger equation, Mem. Amer. Math. Soc. 238 (2015), no. 1124, v+107 pp

2015
[17]

Faou, Linearized Wave Turbulence Convergence Results for Three- Wave Systems , Volume 378, pages 807–849, (2020)

E. Faou, Linearized Wave Turbulence Convergence Results for Three- Wave Systems , Volume 378, pages 807–849, (2020)

2020
[18]

Germain, A

P. Germain, A. D. Ionescu, M.-B. Tran, Optimal local well-posedness theory for the kinetic wave eq uation, J. Funct. Anal. 279 (2020), no. 4, 108570, 28 pp

2020
[19]

Germain, J

P. Germain, J. La, K. Zhang, Local Well-posedness for the Kinetic MMT Model , Preprint arXiv:2310.11893 (2023)

2023 arXiv
[20]

Z. Hani, J. Shatah, H. Zhu, Inhomogeneous turbulence for the Wick nonlinear Schr¨ odin ger equation , Comm. Pure Appl. Math. 77, 11, 4100-4142, (2024)

2024
[21]

Hannani, M

A. Hannani, M. Rosenweig, G. Staﬃlani, M.B. Tran, On the wave turbulence theory for a stochastic KdV type equation – Generalization for the inhomogeneous kinet ic limit , Preprint: arXiv:2210.17445 (2022)

2022 arXiv
[22]

Kenig, G

C. Kenig, G. Ponce, L. Vega, Small solutions to nonlinear Schr¨ odinger equations , Ann. Inst. Henri Poincar´ e Analyse Nonlin´ eaire,10, no. 3, (1993), 255–288

1993
[23]

Kenig, G

C. Kenig, G. Ponce, L. Vega, Smoothing eﬀect and local existence theory for generalized n onlinear Schr¨ odinger equations, Invent. math. 134, (1998), 489–545 24 I. AMPATZOGLOU AND T. L ´EGER

1998
[24]

Ma, Almost sharp wave kinetic theory of multidimensional KdV ty pe equations with d ≥ 3, Preprint: arXiv:2204.06148 (2022)

X. Ma, Almost sharp wave kinetic theory of multidimensional KdV ty pe equations with d ≥ 3, Preprint: arXiv:2204.06148 (2022)

2022 arXiv
[25]

Majda, D.W

A.J. Majda, D.W. McLaughlin, E.G. Tabak, A one-dimensional model for dispersive wave turbulence , J. Nonlinear Sci. 6, pp. 9-44 (1997)

1997
[26]

Menegaki, L2-stability near equilibrium for the 4 waves kinetic equatio n, Kinet

A. Menegaki, L2-stability near equilibrium for the 4 waves kinetic equatio n, Kinet. Relat. Models, 2024, Volume 17, Issue 4: 514-532

2024
[27]

Nazarenko, Wave turbulence, Lecture Notes in Physics, 825

S. Nazarenko, Wave turbulence, Lecture Notes in Physics, 825. Springer, Heidelberg, 2011

2011
[28]

Staﬃlani, M.-B

G. Staﬃlani, M.-B. Tran, On the wave turbulence theory for a stochastic KdV type equat ion, arXiv:2106.09819 (2021)

2021 arXiv
[29]

Y. Zhu, B. Semisalov, G. Krstulovic, S. Nazarenko, Testing wave turbulence theory for the Gross- Pitaevskii system , Physical Review E 106, 014205 (2022) Baruch College, The City University of New York, Newman Vert ical Campus, 55 Lexington A ve New York, NY, 10010, USA Email ...

2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [4]

Ampatzoglou, T

I. Ampatzoglou, T. L´ eger,Scattering theory for the inhomogeneous kinetic wave equat ion, Preprint (2024)

work page 2024

[2] [1]

Ampatzoglou, Global well-posedness of the inhomogeneous kinetic wave eq uation near vacuum , Kinet

I. Ampatzoglou, Global well-posedness of the inhomogeneous kinetic wave eq uation near vacuum , Kinet. Relat. Models, Volume 17, Issue 6: 838-854, (2024)

work page 2024

[3] [2]

Ampatzoglou, C

I. Ampatzoglou, C. Collot, P. Germain, Derivation of the kinetic wave equation for quadratic dispe rsive problems in the inhomogeneous setting , to appear in Amer. J. Math. (2024)

work page 2024

[4] [3]

Ampatzoglou, J.K

I. Ampatzoglou, J.K. Miller, N. Pavlovi´ c, M. Taskovi´ c , Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted L∞ spaces, arXiv:2405.03984 (2024)

arXiv 2024

[5] [5]

B´ enyi, T

A. B´ enyi, T. Oh, The Sobolev inequality on the torus revisited , Publ. Math. Debrecen 83 (2013), no. 3, 359–374

work page 2013

[6] [6]

Buckmaster, P

T. Buckmaster, P. Germain, Z. Hani, J. Shatah, Onset of the wave turbulence description of the longtime behavior of the nonlinear Schr¨ odinger equation, Invent. Math. 225 (2021), no. 3, 787–855

work page 2021

[7] [7]

X. Chen, J. Holmer, Well/ill-posedness bifurcation for the Boltzmann equatio n with constant collision kernel, to appear in Ann. PDE

work page

[8] [8]

Collot, H

C. Collot, H. Dietert, P. Germain, Stability and cascades for the Kolmogorov-Zakharov spectru m of wave turbulence, Arch. Ration. Mech. Anal. 248 (2024), no. 1 , Paper No. 7, 31 pp

work page 2024