Local well-posedness for the periodic Boltzmann equation with constant collision kernel

Chenmin Sun; Engin Ba\c{s}ako\u{g}lu; Nikolay Tzvetkov; Yuzhao Wang

REVIEW 2 major objections 6 minor 31 references

Local well-posedness for the periodic Boltzmann equation with constant collision kernel

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves local well-posedness for the periodic Boltzmann equation with constant collision kernel at spatial regularity $s>\frac{d}{2}-\frac{1}{4}$ in every dimension $d\ge 2$.

desk verdict Solid periodic Strichartz estimate and a plausible LWP threshold, but the contraction step is explicitly missing, so Theorem 1.2 is not fully established as written. read the letter →

arxiv 2411.12140 v1 pith:5U7CNZ5D submitted 2024-11-19 math.AP math-phmath.MP

Engin Başakoğlu , Nikolay Tzvetkov , Chenmin Sun , Yuzhao Wang This is my paper · ORCID

classification math.APmath-phmath.MP MSC 35Q2035A0182C40

keywords Boltzmannequationconstantcollisionkernelperiodicdomainlocalwell-posednessStrichartzestimateshyperbolicSchrödingerFourierrestrictionspaceslatticepointcounting

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

This paper proves that the periodic Boltzmann equation with a constant collision kernel is locally well-posed in the anisotropic spaces $L^{2,r}_v H^s_x$ on $\mathbb{T}^d\times\mathbb{R}^d$ for $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$, in any dimension $d\ge 2$. In the periodic setting, this relaxes the usual $\frac{d}{2}$-type Sobolev regularity threshold by a quarter derivative. The central new instrument is an $L^4$ Strichartz estimate for the linear hyperbolic Schrödinger propagator $e^{it\nabla_\xi\cdot\nabla_x}$, with a derivative loss governed by $\max\{M^d,(MN)^{d-1}\log N\}^{1/4}$. If correct, the Cauchy problem has unique local solutions with uniform continuous dependence on the data, and the same proof extends to Maxwellian molecules under the angular cut-off assumption.

What carries the argument

The engine is the periodic $L^4$ Strichartz estimate for the hyperbolic Schrödinger semigroup $S(t)=e^{it\nabla_x\cdot\nabla_\xi}$: for dyadic spatial frequency $N$ and velocity-frequency scale $M$, $\|S(t)P^x_NP^\xi_M\varphi\|_{L^4(I\times\mathbb{T}^d\times\mathbb{R}^d)}$ is bounded by $\max\{M^d,(MN)^{d-1}\log N\}^{1/4}\|P^x_NP^\xi_M\varphi\|_{L^2}$. Its proof is a modulation-localized bilinear argument, and the decisive estimate is the lattice-point counting Lemma 2.4, which bounds the measure of resonance sets $\{(n,v): C_0\le (v-a)\cdot(n-b)\le C_0+K,\ |v|\lesssim M,\ |n|\lesssim N\}$ by $K\max\{M^d,(MN)^{d-1}\log N\}$ via a one-dimensional harmonic sum. This estimate is then fed through dyadic frequency decompositions and the $X^{s,r,b}$/$Y^{s,r,b}$ restriction spaces, following the standard contraction scheme from nonlinear dispersive PDE.

What would settle it

A concrete numerical check: in dimension $d=2$, take $a=0$, $b=(1/2,1/2)$, and $M=N=2^j$, and compute the supremum over $C_0\in[0,N]$ of the measure $|\{(n,v)\in\mathbb{Z}^2\times\mathbb{R}^2: C_0\le (v-a)\cdot(n-b)\le C_0+1,\ |v|\le M,\ |n|\le N\}|$. If for any $j$ this count exceeds $C\max\{M^2, MN\log N\}$ with a constant independent of $C_0$, Lemma 2.4 is false and the thresholds in Theorem 1.2 would not follow from this proof.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for $d\ge 2$, the Cauchy problem for the periodic Boltzmann equation with constant collision kernel is locally well-posed in $L^{2,r}_vH^s_x$ for $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$. Well-posedness is proved in the strong sense of existence, uniqueness in the Fourier restriction space $Y^{s,r,\frac{1}{2}+}_T$, uniform continuous dependence of the data-to-solution map, and preservation of nonnegativity. The proof passes to the velocity-side Fourier transform, rewrites the collision operator in the Fourier-side product form (1.5), and establishes bilinear spacetime estimates for the loss and gain terms, with the new periodic $L^4$ Strichartz estimate supplying the needed spacetime control. The thresholds are exactly the places where the dyadic summations over frequency scales converge given the Strichartz loss.

Load-bearing premise

The whole argument rests on the counting estimate of Lemma 2.4, which says that for each coordinate the number of integer frequencies $n$ with $|n|\lesssim N$ that make $(v-a)\cdot(n-b)$ fall into an interval of length one is no larger than about $\log N$; if that logarithmic bound failed for some rational displacement $b$, the derivative loss in the Strichartz estimate would grow and the dyadic sums could no longer close.

Editorial extensions

If this is right

For every $d\ge 2$, initial data in $L^{2,r}_vH^s_x$ with $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$ produce a unique local solution, with the time of existence depending only on the size of the data.
The data-to-solution map is uniformly continuous, not merely continuous, and nonnegative initial data yield nonnegative solutions.
The same proof covers Maxwellian molecules with the angular cut-off condition, because the modified gain term obeys the same bilinear estimates (Remark 1.3).
The thresholds $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$ are explicit and determined by the frequency loss in the new Strichartz estimate, so the argument closes with $b=\frac12+$ in the restriction spaces.

Reading between the lines

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

An editor's inference: if the logarithmic factor in Lemma 2.4 could be removed or sharpened for special displacements $b$, the same proof would lower the spatial threshold toward $\frac{d}{2}-\frac12$, the Euclidean Strichartz-admissible level; the paper does not assert this.
An editor's inference: the periodic $L^4$ estimate is independent of the collision kernel's detailed structure, so it should transfer to other kinetic transport equations on $\mathbb{T}^d$ with the same hyperbolic Schrödinger linear part, giving local well-posedness below the standard Sobolev regularity; this is a testable extension.
An editor's inference: the paper proves well-posedness but does not address sharpness, so a natural next step is to determine whether $s=\frac{d}{2}-\frac{1}{4}$ is the actual periodic well/ill-posedness threshold for constant kernels, by adapting concentration arguments from the Euclidean case.

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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 genuinely new piece is Theorem 2.1, the periodic L4 Strichartz estimate for e^{it∇ξ·∇x} on T^d × R^d with loss max{M^d,(MN)^{d−1} log N}^{1/4}, proved from the lattice counting bound in Lemma 2.4. That is a real addition to the dispersive toolset, and the nonlinear estimates in Section 3 are worked out in enough detail that I did not find an obvious error. The threshold s > d/2 − 1/4 is consistent with the loss, and the paper is honest about following the Chen–Holmer–Shen–Zhang program.

But the stress test is right: Theorem 1.2 is not fully proven as written. Section 3.2 shows only that the map Γ sends the ball B to itself. The paper explicitly says the complete treatment—contraction, uniqueness, uniform continuous dependence—is in [8,9,11,13]. That means Definition 1.1(a) and (b) are not established in this manuscript. This is not a cosmetic gap. The thresholds are saturated in the a priori bounds, so the difference estimate needs its own dyadic check, especially for Q(f−g,f) and Q(g,f−g). It is probably fixable with the same machinery, but it is exactly the missing proof of the headline theorem.

Two smaller things. Positivity f ≥ 0 is asserted in Definition 1.1 without proof; that should at least be commented on. And Remark 2.6 invokes an unspecified transference principle to get the arbitrary-center X^{s,r,b} estimate from Corollary 2.2. That is likely standard but should be written out or referenced.

The Lemma 2.4 concern about coordinates with n_i = b_i is less serious than the stress test suggests. The n = b case is bounded separately, and when n ≠ b at least one coordinate differs; a relabeling handles it. A sentence would remove any doubt.

Net: the main new estimate is solid and the paper is worth refereeing. But the stated local well-posedness theorem is conditionally established at best. I would send it to a serious referee and require the contraction/difference argument, or a clearly restated theorem, before publication. Also ask for the positivity comment.

Referee Report

2 major / 6 minor

Summary. The paper studies the periodic Boltzmann equation with constant collision kernel in d ≥ 2 dimensions, after reformulation as a hyperbolic Schrödinger equation via the inverse Fourier transform in velocity. The main result, Theorem 1.2, claims local well-posedness in L^{2,r}_v H^s_x for s > d/2 − 1/4 and r > d/2. The principal new tool is an L^4 Strichartz estimate (Theorem 2.1) with an explicit derivative loss governed by the lattice-point counting bound of Lemma 2.4. The authors then apply this estimate to prove bilinear estimates for the loss and gain terms (Lemmas 3.1 and 3.3) and outline a fixed-point argument in Section 3.2.

Significance. If fully established, the result is a meaningful advance: it moves local well-posedness for the periodic constant-kernel Boltzmann equation below the usual continuity threshold, using a genuinely periodic Strichartz estimate with a controlled derivative loss. The counting lemma (Lemma 2.4) is self-contained and is the main technical novelty. The bilinear estimates in Section 3 are detailed and plausible. However, the paper does not prove the contraction step, which is essential for the uniqueness and continuous-dependence parts of the stated well-posedness, and it asserts positivity without proof. These gaps prevent the paper from fully delivering Theorem 1.2 as written.

major comments (2)

[Definition 1.1(a)] The proof stops after showing that the map Γ sends the ball B = { f ∈ X^{s,r,b} : ||f||_{X^{s,r,b}} ≤ R } into itself. It does not prove that Γ is a contraction on B, nor does it display the difference estimate ||Γf − Γg||_{X^{s,r,b}} ≤ C T^{5/4−b}(||f||_X + ||g||_X)||f−g||_X needed for uniqueness and continuous dependence. The text explicitly states: 'We only show the boundedness of the contraction map on the ball ... for the complete treatment of the conditions of well-posedness for the Boltzmann equation, see [8,9,11,13].' Since Definition 1.1(a) requires uniqueness in Y^{s,r,1/2+}_T and Definition 1.1(b) requires uniform continuous dependence of the data-to-solution map, both of which are consequences of a contraction argument, Theorem 1.2 is not fully established as written. The boundedness of the map alone does not provide uniqueness, and the required difference estimate must be checked with the same dyadic summations used in Lemmas 3.1 and 3.3.
[Definition 1.1(a)] The assertion that f(t,x,v) ≥ 0 whenever f0 ≥ 0 is included in the local well-posedness statement but is never proved in the paper. The construction of solutions via the Fourier restriction norm and the fixed-point operator does not automatically preserve sign; a separate argument, for example via the integral representation and the structure of the collision operator, is needed. As written, this part of the well-posedness claim is unsupported.

minor comments (6)

[Lemma 2.4] The reduction to K = 1 says 'for any K ∈ N', but the statement of the lemma allows any K ≥ 1. The proof should treat non-integer K, for example by replacing K with ⌈K⌉.
[Lemma 2.4] In the definition of h(x), the factor 2 in 'h(x) = 2 ∑ x/|n^{(1)} − b^{(1)}|' is inconsistent with the measure computation for the set 0 ≤ v^{(1)}(n^{(1)} − b^{(1)}) ≤ x; the correct expression has no factor 2. Since the discrepancy is only a constant, the resulting bound is unaffected, but the displayed identity should be corrected.
[Lemma 2.4] The sentence 'Assume that n^{(1)} ≠ b^{(1)}' should be justified by noting that if n ≠ b at least one coordinate differs and coordinates can be permuted; without this remark the reduction is not fully explicit.
[Notation] The symbol b is used both for the angular kernel in Remark 1.3 and for the X^{s,b} index throughout the paper. This overloading is confusing, especially in the statement of Remark 1.3 and in Section 3.2.
[Section 3.2] The contraction estimate (3.16) is written for a general b, but Theorem 1.2 is stated in X^{s,r,1/2+}. The authors should specify the value of b used in the fixed-point argument (for instance b = 1/2+ or b = 1) and consistently state the resulting time factor T^{5/4−b}.
[Throughout] There are numerous typographical issues in the title, abstract, and body (e.g., 'Bol tzmann', 'EQUA TION', 'KERNEL', and misplaced spaces). The manuscript should be carefully proofread.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main Strichartz estimate is proved in-paper from a counting lemma, and the nonlinear estimates are checked directly against the X^{s,b} norms.

full rationale

The paper's central new input, Theorem 2.1, is derived rather than assumed: the modulation-localized estimate Theorem 2.3 reduces the L4 bound to the lattice-point counting Lemma 2.4, whose proof is given in the paper. Corollary 2.5 and Theorem 2.1 then follow by dyadic summation and a local-time argument. The nonlinear gain and loss estimates (Lemmas 3.1 and 3.3) are proven using the X^{s,b} norms, the Strichartz estimate, Bernstein, and Hölder; the only imported lemma, Lemma 3.2 from [3, 11], is a standard inequality external to the claim. No parameter is fitted and no target quantity is fed back as an input. The citation [29] (Takaoka–Tzvetkov) for the 'argument' of Theorem 2.1 is not load-bearing because the counting argument and all estimates are reproduced in this paper; a self-citation without independent verification would be a concern only if the result were imported unproved. The manuscript is explicit that Section 3.2 only shows boundedness of the contraction map on a ball and defers to [8,9,11,13] for the full contraction and continuous-dependence treatment; this is a completeness limitation rather than circularity, since the deferred argument is an external standard scheme and the quoted sentence itself is part of the manuscript and must be weighed. Lemma 2.4 has a possible edge case n_i = b_i that is only crudely bounded, but a proof gap is not circularity. Thus the derivation chain is self-contained at its core, and the circularity score is 0.

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

The proof introduces no free parameters or invented entities. It relies on external results (Bobylev identity, gain-operator Lp estimates, R^d Strichartz) and, importantly, on the assumption that the contraction argument can be transplanted from [8,9,11,13]; the latter is not demonstrated in the paper.

assumptions (5)

standard math Standard Littlewood-Paley theory, Bernstein inequalities, and Sobolev embedding on T^d × R^d
Used throughout Section 3 to localize in frequency and to convert L^∞ bounds into L^4 bounds via N^{d/4} and M^{d/4} factors.
domain assumption The Bobylev identity for the Fourier-transformed collision operator (1.5)
Reduces the quadratic collision term to a product-type operator in ξ; the identity is cited to [6] and is essential for the entire analysis.
domain assumption Lemma 3.2 (Hölder-type estimate for the gain operator in ξ) is imported from [3,11] without proof
All gain-term estimates in Lemma 3.3 rely on this external bound; if it failed, the bilinear estimate would collapse.
ad hoc to paper The contraction mapping, uniqueness, and continuous dependence follow from the bilinear estimates exactly as in [8,9,11,13]
Section 3.2 only shows boundedness of Γ on a ball and defers the complete treatment to references; this is a load-bearing unproved step in the paper as written.
standard math The R^d Strichartz estimates (1.6) from [25] (via [12,13])
Used as motivation and comparison; not directly used in the periodic proof but underlies the claimed analogy.

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Pith. "Pith review of Local well-posedness for the periodic Boltzmann equation with constant collision kernel." pith.science (2026). https://pith.science/paper/5U7CNZ5D

@misc{pith2026241112140,
  author       = {Pith},
  title        = {Pith review of: Local well-posedness for the periodic Boltzmann equation with constant collision kernel},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5U7CNZ5D}},
  note         = {Machine review of arXiv:2411.12140}
}

abstract

We study the Boltzmann equation with the constant collision kernel in the case of spatially periodic domain $\mathbb{T}^d$, $d\geq 2$. Using the existing techniques from nonlinear dispersive PDEs, we prove the local well-posedness result in $L^{2,r}_vH^s_x$ for $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$. To reach the result, the main tool we establish is the $L^4$ Strichartz estimate for solutions to the corresponding linear equation.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

31 extracted references · 23 canonical work pages

[1]

Alexandre, Y

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang. Global existence and full regularity of the Boltzmann equation without angular cutoﬀ. Comm. Math. Phys. , 304(2):513–581, 2011

2011
[2]

Alexandre, Y

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang. Local existence with mild regularity for the Boltzmann equation. Kinetic and Related Models , 6:1011–1041, 2013

work page 2013
[3]

R. J. Alonso and E. Carneiro. Estimates for the Boltzmann collision operator via radial symmetry and Fourier transfo rm. Advances in Mathematics , 223(2):511–528, 2010

work page 2010
[4]

Ampatzoglou, I

I. Ampatzoglou, I. M. Gamba, N. Pavlovi´ c, and M. Taskovi ´ c. Global well-posedness of a binary-ternary Boltzmann equation. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 39(2):327–367, 2022

work page 2022
[5]

D. Arsenio. On the global existence of mild solutions to t he Boltzmann equation for small data in LD. Comm. Math. Phys., 302:453–476, 2011. WELL-POSEDNESS FOR BOLTZMANN EQUATION 17

work page 2011
[6]

A. V. Bobylev. Exact solutions of the nonlinear Boltzman n equation and the theory of relaxation of a Maxwellian gas. Theoretical and Mathematical Physics , 60:280 – 310, 1984

work page 1984
[7]

Bourgain

J. Bourgain. Fourier transform restriction phenomena f or certain lattice subsets and applications to nonlinear ev olution equations. I. Schr¨ odinger equations. Geometric & Functional Analysis GAF A , 3:107–156, 1993

work page 1993
[8]

T. Chen, R. Denlinger, and N. Pavlovi´ c. Local well-pose dness for Boltzmann’s equation and the Boltzmann hierarchy via Wigner Transform. Communications in Mathematical Physics , 368(2):427–465, 2019

work page 2019

Show all 31 references

[9]

T. Chen, R. Denlinger, and N. Pavlovi´ c. Moments and regu larity for a Boltzmann equation via Wigner transform. Discrete and Continuous Dynamical Systems , 39(9):4979–5015, 2019

2019
[10]

T. Chen, R. Denlinger, and N. Pavlovi´ c. Small data glob al well-posedness for a Boltzmann equation via bilinear spa cetime estimates. Archive for Rational Mechanics and Analysis , 240:327–381, 2021

2021
[11]

Chen and J

X. Chen and J. Holmer. W ell/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel. arXiv preprint arXiv:2206.11931, 2022

2022 arXiv
[12]

X. Chen, S. Shen, and Z. Zhang. Sharp global well-posedn ess and scattering of the Boltzmann equation . arXiv preprint arXiv:2311.02008, 2023

2023 arXiv
[13]

X. Chen, S. Shen, and Z. Zhang. W ell/Ill-posedness of th e Boltzmann equation with soft potential. arXiv preprint arXiv:2310.05042, 2023

2023 arXiv
[14]

R. J. DiPerna and P.-L. Lions. On the Cauchy problem for B oltzmann equations: global existence and weak stability. Ann. of Math. , 130(2):321–366, 1989

1989
[15]

R. Duan, F. Huang, Y. W ang, and T. Yang. Global well-pose dness of the Boltzmann equation with large amplitude initial data. Arch. Ration. Mech. Anal. , 225(1):375–424, 2017

2017
[16]

R. Duan, S. Liu, S. Sakamoto, and R. M. Strain. Global mil d solutions of the Landau and non-cutoﬀ Boltzmann equations . Comm. Pure Appl. Math. , 74(5):932–1020, 2021

2021
[17]

R. Duan, S. Liu, and J. Xu. Global well-posedness in spat ially critical Besov space for the Boltzmann equation. Arch. Ration. Mech. Anal. , 220(2):711–745, 2016

2016
[18]

H. Grad. Asymptotic equivalence of the Navier-Stokes a nd nonlinear Boltzmann equation. Proc. Symp. Appl. Math , 17:154–183, 1965

1965
[19]

P. T. Gressman and R. M. Strain. Global classical soluti ons of the Boltzmann equation without angular cut-oﬀ. J. Amer. Math. Soc. , 24(3):771–847, 2011

2011
[20]

Y. Guo. Classical solutions to the Boltzmann equation f or molecules with an angular cutoﬀ. Arch. Ration. Mech. Anal. , 169(4):305–353, 2003

2003
[21]

Y. Guo. Decay and continuity of the Boltzmann equation i n bounded domains. Arch. Ration. Mech. Anal. , 197:713–809, 2010

2010
[22]

He and J

L. He and J. Jiang. W ell-posedness and scattering for th e Boltzmann equations: soft potential with cut-oﬀ. J. Stat. Phys., 168(2):470–481, 2017

2017
[23]

He and J

L. He and J. Jiang. On the Cauchy problem for the cutoﬀ Bol tzmann equation with small initial data. J. Stat. Phys. , 190(3):Paper No. 52, 25, 2023

2023
[24]

Kaniel and M

S. Kaniel and M. Shinbrot. The Boltzmann equation: I. Un iqueness and local existence. Comm. Math. Phys. , 58:65–84, 1978

1978
[25]

Keel and T

M. Keel and T. Tao. Endpoint Strichartz estimates. Amer. J. Math. , 120(5):955–980, 1998

1998
[26]

T. Liu, T. Yang, and S. H. Yu. Energy method for the Boltzm ann equation. Physica D , 188:178–192, 2004

2004
[27]

Nishimura

K. Nishimura. Global existence for the Boltzmann equat ion in Lr vL8 t L8 x spaces. Commun. Pur. Appl. Anal. , 18(4):1769– 1782, 2019

2019
[28]

Sohinger and R

V. Sohinger and R. M. Strain. The Boltzmann equation, Be sov spaces, and optimal time decay rates in Rn x . Adv. Math. , 261:274–332, 2014

2014
[29]

Takaoka and N

H. Takaoka and N. Tzvetkov. On 2d nonlinear Schr¨ odinger equations with data on R ˆ T. Journal of Functional Analysis , 182(2):427–442, 2001

2001
[30]

G. Toscani. Global solution of the initial value proble m for the Boltzmann equation near a local Maxwellian. Arch. Rational Mech. Anal. , 102(3):231–241, 1988

1988
[31]

S. Ukai. On the existence of global solutions of mixed pr oblem for non-linear Boltzmann equation. Proc. Japan Acad. , 50:179–184, 1974. 18 E. BAS ¸AKO ˘GLU, C. SUN, N. TZVETKOV, AND Y. W ANG Institute of Mathematical Sciences, ShanghaiTech Universi ty, Shanghai, 201210, China...

1974

Pith tools

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

[1] [1]

Alexandre, Y

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang. Global existence and full regularity of the Boltzmann equation without angular cutoﬀ. Comm. Math. Phys. , 304(2):513–581, 2011

2011

[2] [2]

Alexandre, Y

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang. Local existence with mild regularity for the Boltzmann equation. Kinetic and Related Models , 6:1011–1041, 2013

work page 2013

[3] [3]

R. J. Alonso and E. Carneiro. Estimates for the Boltzmann collision operator via radial symmetry and Fourier transfo rm. Advances in Mathematics , 223(2):511–528, 2010

work page 2010

[4] [4]

Ampatzoglou, I

I. Ampatzoglou, I. M. Gamba, N. Pavlovi´ c, and M. Taskovi ´ c. Global well-posedness of a binary-ternary Boltzmann equation. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 39(2):327–367, 2022

work page 2022

[5] [5]

D. Arsenio. On the global existence of mild solutions to t he Boltzmann equation for small data in LD. Comm. Math. Phys., 302:453–476, 2011. WELL-POSEDNESS FOR BOLTZMANN EQUATION 17

work page 2011

[6] [6]

A. V. Bobylev. Exact solutions of the nonlinear Boltzman n equation and the theory of relaxation of a Maxwellian gas. Theoretical and Mathematical Physics , 60:280 – 310, 1984

work page 1984

[7] [7]

Bourgain

J. Bourgain. Fourier transform restriction phenomena f or certain lattice subsets and applications to nonlinear ev olution equations. I. Schr¨ odinger equations. Geometric & Functional Analysis GAF A , 3:107–156, 1993

work page 1993

[8] [8]

T. Chen, R. Denlinger, and N. Pavlovi´ c. Local well-pose dness for Boltzmann’s equation and the Boltzmann hierarchy via Wigner Transform. Communications in Mathematical Physics , 368(2):427–465, 2019

work page 2019