Derivation of 2D Vlasov-Poisson for classical particles with Coulomb interactions

Mitia Duerinckx; Pierre-Emmanuel Jabin

REVIEW 2 major objections 4 minor 25 references

Derivation of 2D Vlasov-Poisson for classical particles with Coulomb interactions

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that, for singular interaction forces in $L^{2d/(d+2\gamma)}_{\mathrm{loc}}$ — in particular the 2D Coulomb force — the $k$-particle marginals of the $N$-particle Liouville dynamics converge to $f^{\otimes k}$ in the…

desk verdict First cutoff-free 2D Coulomb mean-field derivation, but the advertised theorem rests on unverified regularity assumptions on the limiting Vlasov solution that need a serious look. read the letter →

arxiv 2608.04104 v1 pith:2HXUBBMM submitted 2026-08-04 math.AP math-phmath.MP

Mitia Duerinckx , Pierre-Emmanuel Jabin This is my paper · ORCID

classification math.APmath-phmath.MP MSC 35Q8382C4035B6582C22

keywords propagationofchaosmean-fieldlimitVlasov-PoissonCoulombinteractionBBGKYhierarchykineticregularizationhypoellipticestimatesaveraginglemma

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 proves a mean-field limit with propagation of chaos for a system of $N$ classical particles interacting through singular forces, in the regime where the limiting equation is Vlasov or Vlasov-Fokker-Planck. Its main theorem (Theorem 1.1) states that if the force kernel lies in $L^{2d/(d+2\gamma)}_{\mathrm{loc}}$ with the stated ranges of $\gamma$ — in particular the two-dimensional Coulomb force $K(x)\simeq x/|x|^2$, which is only in $L^{2,\infty}$ — then every fixed-order marginal $F_{N,k}(t)$ converges to the tensor product $f(t)^{\otimes k}$ in the sense of distributions, on any time interval where the limiting solution $f$ satisfies explicit regularity conditions. This is presented as the first derivation of the 2D Vlasov-Poisson equation from classical particle dynamics without any microscopic cutoff. The same statement covers Brownian particles ($\alpha>0$), arbitrary dimensions, and forces that need not be repulsive or potential-derived.

What carries the argument

The load-bearing object is the dual BBGKY hierarchy for rescaled Hoeffding correlations, equation (2.13): $(\partial_t + L^m_f)\bar C_{N,m} = R_{N,m} + \sqrt{(m+1)(m+2)}\, K_f[\bar C_{N,m+2}]$, where $L^m_f$ is the linearized mean-field operator and $R_{N,m}$ contains all terms with velocity derivatives, formally $O(N^{-1/2})$. The mechanism that makes the hierarchy tractable is kinetic regularization: hypoelliptic regularity (Lemma 3.1) in the diffusive case and a four-particle averaging lemma with a doubling argument (Lemma 4.1, Proposition 4.3) in the non-diffusive case give fractional spatial regularity $|\nabla_x|^\gamma$ on the weighted correlations, replacing the unavailable tensorized regularity and allowing the limit in $K_f$, defined as $K_f(z,z') = (K(x-x') - (K*f)(x))\cdot\nabla_v\log f(z)$. The triangular structure of the limiting hierarchy plus the propagator estimate of Lemma 5.5 closes uniqueness.

What would settle it

For the 2D Coulomb kernel and a sequence of $N$-particle data satisfying (1.3), construct the backward dual observable $\Phi_N$ from (2.8) with a fixed bounded $h$ and compute the quantity $N\int_0^T\int_{D^2} K_f\, \Pi_{N,2}[\Phi_N]\, f^{\otimes2}$ (as in (2.5)); Lemma 2.1 shows that if this quantity has a nonzero limit then $F_{N,k}(T)$ cannot converge to $f(T)^{\otimes k}$, which would refute Theorem 1.1 for that data.

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Extended reading notes

Core claim

The central discovery is that the locally-square-integrable threshold $L^2_{\mathrm{loc}}$ of the earlier dual hierarchical method can be exceeded by exploiting kinetic regularization. The proof works with the backward dual Liouville equation, decomposes bounded observables into Hoeffding components, rescales the second-order correlations as $\bar C_{N,m}=\sqrt{\binom{N}{m}}\,\Pi_{N,m}[\Phi_N]$, and derives a BBGKY-type hierarchy for them. Hypoelliptic estimates in the diffusive case ($\gamma<1/6$) and velocity-averaging estimates in the non-diffusive case ($\gamma\le \gamma_*$, e.g. $1/9+O(1/d)$ in $d\ge 2$) convert losses of velocity derivatives into fractional spatial regularity, which is enough to give meaning to the singular interaction operator $K_f$ and to pass to the limit. Uniqueness of the limiting hierarchy is obtained not by tensorized Sobolev regularity but by an explicit Duhamel expansion that exploits the triangular structure of the hierarchy and the time-integrated kinetic regularity of the propagator. Together with the weak chaos assumption (1.3), this yields the distributional convergence $F_{N,k}(t)\to f(t)^{\otimes k}$.

Load-bearing premise

The load-bearing premise is that a weak solution $f$ of the Vlasov(-Fokker-Planck) equation satisfies the stated regularity assumptions — positive on the whole time interval, with weighted bounds on $f^\delta\nabla_v\log f$ and related velocity Hessians — because the paper neither proves nor cites existence of such solutions for the 2D Coulomb problem.

Editorial extensions

If this is right

The 2D Coulomb force $K(x)\simeq \pm x/|x|^2$ is covered, yielding the first cutoff-free derivation of 2D Vlasov-Poisson in the deterministic case $\alpha=0$.
The result applies without repulsiveness or potential structure, and under an $L^2$ relative-entropy-type chaos condition weaker than exact tensorization.
In the diffusive case $\alpha>0$, the theorem extends the known short-time Vlasov-Poisson-Fokker-Planck derivation to longer times under weaker initial assumptions.
The convergence holds globally in time whenever the limiting mean-field solution satisfies the stated regularity bounds; the statement is not restricted to a short time horizon.
The same convergence statement covers Brownian particles ($\alpha>0$) as well as deterministic Newtonian dynamics.

Reading between the lines

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

Because the theorem is conditional on $f$ satisfying (1.6)-(1.9), the decisive open step for an unconditional no-cutoff derivation is to establish those weighted bounds for global 2D Coulomb solutions; the paper does not do this.
The proof's quantitative estimates suggest the admissible exponent $\gamma$ in Propositions 3.3 and 4.3 is not optimal: Remark 5.3 indicates well-posedness of the limiting hierarchy up to $\gamma=5/6$ (diffusive) and $\gamma=1/2$ (deterministic), so sharper kinetic estimates could widen the range of kernels.
The triangular Duhamel uniqueness argument, which sidesteps the unavailable tensorized Sobolev regularity, is a transferable mechanism for other hierarchies where only one-particle kinetic regularity is available, such as quantum mean-field hierarchies.

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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

The thing to know: this paper really does prove propagation of chaos for the 2D Coulomb force without a microscopic cutoff, provided the limiting Vlasov solution satisfies the regularity conditions in Section 1.2. That is a genuine first, and the machinery is serious. The kinetic regularization ideas—hypoellipticity for α>0 and velocity averaging plus a doubling/mollification argument for α=0—are new and well matched to the difficulty. The estimates are explicit and the paper is honest about the restrictions on γ, including where the restrictions come from and where the method loses ground. Credit is due for that.

The soft spot is exactly the one the stress-test flags, and it is load-bearing. Condition (1.6), f^{-1} ∈ L∞_loc, is not a harmless technicality. For compactly supported f, 1/f blows up at the support boundary, so every compactly supported initial datum is excluded. For full-support data like Maxwellians, the condition is plausible locally, but the weighted velocity-log-derivative bounds (1.7)–(1.9) are not implied by standard 2D Vlasov-Poisson global regularity. The paper neither proves nor cites an existence theorem for solutions satisfying these hypotheses. So the advertised "first derivation" is conditional on a class of solutions that may be smaller than the framing suggests—possibly nonempty, but unverified in the text.

I do not see a fatal flaw in the proof itself. The hierarchy estimates check out at the level of detail given, the Duhamel/iteration uniqueness argument is coherent, and the dependence on the companion preprint [7] is structural but clearly stated. The weak point is that the theorem's hypotheses are not just strong; they are unexamined. A referee should ask: does any nontrivial 2D Coulomb Vlasov solution satisfy (1.6)–(1.9) for a positive time? If yes, the paper is a milestone. If not, the main result is vacuous.

My recommendation: send this to peer review, not desk reject. The proof framework deserves referee time and the missing existence/regularity question is exactly what a good referee should demand. I would not cite the main theorem as a black box until that regularity class is populated, but I would bring it to reading group and would want the authors to address the gap in revision.

Referee Report

2 major / 4 minor

Summary. The paper claims the first cutoff-free derivation of the 2D Vlasov-Poisson equation for classical particles with Coulomb interactions. Starting from the N-particle Liouville equation with interaction kernel K (deterministic case α=0 and Brownian case α>0), the authors prove that the k-particle marginals converge to the k-fold tensor product of a solution f of the Vlasov(-Fokker-Planck) equation, in the sense of distributions, for every fixed k. The proof uses the dual hierarchical method introduced in the companion preprint [7], combined with new kinetic regularity estimates: hypoelliptic regularity in the diffusive case and velocity averaging in the non-diffusive case. The result is stated for kernels K ∈ L^{2d/(d+2γ)}_{loc} with a small positive γ, including the 2D Coulomb kernel, and is conditional on strong regularity assumptions on the limiting mean-field solution f, namely (1.5)-(1.9). The paper also contains a uniqueness result for the limiting dual hierarchy, based on its triangular structure and an explicit Duhamel expansion.

Significance. If the theorem is correct, it is a major advance: it would remove the microscopic cutoff for the 2D Coulomb interaction, a threshold that had remained open for singular kinetic mean-field limits. The methodological contribution is also valuable: the use of kinetic regularization effects within the dual hierarchy, and the triangular-structure uniqueness argument, are genuine innovations. The proof is structural and parameter-free, with explicit estimates in Propositions 3.3 and 4.3. However, the advertised conclusion is conditional on a regularity class for the limiting Vlasov solution that is neither proved nor cited to be nonempty for 2D Coulomb data. Until that gap is addressed, the significance of the 'first derivation' claim remains uncertain; the paper's lasting contribution may be the technical framework rather than the unconditional theorem as stated.

major comments (2)

[§1.2, Theorem 1.1 and assumptions (1.6)-(1.9)] The theorem's regularity assumptions on the limiting mean-field solution are load-bearing and are not known to be satisfiable in the advertised Coulomb case. Condition (1.6) requires f^{-1} ∈ L∞_loc; for any compactly supported f this fails on compact sets intersecting the complement of the support, so the standard global well-posedness class for 2D Vlasov-Poisson is excluded. For full-support data such as Maxwellians, no cited theorem establishes the weighted bounds (1.7)-(1.9), which involve derivatives of the transport flow and can grow in time. Since the abstract claims the first derivation of 2D Vlasov-Poisson without cutoff, the paper should either prove or cite an existence theorem for nontrivial Coulomb initial data satisfying (1.5)-(1.9), or explicitly state that the result is conditional and withdraw the unconditional phrasing of the claim.
[§2.2, Lemmas 2.1-2.3 and the companion preprint [7]] The proof imports the entire weak duality solution theory from the unpublished companion preprint [7]: the definition of 'global weak duality solution' of the Liouville equation (1.1) in the sense of [7, Appendix], the duality relation (2.10), the a priori estimates of Lemma 2.2, and the hierarchy for dual correlations in Lemma 2.3. Since [7] is described as covering kernels in L^2_loc, while Theorem 1.1 requires K ∈ L^{2d/(d+2γ)}_loc with γ>0 (the case that includes Coulomb), the manuscript does not establish that the assumed weak duality solutions exist for the singular kernels under consideration. The relevant definitions and statements from [7] should be reproduced or precisely cited, and their applicability to the larger singularity class should be justified within this paper.

minor comments (4)

[§5.1, proof of Lemma 5.1, Step 2] In the diffusive case α>0, the text says the kinetic regularity estimate (5.2) follows from Proposition 4.3; this should refer to Proposition 3.3.
[References, [7]] The author list of reference [7] is inconsistent with the text: the manuscript refers to 'Bresch, Duerinckx, and Jabin', but the reference lists 'M. Bresch, D. Duerinckx and P.-E. Jabin'; the initials should be corrected.
[§3, equation (3.2)] The summation in the display for the Laplacian term contains a typographical 'n' instead of 'm' in 'α nX_{i=1} △_{v_i}'.
[Abstract and §1.2] The phrase 'first derivation' in the abstract is stronger than what the theorem supports; since the theorem is conditional on the unverified regularity conditions (1.5)-(1.9), the abstract should be rephrased to avoid implying an unconditional result.

Circularity Check

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No significant circularity: the derivation is a conditional proof from the N-particle Liouville equation to Vlasov, with regularity hypotheses on f rather than fitted inputs.

full rationale

The paper's central claim, Theorem 1.1, is a conditional propagation-of-chaos result: for any weak solution f of the Vlasov equation satisfying the stated regularity conditions (1.5)–(1.9), the k-particle marginals of a global weak duality solution of the N-particle Liouville equation converge to f^{⊗k} in the distributional sense. The regularity conditions on f are hypotheses of the theorem, not quantities fitted to particle data, and the conclusion does not feed back into them. No parameter is tuned to force the target convergence: the only small parameter chosen in the proof is the mollification scale ε=N^{-θ} in Proposition 4.3, which is an analytic device for the dual hierarchy, not a microscopic cutoff, and the estimates are uniform in N before taking limits. The proof does rely on the authors' prior framework [7] for the dual reformulation, a priori bounds, and the dual BBGKY hierarchy, but these are cited computational lemmas whose assumptions (K∈L^2_loc) do not include the target result (2D Coulomb belongs only to L^{2,∞}), and the new kinetic-regularity estimates, the uniqueness of the limiting hierarchy, and the passage to the limit are proven here. The cited weak-duality-solution notion is a definitional framework from [7, Appendix], not an imported uniqueness theorem used to forbid alternatives. The skeptic's concern that no nontrivial 2D Coulomb Vlasov solution is known to satisfy (1.5)–(1.9) is a substantive existence/regularity gap that could affect the applicability of the theorem, but it is not a circularity: the theorem's statement is conditional, and the derivation from the particle system to the (assumed regular) mean-field limit is self-contained once those hypotheses hold. The use of the authors' own earlier work is transparent self-citation and is not load-bearing in a circular sense, because the earlier work does not already contain the 2D Coulomb result. Accordingly, the paper does not exhibit any step in which a prediction is equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the companion preprint [7] for the dual hierarchy machinery and on the conditional regularity hypotheses on f. No free parameters are fitted to data; the exponents γ and δ are part of the assumptions, and the range of γ is a limitation, not a fitted value. No invented entities appear.

assumptions (4)

domain assumption The companion framework of [7] supplies the notion of global weak duality solution to the singular Liouville equation (1.1), the derivation of the dual hierarchy (Lemma 2.3), and the a priori estimate Lemma 2.2.
Invoked in Sections 2 and 5.1; the paper does not re-prove these results and they are central. Appears via [7, Appendix], cited in Theorem 1.1 and Lemma 2.1.
domain assumption The limiting Vlasov(-Fokker-Planck) solution f satisfies the regularity conditions (1.5)-(1.9), in particular f^{-1} in L∞_loc and the weighted log-derivative bounds, with the lower bound (1.8) when α>0.
These are hypotheses of Theorem 1.1 and enter Propositions 3.3, 4.3, and 5.2. The paper does not prove existence of such f for the 2D Coulomb case, so the main result is conditional.
domain assumption The interaction kernel K is odd, belongs to L^{2d/(d+2γ)}_loc(R^d), and is bounded at infinity; in particular the 2D Coulomb force K(x)≈±x/|x|^2 is included.
Assumption in Theorem 1.1; this class defines the result's scope and is not derived.
standard math Standard kinetic analysis tools: hypoelliptic regularity (Lemma 3.1), L^2 averaging lemma (Lemma 4.1), Sobolev embedding H^γ ⊂ L^{2d/(d-2γ)}, and Schur test for kernel bounds.
The kinetic lemmas are proved in Appendix A; the embedding and Schur test are invoked throughout Sections 3-5. They are standard background, not part of the target result.

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Pith. "Pith review of Derivation of 2D Vlasov-Poisson for classical particles with Coulomb interactions." pith.science (2026). https://pith.science/paper/2HXUBBMM

@misc{pith2026260804104,
  author       = {Pith},
  title        = {Pith review of: Derivation of 2D Vlasov-Poisson for classical particles with Coulomb interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2HXUBBMM}},
  note         = {Machine review of arXiv:2608.04104}
}

abstract

We obtain the first derivation of the 2D Vlasov-Poisson equation for classical particles with Coulomb interactions without any microscopic cutoff. The proof relies on our recent dual hierarchical approach to mean-field limits, together with a refined analysis of dual BBGKY hierarchies on linearized correlations, based on kinetic regularization effects. The result holds more generally for arbitrary singular interaction forces $K\in L^{2-\eta}_{loc}$ with $\eta>0$ small enough, in any dimension, and it extends to Brownian particles. It holds globally in time as long as the mean-field solution is regular enough.

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Works this paper leans on

25 extracted references · 22 canonical work pages

[7]

Bresch, D

M. Bresch, D. Duerinckx and P.-E. Jabin. A duality method for mean-field limits with singular inter- actions. Preprint, arXiv:2402.04695

arXiv
[1]

Bardos, F

C. Bardos, F. Golse, and N. Mauser. Weak coupling limit of then-particle Schrödinger equation. Methods Appl. Anal., 7, 2000

work page 2000
[2]

Boers and P

N. Boers and P. Pickl. On mean field limits for dynamical systems.Journal of Statistical Physics, 164(1):1–16, 2016

2016
[3]

F. Bouchut. Hypoelliptic regularity in kinetic equations.J. Math. Pures Appl., 81:1135–1159, 2002

work page 2002
[4]

Braun and K

W. Braun and K. Hepp. The Vlasov dynamics and its fluctuations in the1/nlimit of interacting classical particles.Communications in Mathematical Physics, 56(2):101–113, 1977

work page 1977
[5]

Bresch, P.-E

D. Bresch, P.-E. Jabin, and J. Soler. A new approach to the mean-field limit of Vlasov-Fokker-Planck equations.Anal. PDE, 18(4):1037–1064, 2025

work page 2025
[6]

Bresch, P.-E

D. Bresch, P.-E. Jabin, and Z. Wang. Modulated free energy and mean field limit.Séminaire Laurent Schwartz—EDP et applications, pages 1–22, 2019–2020

work page 2019
[8]

J. A. Carrillo, Y.-P. Choi, and S. Salem. Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off.Communications in Contemporary Mathematics, 21(4):1850039, 2019

work page 2019

Show all 25 references

[9]

R. L. Dobrushin. Vlasov equations.Functional Analysis and Its Applications, 13(2):115–123, 1979

1979
[10]

Duerinckx and P.-E

M. Duerinckx and P.-E. Jabin. Singular mean-field limits for fluctuations around equilibrium. Preprint, arXiv:2605.28979

arXiv
[11]

Erdös and H.-T

L. Erdös and H.-T. Yau. Derivation of the nonlinear Schrödinger equation from a many body Coulomb system.Adv. Theor. Math. Phys., 5, 2001

2001
[12]

Feistl-Held and P

M. Feistl-Held and P. Pickl. On the mean-field limit for the Vlasov–Poisson system. Preprint, arXiv:2504.01471

arXiv
[13]

Feistl-Held and P

M. Feistl-Held and P. Pickl. On the mean-field limit for the Vlasov–Poisson system in two dimensions. Preprint, arXiv:2509.17821
[14]

Hauray and P.-E

M. Hauray and P.-E. Jabin.n-particles approximation of the Vlasov equations with singular potential. Archive for Rational Mechanics and Analysis, 183(3):489–524, 2007. 36 M. DUERINCKX AND P.-E. JABIN

2007
[15]

Hauray and P.-E

M. Hauray and P.-E. Jabin. Particle approximation of Vlasov equations with singular forces: propa- gation of chaos.Annales Scientifiques de l’École Normale Supérieure, 48(4):891–940, 2015

2015
[16]

Huang, J.-G

H. Huang, J.-G. Liu, and P. Pickl. On the mean-field limit for the Vlasov–Poisson–Fokker–Planck system.Journal of Statistical Physics, 181(5):1915–1965, 2020

1915
[17]

Jabin, H.-Y

P.-E. Jabin, H.-Y. Lin, and E. Tadmor. Commutator method for averaging lemmas.Anal. PDE, 15(6):1561–1584, 2022

2022
[18]

Jabin and Z

P.-E. Jabin and Z. Wang. Mean field limit and propagation of chaos for Vlasov systems with bounded forces.Journal of Functional Analysis, 271(12):3588–3627, 2016

2016
[19]

Lazarovici and P

D. Lazarovici and P. Pickl. A mean field limit for the Vlasov–Poisson system.Archive for Rational Mechanics and Analysis, 225(3):1201–1231, 2017

2017
[20]

Neunzert and J

H. Neunzert and J. Wick. Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen. In R. Ansorge and W. Törnig, editors,Numerische Behandlung nichtlinearer Integrodifferential- und Differentialgleichungen, volume 395 ofLecture Notes in Mathem...

1974
[21]

Q. H. Nguyen and S. Serfaty. Singular mean-field limits via a multiscale mollification metric. Preprint, arXiv:2607.10686

arXiv
[22]

T. Paul, M. Pulvirenti, and S. Simonella. On the Size of Chaos in the Mean Field Dynamics.Arch. Ration. Mech. Anal., 231(1):285–317, 2019

2019
[23]

S. Serfaty. Mean field limit for coulomb-type flows.Duke Mathematical Journal, 169(15):2887–2935,
[24]

Sznitman

A.-S. Sznitman. Topics in propagation of chaos. InÉcole d’Été de Probabilités de Saint-Flour XIX— 1989, volume 1464 ofLecture Notes in Math., pages 165–251. Springer, Berlin, 1991. M. Duerinckx. Université Libre de Bruxelles, Département de Mathématiques, 1050 Brussels, Belgiu...

1989
[2020]

Duerinckx

With an appendix with M. Duerinckx

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [7]

Bresch, D

M. Bresch, D. Duerinckx and P.-E. Jabin. A duality method for mean-field limits with singular inter- actions. Preprint, arXiv:2402.04695

arXiv

[2] [1]

Bardos, F

C. Bardos, F. Golse, and N. Mauser. Weak coupling limit of then-particle Schrödinger equation. Methods Appl. Anal., 7, 2000

work page 2000

[3] [2]

Boers and P

N. Boers and P. Pickl. On mean field limits for dynamical systems.Journal of Statistical Physics, 164(1):1–16, 2016

2016

[4] [3]

F. Bouchut. Hypoelliptic regularity in kinetic equations.J. Math. Pures Appl., 81:1135–1159, 2002

work page 2002

[5] [4]

Braun and K

W. Braun and K. Hepp. The Vlasov dynamics and its fluctuations in the1/nlimit of interacting classical particles.Communications in Mathematical Physics, 56(2):101–113, 1977

work page 1977

[6] [5]

Bresch, P.-E

D. Bresch, P.-E. Jabin, and J. Soler. A new approach to the mean-field limit of Vlasov-Fokker-Planck equations.Anal. PDE, 18(4):1037–1064, 2025

work page 2025

[7] [6]

Bresch, P.-E

D. Bresch, P.-E. Jabin, and Z. Wang. Modulated free energy and mean field limit.Séminaire Laurent Schwartz—EDP et applications, pages 1–22, 2019–2020

work page 2019

[8] [8]

J. A. Carrillo, Y.-P. Choi, and S. Salem. Propagation of chaos for the Vlasov–Poisson–Fokker–Planck equation with a polynomial cut-off.Communications in Contemporary Mathematics, 21(4):1850039, 2019

work page 2019