Global Stability and Energy Growth in the Sheared Vlasov--Poisson--Boltzmann System

Mingying Zhong; Zhida Chang

REVIEW 2 major objections 5 minor 36 references

Global Stability and Energy Growth in the Sheared Vlasov--Poisson--Boltzmann System

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For small shear, the self-similar sheared VPB profile is globally stable, and total energy grows like $e^{2\beta t}$.

desk verdict First global stability and energy-growth result for sheared Vlasov–Poisson–Boltzmann; the proof is standard machinery assembled well, with the main risk being an imported profile theorem. read the letter →

arxiv 2608.13356 v1 pith:Z66M6XQI submitted 2026-08-13 math.AP

Zhida Chang , Mingying Zhong This is my paper · ORCID

classification math.AP MSC 35Q2035Q8335B3535B40

keywords Vlasov-Poisson-Boltzmannsystemuniformshearflowself-similarprofileglobalstabilityMaxwellmoleculesGradangularcutoffshear-inducedenergygrowthzero-frequencymode

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 sets out to prove that a dilute gas of charged particles on a three-torus, driven by uniform shear and its own self-consistent electric field, converges in self-similar variables to the non-Maxwellian sheared Boltzmann profile rather than to equilibrium or to blow-up. It establishes global existence and uniqueness of small weighted-$L^\infty$ perturbations of that profile, exponential decay of all nonzero spatial derivatives and of the electric field, and uniform boundedness of the renormalized perturbation. It also derives a closed zero-frequency system for a renormalized total energy and suitable second-order moments, yielding the precise large-time law that total energy grows like $e^{2\beta t}$ with a prefactor fixed by the initial shear-stress moment. If correct, this gives a global stability theory for the sheared Vlasov–Poisson–Boltzmann system and a quantitative prediction for shear-induced heating of a plasma.

What carries the argument

Two tools carry the proof. First, the renormalized perturbation is split as $\mu^{1/2}\tilde{g}=g_1+\mu^{1/2}g_2$, with $g_1$ absorbing the growing high-velocity terms and $g_2$ controlled through the spectral-gap coercivity of the linearized collision operator; weighted $L^\infty$ estimates are closed by an $L^\infty$\u2013$L^2$ energy method and by layered time weights $\lambda^m_n=0$ for $|m|=0$ and $\lambda-|n|\beta$ for $|m|>0$, which exactly absorb commutators that convert one velocity derivative into one spatial derivative. Second, the zero-frequency temperature mode is controlled by a closed three-component ODE for $U=(E_\alpha,P_0d_{12},P_0d_{22})$ with constant matrix $A_\alpha$; the paper computes its spectrum, eigenvalues $0$ and $2b_0+3\beta\pm i\omega_\alpha$, which yields uniform control of the temperature mode and, later, the exponential convergence of the renormalized energy.

What would settle it

Solve the stationary sheared Boltzmann profile (1.12) numerically for a fixed cutoff kernel and check whether its moments obey $\alpha^2 b_0=6\beta(b_0+\beta)^2$ and whether the first-order term in $G$ matches $-\alpha v_1v_2\mu/(2b_0)$; alternatively, simulate the USF-VPB system with small shear and measure whether the total energy grows as $e^{2\beta t}$ with the prefactor in (1.24).

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for Maxwell molecules with Grad’s angular cutoff, small shear rate $\alpha$, and initial data within $\varepsilon_0$ of the self-similar profile $G$ in a velocity-weighted $L^\infty$ norm, with matching total mass and momentum, the uniform-shear Vlasov–Poisson–Boltzmann system on the torus has a unique global solution. In self-similar coordinates the renormalized perturbation remains uniformly bounded, while every component involving at least one spatial derivative decays as $e^{-[\lambda-(N-1)\beta]t}$ and the electric field decays as $e^{-\lambda t}$. Theorem 1.2 complements this with a sharp large-time energy law: the renormalized total energy $E_\alpha(t)$ converges exponentially to a limit $E_\alpha(\infty)$, so the true total energy grows like $e^{2\beta t}$, with a prefactor whose leading shear correction is set by the initial $v_1v_2$-stress moment.

Load-bearing premise

The argument inherits, without reproving, the small-shear expansion $G=\mu-\frac{\alpha}{2b_0}v_1v_2\mu+O(\alpha^2)$ and the relation $\alpha^2 b_0=6\beta(b_0+\beta)^2$ from the Boltzmann-only theory; if that expansion fails, the perturbation system, the eigenvalue computation, and both theorems collapse.

Editorial extensions

If this is right

The self-similar sheared profile $G$ is nonlinearly stable for sufficiently small shear: any small weighted-$L^\infty$ perturbation satisfying mass and momentum conservation stays close to $G$ for all time.
All nonzero-spatial-frequency components of the renormalized perturbation decay exponentially at rate $\lambda-(N-1)\beta$, and the self-consistent electric field decays like $e^{-\lambda t}$; taking the shear rate small keeps this rate positive.
The renormalized total energy $E_\alpha(t)$ converges exponentially to $E_\alpha(\infty)$, so the physical total energy grows at the precise rate $e^{2\beta t}$.
Up to a quadratic remainder, the limiting energy state is a shear-corrected second-order moment of the initial perturbation: mainly the $v_1v_2$ stress moment at order $\alpha$, with $v_2^2$ and $|v|^2$ corrections at order $\alpha^2$.
When $\alpha=0$, the theorem reduces to the classical Vlasov–Poisson–Boltzmann dynamics near a Maxwellian, with trivial energy growth and linear stability as a special case.

Reading between the lines

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

Inference: the same three-component zero-frequency closure should extend to higher second-order moments, suggesting that the long-time energy dynamics of sheared VPB is effectively finite-dimensional.
Inference: the relation $\alpha^2 b_0=6\beta(b_0+\beta)^2$ predicts $\beta\approx\alpha^2/(6b_0)$ for tiny shear, a heating law that is testable in kinetic simulations by measuring the growth of $\langle v_1v_2\rangle$.
Inference: increasing $\alpha$ or removing the angular cutoff should eventually destroy the spectral picture; the collision of the eigenvalues $\lambda_+=\lambda_-$ or loss of the zero eigenvalue is a natural place to look for a stability threshold.
Inference: switching off the Poisson coupling and comparing the field-decay rate $\lambda$ with the collision-only rate would isolate how the self-consistent electric field modifies shear stabilization.

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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

Two things to know. First, the main theorem is genuinely new: prior stability work on the sheared Boltzmann equation (Duan–Liu–Shen, arXiv:2509.13921) had no Poisson coupling, and the homogeneous profile theory (Duan–Liu, ARMA 2021) had no spatial inhomogeneity or electric field. This paper closes that gap for Maxwell molecules with Grad cutoff and small shear, proving global weighted-L∞ stability of the self-similar profile, exponential decay of spatial derivatives, and e^{2βt} energy growth with a computed prefactor. Second, the proof is an assembly of established tools—Caflisch decomposition, Guo L∞–L2, macro–micro, zero-frequency spectral analysis—but the assembly is nontrivial, and the closed three-component ODE for (renormalized energy, P0 d12, P0 d22) with eigenvalues 0 and 2b0+3β±iω is a real technical step.

The paper does several things well. The moment identity in Appendix B for L(|v|²v_i μ^{1/2}) is clean. Lemma 3.4, the characteristic change-of-variables with determinant bounds, is the kind of estimate that usually gets hand-waved, and here it is written out. The spectral study of A_α in Lemma 4.6 is internally consistent; using the profile equation itself to produce the eigenvector relation α²b0 = 6β(b0+β)² is neat.

Soft spots, in proportion. The load-bearing dependency is Lemma 2.7, the imported profile expansion from Duan–Liu [15]. Everything downstream—the perturbation system, the layered weights that need β = O(α²), the eigenvector computation—depends on that expansion and on [15] giving C^k weighted bounds uniformly for the l and N used here. The paper cites rather than reproves it. That is standard practice, not circularity, but it means the two theorems are conditional on [15] being exactly as strong as claimed. A referee should check the uniformity in weights and derivative orders.

Second, several estimates are delegated rather than proved. Lemma 2.2 states the collision estimates and says the details are omitted. The nonnegativity step in Theorem 1.1 says 'we omit the repeated details.' And parts of Section 4 lean on 'similarly' and 'same argument' for borderline cases. These are probably fillable—they are standard in this line of work—but they are real gaps, and the referee will need to do some of the work.

Who this is for: kinetic theory people working on shear flows and VPB stability. It deserves a serious referee, not a desk reject. If the delegated estimates check out and the [15] dependency is confirmed, this is a publishable major result.

Recommendation: send to peer review, with a referee brief that focuses on the omitted standard estimates and the compatibility with [15].

Referee Report

2 major / 5 minor

Summary. The paper studies the one-species Vlasov--Poisson--Boltzmann system on the three-dimensional torus under uniform shear flow, with Maxwell molecules and Grad's angular cutoff. In self-similar variables, it proves global-in-time stability of the spatially homogeneous self-similar profile G of the sheared Boltzmann equation for sufficiently small shear rate α, in weighted L∞ spaces, with exponential decay of all components involving spatial derivatives and of the electric field (Theorem 1.1). It also derives a closed zero-frequency ODE for a renormalized total energy and two second-order moments, and uses its spectral decomposition to prove exponential convergence of the renormalized energy and to obtain an explicit shear-corrected representation of the limiting energy (Theorem 1.2). The proof combines self-similar scaling, the Caflisch decomposition, Guo's L∞--L2 method, macro--micro decomposition, and a spectral analysis of the zero-frequency matrix A_α.

Significance. If the proof is completed as indicated below, this is a substantial extension of the sheared-Boltzmann stability theory to the Poisson-coupled VPB system, and appears to be the first global stability and large-time asymptotic result for the sheared VPB system around a nonequilibrium self-similar profile in a weighted L∞ setting. The paper is careful to separate the exponentially decaying nonzero-spatial-derivative modes from the uniformly controlled zero-order modes, and the zero-frequency analysis is concrete: the matrix A_α is explicitly diagonalized, the zero eigenvalue is obtained from moment identities applied to the profile equation, and the relation α²b0 = 6β(b0+β)² is re-derived in Lemma 4.6 rather than only quoted from the external profile theory. The appendices supply a local existence theorem and a moment identity used in Lemma 2.6, adding useful self-contained elements. The principal weaknesses are the delegated nonnegativity argument and the heavy reliance on the external profile theorem [15] without an explicit verification that its hypotheses cover the derivative range and weights used here.

major comments (2)

[Section 5, Proof of Theorem 1.1, Step 2 (nonnegativity)] The proof of the claim F(t,x,v) ≥ 0 is delegated: the paper states that a frozen iteration 'can be carried out locally by the same argument as in the proof of the local existence theorem in Appendix A, and we omit the repeated details.' This is a genuine gap in the written proof. The global solution was obtained by continuation from the fixed point in Lemma A.1, which lives in the weighted space eY_T and does not by itself lie in any nonnegativity cone; the a priori estimates in Sections 3 and 4 do not use positivity. Since nonnegativity is part of the statement of Theorem 1.1, the authors should either give the frozen-iteration argument and prove its convergence to the unique solution in eY_T, or invoke a published theorem that applies directly to the present non-autonomous, shear- and Poisson-coupled system. As written, this step is not a proof.
[Section 2, Lemma 2.7 and Sections 3--5 (external profile theorem)] The main theorems are conditional on [15, Theorem 1.1], which provides the existence, smoothness, normalization, and expansion (2.24) of the self-similar profile G, together with the bound β = O(α²). These imported facts enter the perturbation system (2.6), the G1 bounds (2.25), the smallness of β used throughout, and the spectral computation of A_α. Although Lemma 4.6 re-derives the more precise relation α²b0 = 6β(b0+β)² from the profile equation, the existence and normalization of G are still external. The paper should state explicitly which statements are quoted from [15] and verify that the α0(l), l0, and derivative-order hypotheses of [15, Theorem 1.1] cover the N and l appearing in Theorems 1.1 and 1.2. Without this, a reader cannot check from the present manuscript alone that the imported hypotheses match the setting of the theorems.

minor comments (5)

[Section 1, last paragraph before Section 2] The phrase 'the proof of the local existence result anf the proof' contains a typo: 'anf' should be 'and'.
[Section 4, after equation (4.55)] The sentence 'Combining (4.53) with (4.55), we obtain (4.55) yield' is garbled; it should say something like 'Combining (4.53) with (4.55), we obtain (4.56).'
[Section 2, equations (2.4)--(2.6)] The symbol g̃ is reused for the new unknown after the substitution f̃ = μ^{1/2} g̃, which is a common but potentially confusing abuse of notation. I suggest renaming the new unknown, for example ĝ, to avoid ambiguity.
[Section 2, Lemma 2.2] Lemma 2.2 states two bilinear estimates but its proof is omitted, with the reader referred to analogous arguments in [21,23]. If these estimates are used later in the paper, please include the proof in an appendix or state precisely which existing lemmas imply each inequality; if they are not used, consider removing the lemma to declutter the text.
[Section 5, Proof of Theorem 1.2, equation (5.13)] The passage from (5.13) to the representation (1.24) uses the exact relation α²b0 = 6β(b0+β)² and Remark 1.1, but the algebra is not shown. Please add one or two lines displaying the simplification, or explicitly refer the reader to Remark 1.1 at that point.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the self-similar profile is imported from independent prior work, and the paper's main estimates close without self-referential fitting.

full rationale

The paper's load-bearing input is the self-similar profile G from Duan–Liu [15, Theorem 1.1], including the first-order expansion (2.24), the bound (2.25), and β=O(α²). This is an external, independently established result, not derived from the present theorems, and the authors are not the authors of [15]; hence it is real evidence rather than circularity. The zero-frequency analysis in Lemma 4.6 derives the eigenvector and the relation α²b0=6β(b0+β)² from the profile equation by testing against moments, not from the theorem being proved. The three-component ODE system (4.75)–(4.77) is genuinely closed from the macroscopic equations, and Theorem 1.2's prefactor is obtained from the spectral projection of that system, not from fitting initial data. No parameter is fitted and then called a prediction. Although the paper cites works co-authored by the present authors ([28]–[30]), those citations are background material for the classical VPB system and are not load-bearing for the new results. The derivation chain is therefore self-contained relative to its stated external inputs, and no circular step is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a standard package of kinetic-theory tools and on the prior existence of the self-similar profile G. All axioms are from the cited literature; none are ad hoc to this paper. The proof parameters (l, M, M0, κ, η, ε, λ) are arbitrary constants chosen to close estimates and do not enter the final physical formulas, so they are not free parameters in the sense of fitted values.

assumptions (7)

domain assumption Maxwell molecule collision kernel with Grad's angular cutoff (1.5)-(1.6)
The entire analysis assumes the collision kernel B0(cosθ) ≤ C|cosθ| and no velocity dependence; this restricts the physical regime.
domain assumption Existence, uniqueness, smoothness, and expansion of self-similar profile G (Lemma 2.7, from [15, Theorem 1.1])
The paper imports the profile G and its expansion μ - α/(2b0) v1v2 μ + O(α²), which underlies the perturbation system and the spectral relations in Lemma 4.6.
standard math Spectral gap of linearized collision operator L (Lemma 2.1, from [22])
Used to obtain coercivity of the microscopic part in L2 estimates and the L∞-L2 closure.
standard math Collision operator estimates (Lemmas 2.2, 2.3 from [21, 23, 2])
Provide the nonlinear estimates in weighted L2 and L∞ used throughout Sections 3 and 4.
standard math Smoothing kernel estimates for K (Lemma 2.4 from [14]) and high-velocity smallness (Lemma 2.5 from [15])
Used in the L∞-L2 estimates for g2 and the cutoff decompositions.
standard math Moment identities for the collision operator (Lemma 4.5 from [18])
These identities close the zero-frequency ODE system (4.75).
standard math Elliptic regularity on the torus (Lemma 2.8 from [32])
Used to control the electric potential in terms of the density perturbation.

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Pith. "Pith review of Global Stability and Energy Growth in the Sheared Vlasov--Poisson--Boltzmann System." pith.science (2026). https://pith.science/paper/Z66M6XQI

@misc{pith2026260813356,
  author       = {Pith},
  title        = {Pith review of: Global Stability and Energy Growth in the Sheared Vlasov--Poisson--Boltzmann System},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z66M6XQI}},
  note         = {Machine review of arXiv:2608.13356}
}

abstract

We study the Vlasov--Poisson--Boltzmann system on the three-dimensional torus under uniform shear flow. For Maxwell molecules with Grad's angular cutoff and sufficiently small shear rate, we consider perturbations around the spatially homogeneous self-similar profile of the sheared Boltzmann equation. In self-similar variables, we prove the global stability of this profile, including global existence and uniqueness, the exponential decay of nonzero-spatial derivatives in weighted $L^\infty$ spaces, and the uniform boundedness of the renormalized perturbation. The analysis combines a Caflisch's decomposition, Guo's $L^\infty$--$L^2$ estimates, macro--micro analysis, and a spectral study of the zero-frequency mode. We further derive a closed zero-frequency system for a renormalized total energy and suitable second-order moments, which yields a precise large-time description of the shear-induced energy growth.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

36 extracted references · 35 canonical work pages

[15]

R. J. Duan and S. Q. Liu, The Boltzmann equation for uniform shear flow,Arch. Rational Mech. Anal.242(2021), no. 3, 1947–2002

work page 2021
[1]

Acedo, A

L. Acedo, A. Santos and A. V. Bobylev, On the derivation of a high-velocity tail from the Boltzmann–Fokker–Planck equation for shear flow,J. Stat. Phys.109(2002), no. 5–6, 1027–1050

work page 2002
[2]

Arkeryd, R

L. Arkeryd, R. Esposito and M. Pulvirenti, The Boltzmann equation for weakly inhomogeneous data,Comm. Math. Phys.111(1987), no. 3, 393–407

work page 1987
[3]

B´ enyi and T

´A. B´ enyi and T. Oh, The Sobolev inequality on the torus revisited,Publ. Math. Debrecen83(2013), no. 3, 359–374

work page 2013
[4]

A. V. Bobylev, The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules (Russian),Dokl. Akad. Nauk SSSR225(1975), 1041–1044; translated inSoviet Phys. Dokl.20(1976), 820–822

work page 1975
[5]

A. V. Bobylev, G. L. Caraffini and G. Spiga, On group invariant solutions of the Boltzmann equation,J. Math. Phys.37(1996), no. 6, 2787–2795

work page 1996
[6]

A. V. Bobylev and C. Cercignani, Self-similar solutions of the Boltzmann equation and their applications,J. Stat. Phys.106(2002), no. 5–6, 1039–1071

work page 2002
[7]

A. V. Bobylev, A. Nota and J. J. L. Vel´ azquez, Self-similar asymptotics for a modified Maxwell–Boltzmann equation in systems subject to deformations,Comm. Math. Phys.380(2020), no. 1, 409–448

work page 2020

Show all 36 references

[8]

R. E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous,Comm. Math. Phys. 74(1980), no. 1, 71–95. Z.-D. Chang, M.-Y. Zhong 49

1980
[9]

Cercignani, Existence of homoenergetic affine flows for the Boltzmann equation,Arch

C. Cercignani, Existence of homoenergetic affine flows for the Boltzmann equation,Arch. Rational Mech. Anal.105 (1989), no. 4, 377–387

1989
[10]

Cercignani, Shear flow of a granular material,J

C. Cercignani, Shear flow of a granular material,J. Stat. Phys.102(2001), no. 5–6, 1407–1415

2001
[11]

Cercignani, The Boltzmann equation approach to the shear flow of a granular material,Philos

C. Cercignani, The Boltzmann equation approach to the shear flow of a granular material,Philos. Trans. Roy. Soc. Lond. Ser. A360(2002), no. 1792, 407–414

2002
[12]

R. J. Duan and T. Yang, Stability of the one-species Vlasov–Poisson–Boltzmann system,SIAM J. Math. Anal.41 (2010), 2353–2387

2010
[13]

R. J. Duan and R. M. Strain, Optimal time decay of the Vlasov–Poisson–Boltzmann system inR 3,Arch. Rational Mech. Anal.199(2011), no. 1, 291–328

2011
[14]

R. J. Duan, T. Yang and H.-J. Zhao, The Vlasov–Poisson–Boltzmann system in the whole space: The hard potential case,J. Differential Equations252(2012), 6356–6386

2012
[16]

R. J. Duan, S. Q. Liu, R. M. Strain and A. Yang, The 3D kinetic Couette flow via the Boltzmann equation in the diffusive limit, preprint, arXiv:2409.00311

arXiv
[17]

R. J. Duan and S. Q. Liu, Uniform shear flow via the Boltzmann equation with hard potentials,Discrete Contin. Dyn. Syst.45(2025), no. 10, 4071–4118

2025
[18]

R. J. Duan, S. Q. Liu and S. L. Shen, Global stability of the inhomogeneous sheared Boltzmann equation in torus, preprint, arXiv:2509.13921
[19]

V. S. Galkin, On a class of solutions of Grad’s moment equations,J. Appl. Math. Mech.22(1958), no. 3, 532–536; translated fromPrikl. Mat. Mekh.22(1958), no. 3, 386–389

1958
[20]

Garz´ o and A

V. Garz´ o and A. Santos,Kinetic Theory of Gases in Shear Flows: Nonlinear Transport, Fundamental Theories of Physics, 131, Kluwer Academic Publishers, Dordrecht, 2003

2003
[21]

Guo, The Vlasov–Poisson–Boltzmann system near Maxwellians,Comm

Y. Guo, The Vlasov–Poisson–Boltzmann system near Maxwellians,Comm. Pure Appl. Math.55(2002), no. 9, 1104–1135

2002
[22]

Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation,Comm

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation,Comm. Pure Appl. Math.59(2006), 626–687

2006
[23]

Guo, Decay and continuity of the Boltzmann equation in bounded domains,Arch

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains,Arch. Rational Mech. Anal.197 (2010), no. 3, 713–809

2010
[24]

R. D. James, A. Nota and J. J. L. Vel´ azquez, Self-similar profiles for homoenergetic solutions of the Boltzmann equation: particle velocity distribution and entropy,Arch. Rational Mech. Anal.231(2019), no. 2, 787–843

2019
[25]

R. D. James, A. Nota and J. J. L. Vel´ azquez, Long-time asymptotics for homoenergetic solutions of the Boltzmann equation: collision-dominated case,J. Nonlinear Sci.29(2019), no. 5, 1943–1973

2019
[26]

R. D. James, A. Nota and J. J. L. Vel´ azquez, Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Hyperbolic-dominated case,Nonlinearity33(2020), no. 8, 3781–3815

2020
[27]

Kepka, Self-similar profiles for homoenergetic solutions of the Boltzmann equation for non-cutoff Maxwell molecules,J

B. Kepka, Self-similar profiles for homoenergetic solutions of the Boltzmann equation for non-cutoff Maxwell molecules,J. Stat. Phys.190(2023), no. 2, Paper No. 27, 22 pp

2023
[28]

H.-L. Li, T. Yang and M. Zhong, Spectrum analysis and optimal decay rates of the bipolar Vlasov–Poisson– Boltzmann equations,Indiana Univ. Math. J.65(2016), 665–725

2016
[29]

H.-L. Li, Y. Wang, T. Yang and M. Zhong, Stability of nonlinear wave patterns to the bipolar Vlasov–Poisson– Boltzmann system,Arch. Rational Mech. Anal.228(2018), 39–127

2018
[30]

H.-L. Li, T. Yang and M. Zhong, Spectrum analysis for the Vlasov–Poisson–Boltzmann system,Arch. Rational Mech. Anal.241(2021), 311–355

2021
[31]

J. M. Montanero, A. Santos and V. Garz´ o, Monte Carlo simulation of the Boltzmann equation for uniform shear flow,Phys. Fluids8(1996), no. 7, 1981–1983

1996
[32]

Modena and G

S. Modena and G. Sattig, Convex integration solutions to the transport equation with full dimensional concentration, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire37(2020), no. 5, 1075–1108

2020
[33]

Truesdell, On the pressures and the flux of energy in a gas according to Maxwell’s kinetic theory, II,J

C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell’s kinetic theory, II,J. Rational Mech. Anal.5(1956), 55–128

1956
[34]

Truesdell and R

C. Truesdell and R. G. Muncaster,Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas, Pure and Applied Mathematics, 83, Academic Press, New York–London, 1980

1980
[35]

Yang, H.-J

T. Yang, H.-J. Yu and H.-J. Zhao, Cauchy problem for the Vlasov–Poisson–Boltzmann system,Arch. Rational Mech. Anal.182(2006), 415–470

2006
[36]

Yang and H.-J

T. Yang and H.-J. Zhao, Global existence of classical solutions to the Vlasov–Poisson–Boltzmann system,Comm. Math. Phys.268(2006), 569–605

2006

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [15]

R. J. Duan and S. Q. Liu, The Boltzmann equation for uniform shear flow,Arch. Rational Mech. Anal.242(2021), no. 3, 1947–2002

work page 2021

[2] [1]

Acedo, A

L. Acedo, A. Santos and A. V. Bobylev, On the derivation of a high-velocity tail from the Boltzmann–Fokker–Planck equation for shear flow,J. Stat. Phys.109(2002), no. 5–6, 1027–1050

work page 2002

[3] [2]

Arkeryd, R

L. Arkeryd, R. Esposito and M. Pulvirenti, The Boltzmann equation for weakly inhomogeneous data,Comm. Math. Phys.111(1987), no. 3, 393–407

work page 1987

[4] [3]

B´ enyi and T

´A. B´ enyi and T. Oh, The Sobolev inequality on the torus revisited,Publ. Math. Debrecen83(2013), no. 3, 359–374

work page 2013

[5] [4]

A. V. Bobylev, The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules (Russian),Dokl. Akad. Nauk SSSR225(1975), 1041–1044; translated inSoviet Phys. Dokl.20(1976), 820–822

work page 1975

[6] [5]

A. V. Bobylev, G. L. Caraffini and G. Spiga, On group invariant solutions of the Boltzmann equation,J. Math. Phys.37(1996), no. 6, 2787–2795

work page 1996

[7] [6]

A. V. Bobylev and C. Cercignani, Self-similar solutions of the Boltzmann equation and their applications,J. Stat. Phys.106(2002), no. 5–6, 1039–1071

work page 2002

[8] [7]

A. V. Bobylev, A. Nota and J. J. L. Vel´ azquez, Self-similar asymptotics for a modified Maxwell–Boltzmann equation in systems subject to deformations,Comm. Math. Phys.380(2020), no. 1, 409–448

work page 2020