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Nonlinear quantum Fokker-Planck equation near equilibrium

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Pith's one-line read The paper proves global well-posedness and algebraic decay toward equilibrium for a nonlinear quantum Fokker-Planck equation whose drift and diffusion are self-consistent functionals of the distribution.

desk verdict A correct and original perturbative theory for a genuinely new self-consistent quantum Fokker–Planck equation; the central coercivity result survives scrutiny and the paper deserves a careful referee. read the letter →

arxiv 2607.26433 v1 pith:PJGBNERJ submitted 2026-07-29 math.AP

Young-Pil Choi , Byung-Hoon Hwang , Ju-Hwan Hyun This is my paper · ORCID
classification math.AP MSC 35Q2035A0135B4082C4082C10
keywords quantumFokker-Planckequationself-consistentmacroscopicfieldsglobalwell-posednessmacro-microdecompositionalgebraicdecayratesPauliexclusionprincipleLandauBose-EinsteinandFermi-Diracstatistics
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 studies a quantum kinetic equation in which the collision frequency, bulk velocity, and temperature are not fixed externally but are computed from the distribution itself. It proves that, for small perturbations of the global quantum equilibrium, the Cauchy problem has a unique global strong solution in H^s for s≥4, that the solution preserves nonnegativity (and, in the fermionic case, the Pauli upper bound), and that it decays algebraically to equilibrium when the initial perturbation has negative Sobolev regularity. The significance is structural: the equation conserves mass, momentum, and energy, dissipates a quantum entropy, and arises formally from the quantum Landau operator, so the result provides a global perturbative theory for a fully self-consistent quantum Fokker-Planck system. A sympathetic reader should take away that the self-consistent feedback is tamed by identifying the exact null space of the linearized operator and proving coercivity on its complement.

What carries the argument

The load-bearing object is the linearized collision operator L=L0+P1, where L0 is a dissipative Fokker-Planck part and P1 is a finite-rank correction generated by the quantum-weighted modes p_j η_ℏ√µ_ℏ and (|p|²η_ℏ−3)√µ_ℏ. Using exact moment identities satisfied by the quantum equilibrium weights, the paper shows ker L=N, with N=span{√µ_ℏ, p_i√µ_ℏ, |p|²√µ_ℏ}, and establishes coercivity −⟨Lg,g⟩_{L²_p}≥λ0|(I−P)g|²_D on the microscopic complement. This microscopic coercivity is combined with a macro-micro decomposition: balance laws for the macroscopic coefficients (a,b,c) and a carefully constructed interaction functional recover dissipation of ∇_x(a,b,c), and propagation of a negative Sobolev

What would settle it

Compute the smallest eigenvalue of the self-adjoint quadratic form associated with −L restricted to N⊥ for a fixed θ0 and several ℏ satisfying ℏe^{-θ0}<1; if any nonzero direction has zero dissipation, the coercivity lemma and the global theorem fail. More directly, solve the linearized equation from a nonzero initial datum in N⊥ and check whether its D-norm decays at the claimed rate.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 1.1: for s≥4 and sufficiently small ∥g0∥_{H^s}, the perturbation equation admits a unique global solution g∈C([0,∞);H^s(R^3×R^3)) with sup_t ∥g(t)∥_{H^s}≤C∥g0∥_{H^s}. The corresponding distribution f=F_ℏ+√µ_ℏ g stays nonnegative, and in the fermionic case stays below the Pauli bound 1/ℏ. If the initial perturbation also lies in a negative Sobolev space Λ^{-s̃}_x L^2 with 0<s̃<3/2, the paper proves a hierarchy of algebraic decay estimates ∑_{l≤|α|≤s}∥∂_x^α g∥²_{L²_{x,p}}≤C_l(1+t)^{-(l+s̃)}. The structural discovery behind the theorem is that the linearized collision operator, despite its nonstandard finite-rank correction from self-consis

Load-bearing premise

The argument collapses if the linearized operator fails to be strictly coercive on the orthogonal complement of the five-dimensional null space—specifically, if λ0 in the coercivity estimate is zero or if the kernel is larger than N.

Editorial extensions

If this is right

  • Small H^s perturbations of the quantum equilibrium never grow: sup_{t≥0}∥g(t)∥_{H^s}≤C∥g0∥_{H^s}.
  • Large-time behavior is quantitative: for each integer 0≤l≤s−1, spatial derivatives of order l through s decay like (1+t)^{-(l+s̃)}.
  • Fermionic solutions remain inside the physically admissible interval 0≤f≤1/ℏ, and nonnegativity is propagated for both statistics.
  • The equation conserves mass, momentum, and kinetic energy and dissipates the quantum entropy, so the global stability result applies to a model with the conservation and entropy structure of quantum collisional kinetic theory.
  • The global energy estimate yields integrability in time of the microscopic dissipation and macroscopic gradients, giving a complete equilibration statement for the self-consistent system.

Reading between the lines

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

  • If the weighted Poincaré constant remains controlled as the quantum parameter tends to zero, the same macro-micro closure should reproduce classical nonlinear Fokker-Planck stability in the semiclassical limit; the paper does not pursue that limit.
  • The finite-rank correction structure suggests that on a torus or bounded domain, where a Poincaré inequality is available, the negative-Sobolev machinery could be replaced by exponential decay of the same hierarchy.
  • Because the formal derivation is carried out in general dimension d, the mechanism—exact null-space identification plus coercivity plus macroscopic dissipation—likely transfers to other dimensions with only the moment constants changed.
  • One testable extension is to verify numerically the spectral gap of the linearized operator on the microscopic complement; a nonzero direction with zero dissipation would falsify the coercivity lemma.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. This paper studies the Cauchy problem for the nonlinear quantum Fokker–Planck equation (1.2), in which the collision frequency, bulk velocity, and diffusion temperature are self-consistent nonlinear functionals of the distribution. The authors formally derive the model from the quantum Landau operator in the Maxwellian-molecule case under a radial ansatz (Section 2.1) and establish structural properties: conservation of mass, momentum, and kinetic energy; a quantum entropy dissipation identity; and a Pauli admissible interval for fermions. The main result (Theorem 1.1) asserts that for s ≥ 4 and sufficiently small H^s perturbations of the quantum equilibrium Fℏ, the perturbative equation (1.5) admits a unique global solution with uniform H^s bound; nonnegativity and (for κ = −1) the upper Pauli bound propagate; and, if Λ_x^{−s̃} g0 ∈ L² with 0 < s̃ < 3/2, the hierarchy Σ_{l≤|α|≤s} ||∂_x^α g||²_{L²_{x,p}} ≤ C(1+t)^{−(l+s̃)} holds. The proof combines the linearized analysis of L = L0 + P1 (Section 3), nonlinear estimates for Γ (Section 4), a macro–micro energy method with temporal interaction functionals (Section 5), and negative-Sobolev interpolation (Section 6), with technical details in the appendices.

Significance. This is a substantial and technically demanding contribution. The new analytical difficulty is the fully self-consistent dependence of the collision operator on macroscopic fields, which produces a finite-rank correction P1 whose generating modes are not the canonical collision-invariant modes. The identification ker L = N and the coercivity (3.19) on N^⊥ are the central structural achievements. I audited the reader’s weakest assumption — the microscopic coercivity of Lemma 3.2 — and the concern does not land: the inclusion N ⊂ ker L follows from (3.2) and (3.21); the dissipation identity (3.25) gives ker L ⊂ N; and the contradiction argument for (3.27) is valid. The paper has no free parameters, states explicit falsifiable decay rates, and is honest about limitations (Remark 1.1 on the range of s̃; Remark 2.1 on the formal Landau reduction). The main external input is the weighted Poincaré inequality (3.26) from [40]; it is true and elementary, but not proved in the text.

minor comments (4)
  1. [§3.3, Eq. (3.26)] The weighted Poincaré inequality (3.26) is load-bearing for the coercivity (3.19), but is cited from [40, Cor. 3.4] without stating the hypotheses or giving a proof. Since μℏ is uniformly comparable to a centered Gaussian weight by (3.4), the inequality is true and elementary; please include the statement of the cited corollary or a short proof so Lemma 3.2 is self-contained.
  2. [§4.1, Lemmas 4.3–4.5] The nonlinear estimates rely repeatedly on statements of the form 'the remaining terms are handled in the same way'. Given that these lemmas carry the entire nonlinear closure, it would substantially help verification if the common structure were isolated (for example, an abstract class of admissible flux terms) or if at least the most technical remaining contributions (N_Θ terms in Γ2 and the ℓ-coefficient terms in Lemma B.4) were spelled out.
  3. [Appendix B / Lemma 5.5] Lemma 5.5 is proved by delegation to Appendix B. Please ensure that the constants and signs in Lemmas B.1–B.4 align exactly with the definitions of the coefficient functionals in (5.19)–(5.20), and add a remark explaining how the fixed coefficients from the elliptic estimates are absorbed into the constants appearing in (5.21).
  4. [General presentation] Typographical and notation fixes: the header reads 'EQUA TION'; there are missing spaces in 'ranges ofαin' and 'Γ2'; the notation ∥g∥_{L²_p(H^s_x)} appears in Lemma 5.2 without being included in the notation list of Section 3.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is a self-contained a priori estimate whose assumptions (small H^s data, admissibility, ℏe^{-θ0}<1) do not include the target result.

full rationale

The central claim, Theorem 1.1, is a perturbative well-posedness and decay theorem for the nonlinear quantum Fokker–Planck equation (1.2). The derivation chain is: (i) the equation is rewritten in perturbation form (1.5) via f = F_ℏ + √µ_ℏ g; (ii) Lemma 3.2 establishes ker L = N and the coercivity estimate (3.19); (iii) Sections 4–5 provide local existence, nonlinear estimates, and the macro–micro closure; (iv) Section 6 extends locally to global existence and uses negative Sobolev norms for decay. At no point is a parameter fitted to data, and no predicted quantity reduces by construction to an input. The formal reduction from the quantum Landau equation in Section 2.1 is explicitly labeled non-rigorous in Remark 2.1 ('should not be interpreted as a rigorous asymptotic limit') and is motivational only; it is not used as an input to the existence proof. The weighted Poincaré inequality (3.26) is cited from [40, Cor. 3.4], an external, independently verifiable result by different authors, not a self-citation; even though it is not re-proved in this paper, that is a citation-risk, not circularity. The cited prior works by the authors concern related but different equations and are not relied on to prove the key coercivity or energy closure. The admissibility propagation (Lemma A.3), the energy equivalence (Lemma 5.8), and the smallness bootstrap are proved within the paper. Thus no load-bearing step reduces, by the paper's own equations or by self-citation, to its inputs.

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

The proof rests on standard analytical tools (weighted Poincaré, Sobolev–Moser, HLS), the model assumption ℏe^{-θ0}<1, and the formal radial-symmetry reduction that motivates the equation but is not used in the theorem. No fitted constants or new physical entities are introduced.

assumptions (4)
  • standard math Weighted Poincaré inequality ∫|ψ|²µ dp ≤ C∫|∇ψ|²µ dp for µ comparable to a Gaussian weight.
    Invoked in Lemma 3.2, Eq. (3.26), cited to [40, Cor. 3.4]; this is the backbone of the microscopic coercivity and compactness argument.
  • domain assumption Assumption ℏe^{-θ0}<1 (Eq. (1.4)) so that η_ℏ = 1+2ℏκF_ℏ is uniformly positive.
    Ensures the D-norm controls the L²_p norm and that denominators in the perturbative expansions do not vanish.
  • standard math Sobolev–Moser and Hardy–Littlewood–Sobolev inequalities in the x-variable.
    Used throughout the nonlinear moment estimates (Lemma 4.1, Lemma 6.1) and for negative-Sobolev decay.
  • domain assumption Radial symmetry ansatz f = Φ(|p−Ũ|²/(2Θ̃)) in the formal derivation from the quantum Landau operator.
    This ansatz is used only to motivate the model (Section 2.1) and is explicitly labeled formal in Remark 2.1; it is not an input to the main theorem.

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Pith. "Pith review of Nonlinear quantum Fokker-Planck equation near equilibrium." pith.science (2026). https://pith.science/paper/PJGBNERJ

@misc{pith2026260726433,
  author       = {Pith},
  title        = {Pith review of: Nonlinear quantum Fokker-Planck equation near equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJGBNERJ}},
  note         = {Machine review of arXiv:2607.26433}
}
read the original abstract

We investigate a nonlinear quantum Fokker--Planck equation with self-consistent collision frequency, bulk velocity, and temperature. In contrast to quantum Fokker--Planck equations with prescribed diffusion and friction coefficients, the macroscopic quantities are nonlinear functionals of the distribution function. The equation preserves mass, momentum, and kinetic energy, admits a quantum entropy dissipation structure, and propagates the Pauli admissible range in the fermionic case. Its collision operator is also formally connected to the quantum Landau equation. For the Cauchy problem in the three-dimensional whole space, we prove the global-in-time existence and uniqueness of strong solutions near a global quantum equilibrium. The proof is based on a perturbative macro--micro energy method that combines microscopic coercivity, estimates for nonlinear velocity moments, and a macroscopic dissipation argument. We further establish the propagation of nonnegativity and the fermionic Pauli upper bound. Under an additional negative Sobolev assumption on the initial perturbation, we obtain algebraic decay rates toward equilibrium.

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

  1. Incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation

    math.AP 2026-07 conditional novelty 7.0 of 10

    A nonlinear quantum Fokker-Planck equation with self-consistent fields converges in the diffusive limit to the incompressible Navier-Stokes-Fourier system, retaining quantum statistics in the transport coefficients.

Reference graph

Works this paper leans on

56 extracted references · 3 linked inside Pith · cited by 1 Pith paper

  1. [40]

    M. Lemou. Linearized quantum and relativistic Fokker-Planck-Landau equations.Math. Methods Appl. Sci., 23(12):1093– 1119, 2000

    2000

  2. [1]

    Alonso, V

    R. Alonso, V. Bagland, L. Desvillettes, and B. Lods. About the use of entropy production for the Landau-Fermi-Dirac equation.J. Stat. Phys., 183(1):Paper No. 10, 27, 2021

    2021

  3. [2]

    Alonso, V

    R. Alonso, V. Bagland, L. Desvillettes, and B. Lods. About the Landau-Fermi-Dirac equation with moderately soft potentials.Arch. Ration. Mech. Anal., 244(3):779–875, 2022

    2022

  4. [3]

    Alonso, V

    R. Alonso, V. Bagland, and B. Lods. Long time dynamics for the Landau-Fermi-Dirac equation with hard potentials.J. Differential Equations, 270:596–663, 2021

    2021

  5. [4]

    Arnold, F

    A. Arnold, F. Fagnola, and L. Neumann. Quantum Fokker-Planck models: the Lindblad and Wigner approaches. In Quantum probability and related topics, volume 23 ofQP–PQ: Quantum Probab. White Noise Anal., pages 23–48. World Sci. Publ., Hackensack, NJ, 2008

    2008

  6. [5]

    Arnold, I

    A. Arnold, I. M. Gamba, M. P. Gualdani, S. Mischler, C. Mouhot, and C. Sparber. The Wigner-Fokker-Planck equation: stationary states and large time behavior.Math. Models Methods Appl. Sci., 22(11):1250034, 31, 2012

    2012

  7. [6]

    Arnold, M

    A. Arnold, M. Pirner, and G. Toshpulatov. Long time behavior of Fokker–Planck equations for bosons and fermions, arXiv:2601.17594

    arXiv

  8. [7]

    G.-C. Bae, J. W. Jang, and S.-B. Yun. The relativistic quantum Boltzmann equation near equilibrium.Arch. Ration. Mech. Anal., 240(3):1593–1644, 2021

    2021

Show all 56 references
  1. [8]

    V. Bagland. Well-posedness for the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials.Proc. Roy. Soc. Edinburgh Sect. A, 134(3):415–447, 2004

    2004

  2. [9]

    Bedrossian

    J. Bedrossian. Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker- Planck equation.Ann. PDE, 3(2):Paper No. 19, 66, 2017

    2017

  3. [10]

    Bedrossian, W

    J. Bedrossian, W. Zhao, and R. Zi. Landau damping, collisionless limit, and stability threshold for the Vlasov-Poisson equation with nonlinear Fokker-Planck collisions.Comm. Math. Phys., 406(7):Paper No. 166, 87, 2025. NONLINEAR QUANTUM FOKKER–PLANCK EQUATION NEAR EQUILIBRIUM 65

    2025

  4. [11]

    F. c. Bouchut, F. c. Golse, and M. Pulvirenti.Kinetic equations and asymptotic theory, volume 4 ofSeries in Applied Mathematics (Paris). Gauthier-Villars, ´Editions Scientifiques et M´ edicales Elsevier, Paris, 2000. Edited and with a foreword by Beno ˆ ıt Perthame and Laurent...

    2000

  5. [12]

    J. A. Ca˜ nizo, J. A. Carrillo, P. Lauren¸ cot, and J. Rosado. The Fokker-Planck equation for bosons in 2D: well-posedness and asymptotic behavior.Nonlinear Anal., 137:291–305, 2016

    2016

  6. [13]

    E. A. Carlen and W. Gangbo. Solution of a model Boltzmann equation via steepest descent in the 2-Wasserstein metric. Arch. Ration. Mech. Anal., 172(1):21–64, 2004

    2004

  7. [14]

    J. A. Carrillo, M. G. Delgadino, and J. Wu. Boltzmann to Landau from the gradient flow perspective.Nonlinear Anal., 219:Paper No. 112824, 49, 2022

    2022

  8. [15]

    J. A. Carrillo, P. Lauren¸ cot, and J. Rosado. Fermi-Dirac-Fokker-Planck equation: well-posedness & long-time asymptotics. J. Differential Equations, 247(8):2209–2234, 2009

    2009

  9. [16]

    J. A. Carrillo, J. Rosado, and F. Salvarani. 1D nonlinear Fokker-Planck equations for fermions and bosons.Appl. Math. Lett., 21(2):148–154, 2008

    2008

  10. [17]

    Chavanis

    P.-H. Chavanis. Generalized thermodynamics and Fokker-Planck equations: Applications to stellar dynamics and two- dimensional turbulence.Phys. Rev. E, 68:036108, Sep 2003

    2003

  11. [18]

    Chavanis

    P.-H. Chavanis. Generalized Fokker-Planck equations and effective thermodynamics.Phys. A, 340(1-3):57–65, 2004. News and expectations in thermostatistics

    2004

  12. [19]

    Chavanis

    P.-H. Chavanis. Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski. Phys. A, 332(1-4):89–122, 2004

    2004

  13. [20]

    Y.-P. Choi. Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces.Nonlinearity, 29(7):1887–1916, 2016

    1916

  14. [21]

    Choi, B.-H

    Y.-P. Choi, B.-H. Hwang, and Y. Yoo. Global existence of weak solutions to the nonlinear Vlasov-Fokker-Planck equation. J. Differential Equations, 444:Paper No. 113573, 53, 2025

    2025

  15. [22]

    Choi and S

    Y.-P. Choi and S. Song. Global weak solutions to nonlinear kinetic Fokker–Planck equations in bounded domains under physical initial data, arXiv:2510.06656

  16. [23]

    Desvillettes and V

    L. Desvillettes and V. Ricci. A rigorous derivation of a linear kinetic equation of Fokker-Planck type in the limit of grazing collisions.J. Statist. Phys., 104(5-6):1173–1189, 2001

    2001

  17. [24]

    Desvillettes and C

    L. Desvillettes and C. Villani. On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation.Comm. Pure Appl. Math., 54(1):1–42, 2001

    2001

  18. [25]

    Dolbeault, C

    J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for linear kinetic equations conserving mass.Trans. Amer. Math. Soc., 367(6):3807–3828, 2015

    2015

  19. [26]

    R. Duan. On the Cauchy problem for the Boltzmann equation in the whole space: global existence and uniform stability inL 2 ξ(H N x ).J. Differential Equations, 244(12):3204–3234, 2008

    2008

  20. [27]

    R. T. Glassey.The Cauchy problem in kinetic theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996

    1996

  21. [28]

    Golding, M

    W. Golding, M. P. Gualdani, and N. Zamponi. Existence of smooth solutions to the Landau-Fermi-Dirac equation with Coulomb potential.Commun. Math. Sci., 20(8):2315–2365, 2022

    2022

  22. [29]

    Golding, C

    W. Golding, C. Henderson, and L. Silvestre. Pointwise bounds and obstructions to blowup for the Landau and Boltzmann equations, arXiv:2605.20426

    arXiv

  23. [30]

    T. Goudon. On Boltzmann equations and Fokker-Planck asymptotics: influence of grazing collisions.J. Statist. Phys., 89(3-4):751–776, 1997

    1997

  24. [31]

    M. P. Gualdani, N. Pavlovi´ c, J. Toyota, and D. Wynter. From the quantum Boltzmann operator to the quantum Landau operator, arXiv:2511.19823

  25. [32]

    Guillen and L

    N. Guillen and L. Silvestre. The Landau equation does not blow up.Acta Math., 234(2):315–375, 2025

    2025

  26. [33]

    Y. Guo. The Boltzmann equation in the whole space.Indiana Univ. Math. J., 53(4):1081–1094, 2004

    2004

  27. [34]

    Guo and Y

    Y. Guo and Y. Wang. Decay of dissipative equations and negative Sobolev spaces.Comm. Partial Differential Equations, 37(12):2165–2208, 2012

    2012

  28. [35]

    J. Hu, S. Jin, and B. Yan. A numerical scheme for the quantum Fokker-Planck-Landau equation efficient in the fluid regime.Commun. Comput. Phys., 12(5):1541–1561, 2012

    2012

  29. [36]

    Kaniadakis

    G. Kaniadakis. Generalized Boltzmann equation describing the dynamics of bosons and fermions.Phys. Lett. A, 203(4):229– 234, 1995

    1995

  30. [37]

    Kaniadakis.H-theorem and generalized entropies within the framework of nonlinear kinetics.Phys

    G. Kaniadakis.H-theorem and generalized entropies within the framework of nonlinear kinetics.Phys. Lett. A, 288(5- 6):283–291, 2001

    2001

  31. [38]

    Kaniadakis and P

    G. Kaniadakis and P. Quarati. Kinetic equation for classical particles obeying an exclusion principle.Phys. Rev. E, 48:4263– 4270, Dec 1993

    1993

  32. [39]

    Kaniadakis and P

    G. Kaniadakis and P. Quarati. Classical model of bosons and fermions.Phys. Rev. E, 49:5103–5110, Jun 1994

    1994

  33. [41]

    Liao and X

    J. Liao and X. Yang. Stability of global Maxwellian for fully nonlinear Fokker-Planck equations.J. Stat. Phys., 185(3):Paper No. 23, 27, 2021

    2021

  34. [42]

    T.-P. Liu, T. Yang, and S.-H. Yu. Energy method for Boltzmann equation.Phys. D, 188(3-4):178–192, 2004. 66 CHOI, HW ANG, AND HYUN

    2004

  35. [43]

    X. Lu. On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles.J. Statist. Phys., 105(1-2):353–388, 2001

    2001

  36. [44]

    Lu and B

    X. Lu and B. Wennberg. On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles.Arch. Ration. Mech. Anal., 168(1):1–34, 2003

    2003

  37. [45]

    Luo and X

    L. Luo and X. Zhang. Global classical solutions for quantum kinetic Fokker-Planck equations.Acta Math. Sci. Ser. B (Engl. Ed.), 35(1):140–156, 2015

    2015

  38. [46]

    Majda.Compressible fluid flow and systems of conservation laws in several space variables, volume 53 ofApplied Mathematical Sciences

    A. Majda.Compressible fluid flow and systems of conservation laws in several space variables, volume 53 ofApplied Mathematical Sciences. Springer-Verlag, New York, 1984

    1984

  39. [47]

    Neumann and C

    L. Neumann and C. Schmeiser. Convergence to global equilibrium for a kinetic fermion model.SIAM J. Math. Anal., 36(5):1652–1663, 2005

    2005

  40. [48]

    Neumann and C

    L. Neumann and C. Sparber. Stability of steady states in kinetic Fokker-Planck equations for bosons and fermions.Com- mun. Math. Sci., 5(4):765–777, 2007

    2007

  41. [49]

    Ouyang and L

    Z. Ouyang and L. Wu. On the quantum Boltzmann equation near Maxwellian and vacuum.J. Differential Equations, 316:471–551, 2022

    2022

  42. [50]

    Pazy.Semigroups of linear operators and applications to partial differential equations, volume 44 ofApplied Mathemat- ical Sciences

    A. Pazy.Semigroups of linear operators and applications to partial differential equations, volume 44 ofApplied Mathemat- ical Sciences. Springer-Verlag, New York, 1983

    1983

  43. [51]

    P. Sampaio. Global solutions to the Landau–Fermi–Dirac equation, arXiv:2410.12681

    arXiv

  44. [52]

    P. Sampaio. On the semi-classical limit for the Landau–Fermi–Dirac equation, arXiv:2501.13800

    arXiv

  45. [53]

    C. Villani. On the Cauchy problem for Landau equation: sequential stability, global existence.Adv. Differential Equations, 1(5):793–816, 1996

    1996

  46. [54]

    C. Villani. On the spatially homogeneous Landau equation for Maxwellian molecules.Math. Models Methods Appl. Sci., 8(6):957–983, 1998

    1998

  47. [55]

    C. Villani. A review of mathematical topics in collisional kinetic theory. InHandbook of mathematical fluid dynamics, Vol. I, pages 71–305. North-Holland, Amsterdam, 2002

    2002

  48. [56]

    Y. Wang, C. Xiao, and Y. Zhang. Global existence and large time behavior of the quantum Boltzmann equation with small relative entropy.Kinet. Relat. Models, 16(1):133–177, 2023. (Young-Pil Choi) Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03...

    2023

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