Weak-strong uniqueness for the Landau equation by a relative entropy method

C\^ome Tabary

REVIEW 1 major objections 4 minor 1 cited by

Weak-strong uniqueness for the Landau equation by a relative entropy method

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Relative entropy locks weak and strong Landau solutions together

desk verdict Solid new relative-entropy weak-strong uniqueness for Landau soft potentials, with a repairable gap in the stated Gronwall rate. read the letter →

arxiv 2505.21120 v1 pith:HTBXZ2LR submitted 2025-05-27 math.AP

Côme Tabary This is my paper · ORCID

classification math.AP MSC 35Q2035A0282C4035B65

keywords LandauequationCoulombpotentialsoftpotentialsweak-stronguniquenessrelativeentropyH-solutionsproductionlogarithmicbounds

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 that no weak solution of the space-homogeneous Landau equation can drift away from a sufficiently regular solution. The tool is relative entropy: the paper shows that the relative entropy between any H-solution $f$ and a classical solution $g$ stays bounded by its initial value times an exponential factor whose rate is controlled by the polynomial growth of $\nabla \ln g$ and $\nabla^2 \ln g$. In particular, if $f$ and $g$ start from the same data, they coincide for all times. This gives a weak-strong uniqueness principle in the soft-potentials range $\gamma \in [-3,0)$, including the physically important Coulomb case $\gamma = -3$, where classical uniqueness for very weak solutions is otherwise out of reach. A second theorem supplies explicit initial-data conditions under which such a strong solution exists, so the principle is not vacuous.

What carries the argument

The argument is carried by the relative entropy $H(f|g)=\int f \ln(f/g)$ and by the identity $\ln g$ as a legitimate test function in the weak formulation. The evolution of $H(f_t|g_t)$ splits into a good term $G$, a relative version of entropy production, and a bad term $B$ that is a 'tensorization defect' quadratic in $(f-g)(f'-g')$. A cut-off version of the Landau kernel $a(z)=|z|^{\gamma+2}\Pi(z)$ is elliptic after convolution with $f$, so $G$ controls a weighted relative Fisher information $\int |\nabla \ln(f/g)|^2 \langle v\rangle^\gamma f\,dv$; the bad term is then absorbed by that same Fisher information up to an error proportional to $H(f|g)$. Gronwall's lemma closes the estimate. The logarithmic bounds on $g$ are exactly what makes $\ln g$ admissible as a test function and keeps the error integrable.

What would settle it

A concrete way to test the claim is to simulate the space-homogeneous Landau-Coulomb equation from a Maxwellian-tailed initial datum satisfying the hypotheses of Theorem 1.2, run a very weak numerical scheme beside a high-accuracy classical one from the same datum, and measure $H(f_t|g_t)$. Finding a positive time where the relative entropy crosses the exponential bound in Theorem 1.1 would falsify the estimate; finding two distinct H-solutions from the same initial datum would falsify its uniqueness corollary.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for $\gamma \in [-3,0)$, if $g$ is a classical solution whose logarithm satisfies the polynomial growth bounds $|\nabla \ln g_t| \leq K_{g1}(t)\langle v\rangle^\kappa$, $|\partial_t \ln g_t| \leq K_{g2}(t)\langle v\rangle^\nu$ and $\|\nabla^2 \ln g_t\| \leq K_{g3}(t)\langle v\rangle^\zeta$ with $\int_0^T (K_{g1}^2 + K_{g2} + K_{g3}^2)dt$ finite, and $f$ is any H-solution with enough initial moments, then $$H(f_t|g_t) \leq H(f_0|g_0) \exp\Bigl(C \int_0^t \bigl(\|\langle v\$rangle^{{-\kappa}}$ \nabla \ln g_s\|_\$infty^{2}$ + \|\langle v\$rangle^{{-\zeta}}$ \$nabla^{2}$ \ln g_s\|_\$infty^{2}$\bigr) ds\Bigr).$$ Taking $f_0 = g_0$ gives uniqueness of H-solutions inside the class of solutions that admit a sufficiently regular one. The paper also proves Theorem 1.2, showing for very soft potentials $\gamma \in [-3,-2]$ that Maxwellian-tailed initial data with a locally $C^{2+\delta}$ logarithm generate a classical solution satisfying the needed bounds, so the hypothesis is actually reachable.

Load-bearing premise

The load-bearing premise is that the smooth solution's logarithm has controlled growth: its first and second derivatives in velocity may grow like a fixed power of $\langle v\rangle$, and the squared norms of those two derivatives must be integrable in time; if these bounds fail, the relative-entropy estimate is not proved.

Editorial extensions

If this is right

If a sufficiently regular classical solution exists, all H-solutions with the same initial data coincide with it: weak-strong uniqueness holds in the soft-potentials range including Coulomb interactions.
The estimate is quantitative: initial closeness in relative entropy (hence in $L^1$ by Pinsker's inequality) is propagated with at most exponential-in-time growth, giving a stability statement, not just uniqueness.
For $\gamma = -3$, Theorem 1.2 applies to Maxwellian-tailed $C^3$ initial data and H-solutions with a moment of order strictly greater than 31, so the uniqueness conclusion covers a concrete class of very weak solutions in the Coulomb case.
No probabilistic interpretation, Wasserstein coupling, or extra $L^p$ integrability of the weak solution is needed; only the entropy-production estimate built into the notion of H-solution.

Reading between the lines

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

The time-derivative bound on $\ln g$ appears only in passing to the limit when $\ln g$ is used as a test function, and it does not show up in the final estimate, so a better approximation argument should be able to remove it and enlarge the class of strong solutions covered.
The same relative-entropy strategy may transfer to other collision kernels whose matrix $a(z)$ is non-negative and elliptic after convolution, whenever the strong solution's logarithm satisfies analogous tail bounds.
The moment threshold in the Coulomb case (order strictly greater than 31) is likely not optimal: since the proof tracks polynomial weights through Schauder estimates rather than optimizing them, a finer handling of the constants could lower the required moment while keeping the same mechanism.

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

This is a serious, well-written paper. It is the first relative entropy weak-strong uniqueness for the Landau equation in the soft potentials range, including Coulomb, and the main computation (Lemma 2.4) is clean. The split into a good term (relative Fisher information) and a bad term (tensorization defect) is natural, and the appendix Schauder estimates with explicit polynomial weights are a real service. The paper deserves a careful referee.

The soft spot is in Proposition 2.12. Combining Lemmas 2.9 and 2.11 gives a differential inequality whose H-coefficient includes a constant term: C[(1+K_{g1}^2)(M_f+M_g) + c0^{-1}(K_{g1}^2+K_{g3}^2)(M_f^2+M_g^2)]. The text then drops the '+1' and concludes the rate exp(C∫(K_{g1}^2+K_{g3}^2)). That is not justified as written. The correct conclusion is exp(Ct + C∫(K_{g1}^2+K_{g3}^2)) unless one adds a lower bound on K_{g1}. The uniqueness corollary is unaffected because H(f_0|g_0)=0 kills any prefactor, and the gap is probably repairable: for a probability density, ∫ v·∇ ln g g dv = -3, so K_{g1}(t) cannot be too small in terms of moments. But that argument is absent, and a referee should ask for it.

The other limitations are acknowledged by the author and are not credibility problems. The time-derivative condition (7) is admittedly too strong and only used to pass to the limit; the moment threshold of order >31 in the Coulomb case is far from sharp. The paper is honest about these, and the main theorem is stated as a conditional result.

My recommendation: send to peer review. The idea is new, the proof is largely self-contained, and the gap is localized and fixable. With a corrected Gronwall step or a modified statement, this is a solid contribution worth publishing.

Referee Report

1 major / 4 minor

Summary. This paper establishes a weak-strong stability and uniqueness principle for the space-homogeneous Landau equation in the soft-potential range γ ∈ [−3,0), including Coulomb interactions. The distance between an H-solution f and a classical solution g is measured by relative entropy. Under conditional logarithmic growth bounds on ∇ ln g, ∂t ln g, and ∇² ln g, Theorem 1.1 proves H(ft|gt) ≤ H(f0|g0) exp(C ∫0^t (∥⟨v⟩^{−κ}∇ ln g_s∥_{L∞}² + ∥⟨v⟩^{−ζ}∇² ln g_s∥_{L∞}²) ds). The proof is organized as a relative-entropy production identity, a control of the bad term by a weighted relative Fisher information plus a relative-entropy error, and a Gronwall argument. Theorem 1.2 gives sufficient initial-data conditions in the very soft Coulomb-range case, propagating Maxwellian bounds and C^{2+δ} regularity of ln g0 to the logarithmic bounds; the parabolic Schauder estimates and maximum principle are proved in appendices.

Significance. If the proof is repaired at one point, the result is a significant contribution: it provides the first relative-entropy-based stability estimate for soft-potential Landau, gives weak-strong uniqueness in the class of H-solutions, and avoids circularity by making the logarithmic bounds hypotheses in Theorem 1.1 and independently verifying them in Theorem 1.2. The paper is careful with constants, states its limitations (the time-derivative bound is acknowledged as too strong), and contains self-contained proofs of the auxiliary parabolic estimates. These strengths make the main claim credible.

major comments (1)

[§2.4, Proposition 2.12] The Gronwall step is not proved as written. After combining Lemma 2.9 and Lemma 2.11, the coefficient of H(fs|gs) is C[(1+K_g1²)(M_f+M_g) + c0^{-1}(K_g1²+K_g3²)(M_f²+M_g²)], and the text then asserts that this is ≤ C(K_g1²+K_g3²). That inequality is false for arbitrary bounded K's; taking K_g1=K_g3=0 makes the left-hand side positive and the right-hand side zero. Consequently the rate exp(C∫(K_g1²+K_g3²)) stated in Proposition 2.12 and Theorem 1.1 is not established by the displayed argument. The gap is repairable: for a probability density g, ∫ v·∇ ln g g dv = −3, and the moment bounds imply K_g1(t) ≥ c(M_g) > 0 pointwise, so 1 ≤ C' K_g1² and the constant can be absorbed into K_g1² up to a larger constant. This argument should be inserted; alternatively the rate should be stated with an additional exp(Ct) factor. Because this step is the final estimate on which the main theorem rests, it is load-bearing.

minor comments (4)

[§2.3, Lemma 2.9] The statement of Lemma 2.9 contains K_g2 in the error term, but the proof and all later applications require K_g1, the gradient bound. The displayed inequality (14) should be corrected to use K_g1.
[§3.2, Lemma 3.3] The stated formula for p appears inconsistent with the proof and with Remark 3.4. The proof yields p = −3γ/(γ+5), and the extra '−γ' in the statement should be removed.
[Theorem 1.2] The statement says 'an associated weak solution', but the proof invokes Theorem 1.1, which requires f to be an H-solution. The theorem should explicitly state that f is an H-solution, or the conclusion should be restricted accordingly.
[Throughout] There are several typographical errors, including repeated 'Thereom' for 'Theorem' and 'wether' for 'whether'. These should be corrected in a final polish.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.1 is a conditional estimate whose hypotheses on the strong solution are not derived from the target conclusion, and Theorem 1.2 verifies those hypotheses from initial data using independent PDE results.

full rationale

The derivation is self-contained and conditional rather than circular. Theorem 1.1 states an estimate for H(ft|gt) under explicit hypotheses (6)-(8) on the strong solution g; those hypotheses are assumptions about g's logarithmic derivatives, not consequences of the estimate being proved. The proof computes the relative-entropy evolution in Lemma 2.4, separates it into a non-negative good term G and a bad term B, and then bounds B by the refined Pinsker inequality plus the assumed gradient and Hessian controls (Lemmas 2.7, 2.11). No parameter appearing in the exponential rate is fitted to the data whose difference is predicted; K_g1 and K_g3 enter as upper bounds on g, not as fitted constants. The ellipticity lemma (2.8) and entropy-production estimate (2.2) are quoted from external sources [15] and [8], and the smooth solution used in Theorem 1.2 is supplied by Guillen-Silvestre [17] with Schauder/Holder tools [19], none of which is authored by or unique to this paper. Taking f0=g0 gives H(f0|g0)=0, so uniqueness follows from the same quantitative estimate without invoking any imported uniqueness theorem. The paper's own remarks that the time-derivative assumption is 'too strong' and that the initial-data regularity is 'still too strong' are candid applicability caveats, not circular moves. One non-circular caveat: in Proposition 2.12 the text drops the '+1' from (1+K_g1^2) when passing to Gronwall, which is not justified by the displayed estimates and looks like a repairable quantitative gap; because this concerns the correctness of the stated rate rather than a reduction of the conclusion to its inputs, it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data; all constants are explicit functions of the hypotheses. The axioms listed are either standard results from the cited literature (entropy production, moment propagation, ellipticity, Guillen-Silvestre existence) or the explicit hypotheses of the theorem (logarithmic tail control of the strong solution). The paper introduces no new physical entities.

assumptions (6)

domain assumption The weak solution f is a Villani H-solution, i.e., it satisfies the entropy production estimate (5) and the weak formulation (Definition 1.5).
This is the class of weak solutions the theorem applies to; the entropy production estimate is used throughout Section 2 to give meaning to grad ln f and to control terms via the entropy inequality.
standard math The entropy production estimate of Desvillettes (Lemma 2.2): integral |grad sqrt(f)|^2 <v>^gamma dv <= c(D(f)+1) holds for probability densities with finite energy and entropy.
Invoked in Section 2.1 to deduce that grad sqrt(f) lies in a weighted L2 space and grad f is a weighted L1 function, which justifies the manipulations with gradients of logarithms.
standard math The ellipticity estimate for the cut-off kernel (Lemma 2.8): a_tilde star f >= c0 <v>^gamma Id, with c0 depending only on the energy and entropy of f0.
Used in Lemma 2.9 to extract a coercive relative Fisher information term from the good term G; this is a classical estimate (see [15, Lemma 2.1]).
domain assumption The assumption that the strong solution g satisfies the logarithmic tail control (6)-(8) with finite weighted norms over [0,T].
This is the main hypothesis of Theorem 1.1; it is needed to use ln g as a test function (Lemma 2.3) and to control the bad term B (Lemma 2.11). The author notes the time-derivative part is likely too strong.
standard math The moment propagation result for H-solutions (Lemma 2.1, from [4, Lemmas 7-8]): moments up to order rho-gamma are bounded on [0,T] by constants depending only on initial data, gamma, T, and rho.
Used in Lemmas 2.9 and 2.11 to bound error terms involving M_f and M_g.
standard math Guillen-Silvestre's theorem [17, Theorem 1.2] guarantees existence of a classical solution for initial data with finite Fisher information and Maxwellian-type bounds.
Used in Proposition 3.1 to build the strong solution g for Theorem 1.2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Weak-strong uniqueness for the Landau equation by a relative entropy method." pith.science (2026). https://pith.science/paper/HTBXZ2LR

@misc{pith2026250521120,
  author       = {Pith},
  title        = {Pith review of: Weak-strong uniqueness for the Landau equation by a relative entropy method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HTBXZ2LR}},
  note         = {Machine review of arXiv:2505.21120}
}

read the original abstract

We derive a weak-strong uniqueness and stability principle for the Landau equation in the soft potentials case (including Coulomb interactions). The distance between two solutions is measured by their relative entropy, which to our knowledge was never used before in stability estimates. The logarithm of the strong solution is required to have polynomial growth while the weak solution can be any H-solution with sufficiently many moments at initial time. Since we require a substantial amount of regularity on the strong solution, we also provide an example of sufficient conditions on the initial data that ensure this regularity in the Coulomb (and very soft potentials) case.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Kac's Program for the Landau Equation
math.AP 2025-06 conditional novelty 8.0 of 10

The k-particle velocity marginals of Kac's particle system converge to the factorized law of the Landau solution for all power-law potentials, including Coulomb collisions, in weak, Wasserstein, entropic, and strong L...

Reference graph

Works this paper leans on

33 extracted references · 22 canonical work pages · cited by 1 Pith paper

[1]

A Priori Estimates for Solu- tions to Landau Equation Under Prodi–Serrin Like Criteria

R. Alonso, V . Bagland, L. Desvillettes, and B. Lods. “A Priori Estimates for Solu- tions to Landau Equation Under Prodi–Serrin Like Criteria”. In:Archive for Rational Mechanics and Analysis 248.3 (May 2024), p. 42. DOI: 10 . 1007 / s00205 - 024 - 01992-y (cit. on pp. 4, 5)

work page 2024
[2]

Interior Schauder Estimates for Parabolic Differential- (or Difference-) Equations via the Maximum Principle

A. Brandt. “Interior Schauder Estimates for Parabolic Differential- (or Difference-) Equations via the Maximum Principle”. In: Israel Journal of Mathematics 7.3 (Sept. 1969), pp. 254–262. DOI: 10.1007/BF02787619 (cit. on pp. 26, 35)

work page doi:10.1007/bf02787619 1969
[3]

Global a Priori Estimates for the Inhomogeneous Landau Equation with Moderately Soft Potentials

Stephen Cameron, Luis Silvestre, and Stanley Snelson. “Global a Priori Estimates for the Inhomogeneous Landau Equation with Moderately Soft Potentials”. In:An- nales de l’Institut Henri Poincaré C, Analyse non linéaire 35.3 (May 2018), pp. 625–642. DOI: 10.1016/j.anihpc.2017.07.001 (cit. on p. 21)

doi:10.1016/j.anihpc.2017.07.001 2018
[4]

Estimates for the large time behavior of the Landau equation in the Coulomb case

Kleber Carrapatoso, Laurent Desvillettes, and Lingbing He. Estimates for the Large Time Behavior of the Landau Equation in the Coulomb Case. Oct. 2015. DOI: 10.48550/ arXiv.1510.08704. arXiv: 1510.08704 [math] (cit. on p. 8). 41

work page Pith review arXiv doi:10.48550/arxiv.1510.08704 2015
[5]

Relative Entropy Method for Particle Approximation of the Landau Equa- tion for Maxwellian Molecules

José Antonio Carrillo, Xuanrui Feng, Shuchen Guo, Pierre-Emmanuel Jabin, and Zhenfu Wang. Relative Entropy Method for Particle Approximation of the Landau Equa- tion for Maxwellian Molecules . Sept. 2024. DOI: 10.48550/arXiv.2408.15035 . arXiv: 2408.15035 [math] (cit. on pp. 3, 6)

arXiv doi:10.48550/arxiv.2408.15035 2024
[6]

Uniqueness of higher integrable solution to the Landau equation with Coulomb interactions

Jann-Long Chern and Maria Gualdani. Uniqueness of Higher Integrable Solution to the Landau Equation with Coulomb Interactions . July 2022. arXiv: 1910.06216 (cit. on pp. 4, 5)

work page Pith review arXiv 2022
[7]

The Second Law of Thermodynamics and Stability

C. M. Dafermos. “The Second Law of Thermodynamics and Stability”. In: Archive for Rational Mechanics and Analysis 70.2 (June 1979), pp. 167–179. DOI: 10.1007/ BF00250353 (cit. on p. 5)

work page 1979
[8]

Entropy Dissipation Estimates for the Landau Equation in the Coulomb Case and Applications

Laurent Desvillettes. Entropy Dissipation Estimates for the Landau Equation in the Coulomb Case and Applications . Oct. 2014. DOI: 10 . 48550 / arXiv . 1408 . 6025. arXiv: 1408.6025 [math] (cit. on pp. 4, 9)

arXiv doi:10.48550/arxiv.1408.6025 2014

Show all 33 references

[9]

Pro- duction of the Fisher Information for the Landau-Coulomb Equation with L1 Initial Data

Laurent Desvillettes, William Golding, Maria Pia Gualdani, and Amelie Loher. Pro- duction of the Fisher Information for the Landau-Coulomb Equation with L1 Initial Data . Oct. 2024. DOI: 10.48550/arXiv.2410.10765 . arXiv: 2410.10765 [math] (cit. on p. 5)

2024 doi arXiv
[10]

Uniqueness of Solutions to Hyperbolic Conservation Laws

Ronald J. Diperna. “Uniqueness of Solutions to Hyperbolic Conservation Laws”. In: Indiana University Mathematics Journal 28.1 (1979), pp. 137–188. JSTOR: 24892290 (cit. on p. 5)

1979
[11]

Particle Approximation of Some Landau Equations

Nicolas Fournier. Particle Approximation of Some Landau Equations . Nov. 2008. DOI: 10.48550/arXiv.0811.2688. arXiv: 0811.2688 (cit. on pp. 4, 5)

2008 doi arXiv
[12]

Uniqueness of Bounded Solutions for the Homogeneous Lan- dau Equation with a Coulomb Potential

Nicolas Fournier. “Uniqueness of Bounded Solutions for the Homogeneous Lan- dau Equation with a Coulomb Potential”. In: Communications in Mathematical Physics 299.3 (Nov. 2010), pp. 765–782. DOI: 10 . 1007 / s00220 - 010 - 1113 - 9. arXiv: 0909.0647 [math] (cit. on pp. 4, 5)

2010 arXiv
[13]

Propagation of Chaos for the Landau Equation with Moderately Soft Potentials

Nicolas Fournier and Maxime Hauray. Propagation of Chaos for the Landau Equation with Moderately Soft Potentials. Jan. 2015. arXiv: 1501.01802 [math] (cit. on p. 5)

2015 arXiv
[14]

Trudinger

David Gilbarg and Neil S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer Science & Business Media, Jan. 2001 (cit. on p. 41)

2001
[15]

François Golse, Cyril Imbert, Sehyun Ji, and Alexis F. Vasseur. Local Regularity for the Space-Homogeneous Landau Equation with Very Soft Potentials. Jan. 2024. DOI: 10. 48550/arXiv.2206.05155. arXiv: 2206.05155 [math] (cit. on pp. 15, 16)

2024 doi arXiv
[16]

Well-Posedness of the Spatially Homogeneous Landau Equation for Soft Potentials

Hélène Guerin and Nicolas Fournier. Well-Posedness of the Spatially Homogeneous Landau Equation for Soft Potentials. June 2008. DOI: 10.48550/arXiv.0806.3379. arXiv: 0806.3379 [math] (cit. on pp. 4, 5)

2008 doi arXiv
[17]

The Landau Equation Does Not Blow Up

Nestor Guillen and Luis Silvestre. The Landau Equation Does Not Blow Up. Nov. 2023. DOI: 10.48550/arXiv.2311.09420. arXiv: 2311.09420 [math] (cit. on pp. 5, 20)

2023 doi arXiv
[18]

C-Infinity Smoothing for Weak So- lutions of the Inhomogeneous Landau Equation

Christopher Henderson and Stanley Snelson. “C-Infinity Smoothing for Weak So- lutions of the Inhomogeneous Landau Equation”. In: (Nov. 2019). DOI: 10.1007/ s00205-019-01465-7 (cit. on pp. 6, 21, 28)

2019
[19]

Local Solutions of the Landau Equation with Rough, Slowly Decaying Initial Data

Christopher Henderson, Stanley Snelson, and Andrei Tarfulea. Local Solutions of the Landau Equation with Rough, Slowly Decaying Initial Data . Sept. 2019. DOI: 10. 48550/arXiv.1909.05914. arXiv: 1909.05914 [math] (cit. on pp. 6, 20, 28). 42

2019 doi arXiv
[20]

Kinetic Schauder Estimates with Time-Irregular Coefficients and Uniqueness for the Landau Equation

Christopher Henderson and Weinan Wang. “Kinetic Schauder Estimates with Time-Irregular Coefficients and Uniqueness for the Landau Equation”. In: Discrete and Continuous Dynamical Systems 44.4 (Mon Apr 01 04:00:00 UTC 2024), pp. 1026–

2024
[21]

The Schauder estimate in kinetic theory with application to a toy nonlinear model

Cyril Imbert and Clément Mouhot. “The Schauder estimate in kinetic theory with application to a toy nonlinear model”. In: Annales Henri Lebesgue 4 (2021), pp. 369–

2021
[22]

Dissipation Estimates of the Fisher Information for the Landau Equation

Sehyun Ji. Dissipation Estimates of the Fisher Information for the Landau Equation. Oct
[23]

Entropy Dissipation Estimates for the Landau Equation with Coulomb Po- tentials

Sehyun Ji. Entropy Dissipation Estimates for the Landau Equation with Coulomb Po- tentials. Aug. 2024. DOI: 10.48550/arXiv.2305.09841 . arXiv: 2305.09841 [math] (cit. on pp. 4, 16)

2024 doi arXiv
[24]

Lectures on Elliptic and Parabolic Equations in Hölder Spaces

N. Krylov. “Lectures on Elliptic and Parabolic Equations in Hölder Spaces”. In: vol. 12. Graduate Studies in Mathematics. Providence, Rhode Island: American Mathematical Society, Sept. 1996. DOI: 10.1090/gsm/012 (cit. on p. 41)

1996 doi
[25]

Lifšic and Lev P

Evgenij M. Lifšic and Lev P . Pitaevskij, eds. Physical Kinetics. Course of Theoretical Physics v. 10. Amsterdam Boston: Elsevier, 2008 (cit. on p. 2)

2008
[26]

From a Kac-like Particle System to the Lan- dau Equation for Hard Potentials and Maxwell Molecules

Nicolas Fournier and Arnaud Guillin. “From a Kac-like Particle System to the Lan- dau Equation for Hard Potentials and Maxwell Molecules”. In:Annales scientifiques de l’École normale supérieure 50.1 (2017), pp. 157–199. DOI: 10.24033/asens.2318 (cit. on p. 5)

2017 doi
[27]

Upper Bounds for Parabolic Equations and the Landau Equation

Luis Silvestre. Upper Bounds for Parabolic Equations and the Landau Equation . Aug
[28]

On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations

Cédric Villani. “On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations”. In: Archive for Rational Mechanics and Analysis 143.3 (Sept. 1998), pp. 273–307. DOI: 10.1007/s002050050106 (cit. on pp. 2, 4, 6)

1998 doi
[29]

Weak-Strong Uniqueness in Fluid Dynamics

Emil Wiedemann. Weak-Strong Uniqueness in Fluid Dynamics . May 2017. DOI: 10. 48550/arXiv.1705.04220. arXiv: 1705.04220 [math] (cit. on p. 6). 43

2017 doi arXiv
[405]

DOI: 10.5802/ahl.75 (cit. on p. 6)

doi
[1072]

DOI: 10.3934/dcds.2023137 (cit. on p. 6)

doi
[2016]

arXiv: 1511.03248 [math] (cit

DOI: 10.48550/arXiv.1511.03248. arXiv: 1511.03248 [math] (cit. on p. 21)

doi arXiv
[2024]

arXiv: 2410.09035 (cit. on p. 5)

arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

A Priori Estimates for Solu- tions to Landau Equation Under Prodi–Serrin Like Criteria

R. Alonso, V . Bagland, L. Desvillettes, and B. Lods. “A Priori Estimates for Solu- tions to Landau Equation Under Prodi–Serrin Like Criteria”. In:Archive for Rational Mechanics and Analysis 248.3 (May 2024), p. 42. DOI: 10 . 1007 / s00205 - 024 - 01992-y (cit. on pp. 4, 5)

work page 2024

[2] [2]

Interior Schauder Estimates for Parabolic Differential- (or Difference-) Equations via the Maximum Principle

A. Brandt. “Interior Schauder Estimates for Parabolic Differential- (or Difference-) Equations via the Maximum Principle”. In: Israel Journal of Mathematics 7.3 (Sept. 1969), pp. 254–262. DOI: 10.1007/BF02787619 (cit. on pp. 26, 35)

work page doi:10.1007/bf02787619 1969

[3] [3]

Global a Priori Estimates for the Inhomogeneous Landau Equation with Moderately Soft Potentials

Stephen Cameron, Luis Silvestre, and Stanley Snelson. “Global a Priori Estimates for the Inhomogeneous Landau Equation with Moderately Soft Potentials”. In:An- nales de l’Institut Henri Poincaré C, Analyse non linéaire 35.3 (May 2018), pp. 625–642. DOI: 10.1016/j.anihpc.2017.07.001 (cit. on p. 21)

doi:10.1016/j.anihpc.2017.07.001 2018

[4] [4]

Estimates for the large time behavior of the Landau equation in the Coulomb case

Kleber Carrapatoso, Laurent Desvillettes, and Lingbing He. Estimates for the Large Time Behavior of the Landau Equation in the Coulomb Case. Oct. 2015. DOI: 10.48550/ arXiv.1510.08704. arXiv: 1510.08704 [math] (cit. on p. 8). 41

work page Pith review arXiv doi:10.48550/arxiv.1510.08704 2015

[5] [5]

Relative Entropy Method for Particle Approximation of the Landau Equa- tion for Maxwellian Molecules

José Antonio Carrillo, Xuanrui Feng, Shuchen Guo, Pierre-Emmanuel Jabin, and Zhenfu Wang. Relative Entropy Method for Particle Approximation of the Landau Equa- tion for Maxwellian Molecules . Sept. 2024. DOI: 10.48550/arXiv.2408.15035 . arXiv: 2408.15035 [math] (cit. on pp. 3, 6)

arXiv doi:10.48550/arxiv.2408.15035 2024

[6] [6]

Uniqueness of higher integrable solution to the Landau equation with Coulomb interactions

Jann-Long Chern and Maria Gualdani. Uniqueness of Higher Integrable Solution to the Landau Equation with Coulomb Interactions . July 2022. arXiv: 1910.06216 (cit. on pp. 4, 5)

work page Pith review arXiv 2022

[7] [7]

The Second Law of Thermodynamics and Stability

C. M. Dafermos. “The Second Law of Thermodynamics and Stability”. In: Archive for Rational Mechanics and Analysis 70.2 (June 1979), pp. 167–179. DOI: 10.1007/ BF00250353 (cit. on p. 5)

work page 1979

[8] [8]

Entropy Dissipation Estimates for the Landau Equation in the Coulomb Case and Applications

Laurent Desvillettes. Entropy Dissipation Estimates for the Landau Equation in the Coulomb Case and Applications . Oct. 2014. DOI: 10 . 48550 / arXiv . 1408 . 6025. arXiv: 1408.6025 [math] (cit. on pp. 4, 9)

arXiv doi:10.48550/arxiv.1408.6025 2014