pith. sign in

arxiv: 2510.17455 · v2 · submitted 2025-10-20 · 🧮 math.AP

A unified relative entropy framework for macroscopic limits of Vlasov--Fokker--Planck equations

Young-Pil Choi , Jinwook Jung This is my paper

Pith reviewed 2026-05-18 06:24 UTC · model grok-4.3

classification 🧮 math.AP
keywords relative entropyVlasov-Fokker-Planck equationmacroscopic limitdiffusive limithigh-field limitstrong magnetic field limitaggregation equationsurface quasi-geostrophic equation
0
0 comments X

The pith

Relative entropy provides a unified framework for proving both strong and weak convergence in singular macroscopic limits of Vlasov-Fokker-Planck equations

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a single relative entropy method to derive three distinct macroscopic limits from Vlasov-Fokker-Planck equations that include Riesz-type interactions and Fokker-Planck relaxation. The regimes treated are the diffusive limit that produces a drift-diffusion equation, the high-field limit that produces the repulsive aggregation equation, and the strong magnetic field limit that produces a generalized surface quasi-geostrophic equation. The framework merges entropy dissipation estimates with Fisher-information bounds and modulated interaction energies to obtain stability controls. A sympathetic reader would care because the same object supplies quantitative rates for strong convergence when initial data are well-prepared and also supplies weaker but still quantitative controls when data are only mildly prepared or the entropy itself diverges with the singular parameter.

Core claim

The authors establish that a relative entropy functional, when combined with entropy dissipation, Fisher-information control, and modulated interaction energies, yields a robust stability theory. This theory produces quantitative relative entropy estimates that imply strong convergence to the three target macroscopic equations under well-prepared initial data, and it produces complementary weak convergence results: sharper weak-topology estimates consistent with optimal scaling in the diffusive regime, bounded Lipschitz stability for mildly prepared data in the high-field regime even when relative entropy diverges, and bounded Lipschitz control on rescaled momentum together with negative Sob

What carries the argument

The unified relative entropy framework that integrates entropy dissipation, Fisher-information control, and modulated interaction energies to obtain stability estimates across the three singular scaling regimes

If this is right

  • In the diffusive regime the method yields sharper quantitative estimates in weak topologies that match the formally optimal scaling.
  • In the high-field regime bounded Lipschitz stability propagates for mildly prepared initial data even when relative entropy diverges with the scaling parameter.
  • In the strong magnetic field regime quantitative weak estimates include bounded Lipschitz control of the rescaled momentum and negative Sobolev control of the density.
  • Relative entropy serves as a mechanism that treats both strong convergence and low-regularity or mildly prepared regimes within one argument.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same combination of controls could be tested on other singular limits such as those arising in Vlasov-Poisson or Landau equations.
  • Numerical schemes that discretely preserve a discrete relative entropy might inherit the stability conclusions for singular-parameter regimes.
  • The framework suggests that modulated energies could be used to obtain rates in mean-field derivations of aggregation or quasi-geostrophic models from particle systems.
  • Extensions to time-dependent or inhomogeneous magnetic fields would require only local adjustments to the modulated interaction term.

Load-bearing premise

Strong convergence results require initial data that are well-prepared with respect to the singular scaling parameter.

What would settle it

An explicit family of initial data that is not well-prepared for which the relative entropy distance fails to vanish in the high-field limit while the weak limit still exists.

read the original abstract

We develop a unified relative entropy framework for macroscopic limits of kinetic equations with Riesz-type interactions and Fokker-Planck relaxation. Our analysis covers three prototypical singular regimes: the diffusive limit leading to a drift-diffusion equation, the high-field limit yielding the aggregation equation in the repulsive regime, and the strong magnetic field limit producing a generalized surface quasi-geostrophic equation. The method combines entropy dissipation, Fisher-information control, and modulated interaction energies into a robust stability theory yielding both strong and weak convergence results. For the strong convergence, we establish quantitative relative entropy estimates toward macroscopic limits under well-prepared initial data, extending the scope of the method to settings where nonlocal forces and singular scalings play a decisive role. For the weak convergence, our approach captures three complementary phenomena: in the diffusive regime, it yields sharper quantitative estimates in weak topologies consistent with the formally optimal scaling; in the high-field regime, it propagates bounded Lipschitz stability for a class of mildly prepared initial data, even when the relative entropy diverges with respect to the singular scaling parameter; and in the strong magnetic field regime, it provides quantitative weak estimates, including bounded Lipschitz control of the rescaled momentum and negative Sobolev control of the density. This broader perspective shows that relative entropy provides not only a tool for strong convergence, but also a mechanism for treating low-regularity and mildly prepared regimes. The analysis highlights the unifying role of relative entropy in connecting microscopic dissipation with both strong and weak macroscopic convergence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a unified relative entropy framework for macroscopic limits of Vlasov--Fokker--Planck equations with Riesz-type interactions and Fokker-Planck relaxation. It addresses three prototypical singular regimes: the diffusive limit to a drift-diffusion equation, the high-field limit to the aggregation equation in the repulsive regime, and the strong magnetic field limit to a generalized surface quasi-geostrophic equation. The method combines entropy dissipation, Fisher-information control, and modulated interaction energies to yield both strong convergence under well-prepared initial data and weak convergence results for mildly prepared data.

Significance. If the results hold, this framework provides a robust stability theory for connecting microscopic dissipation to macroscopic convergence in settings with singular scalings and nonlocal forces. The ability to handle both strong and weak regimes, including sharper quantitative weak estimates and bounded Lipschitz stability, is a significant advancement in the field of kinetic theory and macroscopic limits. The unifying role of relative entropy is well-highlighted.

major comments (1)
  1. [Abstract and Section on Strong Convergence] The quantitative relative entropy estimates for strong convergence in the diffusive, high-field, and strong magnetic field limits are established only under well-prepared initial data with respect to the singular scaling parameter. This assumption is central to controlling the initial term in the relative entropy and allowing the dissipation to dominate via Gronwall. The paper should clarify whether this preparation is necessary for the quantitative rates or if it can be relaxed while maintaining the strong convergence conclusion.
minor comments (2)
  1. [Notation and Definitions] Ensure that the scaling parameters and their singular limits are consistently denoted throughout the manuscript to avoid confusion in the different regimes.
  2. [References] Consider adding citations to related works on relative entropy methods for Vlasov equations with singular interactions to better contextualize the novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for the constructive comment. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and Section on Strong Convergence] The quantitative relative entropy estimates for strong convergence in the diffusive, high-field, and strong magnetic field limits are established only under well-prepared initial data with respect to the singular scaling parameter. This assumption is central to controlling the initial term in the relative entropy and allowing the dissipation to dominate via Gronwall. The paper should clarify whether this preparation is necessary for the quantitative rates or if it can be relaxed while maintaining the strong convergence conclusion.

    Authors: We agree with the referee that the well-prepared initial data assumption is necessary for the quantitative relative entropy estimates that yield strong convergence. The initial relative entropy term must vanish with the singular scaling parameter for the dissipation and Gronwall argument to produce a rate that tends to zero; without well-prepared data this term generally remains of order one, so the quantitative strong convergence cannot hold. Our framework already separates this from the weak convergence results, which apply to mildly prepared data in each regime. In the revised version we will add an explicit remark in the abstract and in the sections treating strong convergence to clarify that the quantitative rates require well-prepared data and cannot be relaxed while preserving the quantitative strong-convergence conclusion. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation relies on standard entropy identities and external operator properties

full rationale

The paper constructs its unified relative entropy framework from entropy dissipation, Fisher-information bounds, and modulated interaction energies applied to Vlasov-Fokker-Planck equations with Riesz interactions. These are standard tools in kinetic theory whose properties (dissipation rates, control of nonlocal forces) are invoked from the literature on Fokker-Planck operators and Riesz potentials rather than defined in terms of the target macroscopic limits. Quantitative strong-convergence estimates are explicitly conditioned on well-prepared initial data with respect to the singular scaling parameter; this is an input assumption, not a quantity fitted or redefined by the paper itself. Weak-convergence results for mildly prepared data are obtained by separate Gronwall-type arguments that do not reduce to the strong-convergence case. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that close the central argument appear in the derivation chain. The framework therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard domain assumptions from kinetic theory and PDE analysis rather than new free parameters or invented entities.

axioms (2)
  • domain assumption Riesz-type interaction kernels satisfy the necessary integrability and positivity properties for entropy dissipation estimates
    Invoked when defining the modulated interaction energies for the three regimes
  • domain assumption Fokker-Planck relaxation produces the expected entropy dissipation and Fisher-information bounds
    Used to close the relative entropy estimates in all three limits

pith-pipeline@v0.9.0 · 5798 in / 1330 out tokens · 38384 ms · 2026-05-18T06:24:03.779220+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages

  1. [1]

    Ambrosio, N

    L. Ambrosio, N. Gigli, and G. Savar´ e.Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel, second edition, 2008

    work page 2008

  2. [2]

    Arnold, J

    A. Arnold, J. A. Carrillo, I. Gamba, and C.-W. Shu. Low and high field scaling limits for the Vlasov- and Wigner-Poisson- Fokker-Planck systems. volume 30, pages 121–153. 2001. The Sixteenth International Conference on Transport Theory, Part I (Atlanta, GA, 1999)

    work page 2001

  3. [3]

    Bardos, F

    C. Bardos, F. c. Golse, and C. D. Levermore. Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation.Comm. Pure Appl. Math., 46(5):667–753, 1993

    work page 1993

  4. [4]

    Bardos, F

    C. Bardos, F. c. Golse, and D. Levermore. Fluid dynamic limits of kinetic equations. I. Formal derivations.J. Statist. Phys., 63(1-2):323–344, 1991

    work page 1991

  5. [5]

    Ben-Porat, M

    I. Ben-Porat, M. Iacobelli, and A. Rege. Derivation of Yudovich solutions of incompressible Euler from the Vlasov-Poisson system.SIAM J. Math. Anal., 57(1):886–906, 2025

    work page 2025

  6. [6]

    Berthelin and A

    F. Berthelin and A. Vasseur. From kinetic equations to multidimensional isentropic gas dynamics before shocks.SIAM J. Math. Anal., 36(6):1807–1835, 2005

    work page 2005

  7. [7]

    M. Bostan. The Vlasov-Maxwell system with strong initial magnetic field: guiding-center approximation.Multiscale Model. Simul., 6(3):1026–1058, 2007

    work page 2007

  8. [8]

    Bostan and T

    M. Bostan and T. Goudon. Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system; the one and one half dimensional case.Kinet. Relat. Models, 1(1):139–170, 2008

    work page 2008

  9. [9]

    Bostan and A.-T

    M. Bostan and A.-T. Vu. Asymptotic behavior of the two-dimensional Vlasov-Poisson-Fokker-Planck equation with a strong external magnetic field.Kinet. Relat. Models, 18(1):101–147, 2025

    work page 2025

  10. [10]

    F. c. Bouchut. Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions.J. Funct. Anal., 111(1):239–258, 1993. 40 CHOI AND JUNG

    work page 1993

  11. [11]

    Y. Brenier. Convergence of the Vlasov-Poisson system to the incompressible Euler equations.Comm. Partial Differential Equations, 25(3-4):737–754, 2000

    work page 2000

  12. [12]

    Caffarelli and L

    L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian.Comm. Partial Differential Equa- tions, 32(7-9):1245–1260, 2007

    work page 2007

  13. [13]

    R. E. Caflisch. The fluid dynamic limit of the nonlinear Boltzmann equation.Comm. Pure Appl. Math., 33(5):651–666, 1980

    work page 1980

  14. [14]

    J. A. Carrillo and Y.-P. Choi. Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 37(4):925–954, 2020

    work page 2020

  15. [15]

    J. A. Carrillo and Y.-P. Choi. Mean-field limits: from particle descriptions to macroscopic equations.Arch. Ration. Mech. Anal., 241(3):1529–1573, 2021

    work page 2021

  16. [16]

    J. A. Carrillo, Y.-P. Choi, and J. Jung. Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forces.Math. Models Methods Appl. Sci., 31(2):327–408, 2021

    work page 2021

  17. [17]

    J. A. Carrillo, Y.-P. Choi, and Y. Peng. Large friction-high force fields limit for the nonlinear Vlasov-Poisson-Fokker-Planck system.Kinet. Relat. Models, 15(3):355–384, 2022

    work page 2022

  18. [18]

    J. A. Carrillo, Y.-P. Choi, and O. Tse. Convergence to equilibrium in Wasserstein distance for damped Euler equations with interaction forces.Comm. Math. Phys., 365(1):329–361, 2019

    work page 2019

  19. [19]

    J. A. Carrillo and J. Soler. On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in Lp spaces.Math. Methods Appl. Sci., 18(10):825–839, 1995

    work page 1995

  20. [20]

    Cercignani, R

    C. Cercignani, R. Illner, and M. Pulvirenti.The mathematical theory of dilute gases, volume 106 ofApplied Mathematical Sciences. Springer-Verlag, New York, 1994

    work page 1994

  21. [21]

    Y.-P. Choi. Large friction limit of pressureless Euler equations with nonlocal forces.J. Differential Equations, 299:196–228, 2021

    work page 2021

  22. [22]

    Y.-P. Choi, S. Fagioli, and V. Iorio. Small inertia limit for coupled kinetic swarming models.J. Nonlinear Sci., 35(2):Paper No. 39, 52, 2025

    work page 2025

  23. [23]

    Choi and I.-J

    Y.-P. Choi and I.-J. Jeong. Classical solutions for fractional porous medium flow.Nonlinear Anal., 210:Paper No. 112393, 13, 2021

    work page 2021

  24. [24]

    Choi and I.-J

    Y.-P. Choi and I.-J. Jeong. Relaxation to fractional porous medium equation from Euler-Riesz system.J. Nonlinear Sci., 31(6):Paper No. 95, 28, 2021

    work page 2021

  25. [25]

    Choi and I.-J

    Y.-P. Choi and I.-J. Jeong. Global-in-time existence of weak solutions for Vlasov-Manev-Fokker-Planck system.Kinet. Relat. Models, 16(1):41–53, 2023

    work page 2023

  26. [26]

    Choi, I.-J

    Y.-P. Choi, I.-J. Jeong, and K. Kang. Global Cauchy problem for the Vlasov-Riesz-Fokker-Planck system near the global Maxwellian.J. Evol. Equ., 24(3):Paper No. 67, 25, 2024

    work page 2024

  27. [27]

    Choi and J

    Y.-P. Choi and J. Jung. Modulated energy estimates for singular kernels and their applications to asymptotic analyses for kinetic equations.SIAM J. Math. Anal., 56(2):1525–1559, 2024

    work page 2024

  28. [28]

    Choi and O

    Y.-P. Choi and O. Tse. Quantified overdamped limit for kinetic Vlasov-Fokker-Planck equations with singular interaction forces.J. Differential Equations, 330:150–207, 2022

    work page 2022

  29. [29]

    P. Degond. Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions.Ann. Sci. ´Ecole Norm. Sup. (4), 19(4):519–542, 1986

    work page 1986

  30. [30]

    Dembo and O

    A. Dembo and O. Zeitouni.Large deviations techniques and applications, volume 38 ofApplications of Mathematics (New York). Springer-Verlag, New York, second edition, 1998

    work page 1998

  31. [31]

    M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. I. II. Comm. Pure Appl. Math., 28:1–47; ibid. 28 (1975), 279–301, 1975

    work page 1975

  32. [32]

    M. H. Duong, A. Lamacz, M. A. Peletier, A. Schlichting, and U. Sharma. Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics.Nonlinearity, 31(10):4517–4566, 2018

    work page 2018

  33. [33]

    M. H. Duong, A. Lamacz, M. A. Peletier, and U. Sharma. Variational approach to coarse-graining of generalized gradient flows.Calc. Var. Partial Differential Equations, 56(4):Paper No. 100, 65, 2017

    work page 2017

  34. [34]

    Dupuis and R

    P. Dupuis and R. S. Ellis.A weak convergence approach to the theory of large deviations. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, 1997. A Wiley-Interscience Publication

    work page 1997

  35. [35]

    El Ghani and N

    N. El Ghani and N. Masmoudi. Diffusion limit of the Vlasov-Poisson-Fokker-Planck system.Commun. Math. Sci., 8(2):463– 479, 2010

    work page 2010

  36. [36]

    R. C. Fetecau and W. Sun. First-order aggregation models and zero inertia limits.J. Differential Equations, 259(11):6774– 6802, 2015

    work page 2015

  37. [37]

    Figalli and M.-J

    A. Figalli and M.-J. Kang. A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment.Anal. PDE, 12(3):843–866, 2019

    work page 2019

  38. [38]

    Freidlin

    M. Freidlin. Some remarks on the Smoluchowski-Kramers approximation.J. Statist. Phys., 117(3-4):617–634, 2004

    work page 2004

  39. [39]

    F. c. Golse and L. Saint-Raymond. The Vlasov-Poisson system with strong magnetic field.J. Math. Pures Appl. (9), 78(8):791–817, 1999

    work page 1999

  40. [40]

    T. Goudon. Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: analysis of the two-dimensional case.Math. Models Methods Appl. Sci., 15(5):737–752, 2005

    work page 2005

  41. [41]

    Goudon, J

    T. Goudon, J. Nieto, F. Poupaud, and J. Soler. Multidimensional high-field limit of the electrostatic Vlasov-Poisson- Fokker-Planck system.J. Differential Equations, 213(2):418–442, 2005

    work page 2005

  42. [42]

    L. Gross. Logarithmic Sobolev inequalities.Amer. J. Math., 97(4):1061–1083, 1975

    work page 1975

  43. [43]

    Han-Kwan

    D. Han-Kwan. Quasineutral limit of the Vlasov-Poisson system with massless electrons.Comm. Partial Differential Equa- tions, 36(8):1385–1425, 2011. RELATIVE ENTROPY ESTIMATES FOR VLASOV–FOKKER–PLANCK EQUATIONS 41

    work page 2011

  44. [44]

    P.-E. Jabin. Macroscopic limit of Vlasov type equations with friction.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 17(5):651–672, 2000

    work page 2000

  45. [45]

    H. A. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions.Physica, 7:284–304, 1940

    work page 1940

  46. [46]

    J. Y. Lee, J. W. Jang, and H. J. Hwang. The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson- Nernst-Planck systemviathe deep neural network approach.ESAIM Math. Model. Numer. Anal., 55(5):1803–1846, 2021

    work page 2021

  47. [47]

    D. Li. On Kato-Ponce and fractional Leibniz.Rev. Mat. Iberoam., 35(1):23–100, 2019

    work page 2019

  48. [48]

    E. Miot. The gyrokinetic limit for the Vlasov-Poisson system with a point charge.Nonlinearity, 32(2):654–677, 2019

    work page 2019

  49. [49]

    Nelson.Dynamical theories of Brownian motion

    E. Nelson.Dynamical theories of Brownian motion. Princeton University Press, Princeton, NJ, 1967

    work page 1967

  50. [50]

    Nguyen, M

    Q.-H. Nguyen, M. Rosenzweig, and S. Serfaty. Mean-field limits of Riesz-type singular flows.Ars Inven. Anal., pages Paper No. 4, 45, 2022

    work page 2022

  51. [51]

    Nieto, F

    J. Nieto, F. Poupaud, and J. Soler. High-field limit for the Vlasov-Poisson-Fokker-Planck system.Arch. Ration. Mech. Anal., 158(1):29–59, 2001

    work page 2001

  52. [52]

    Petrache and S

    M. Petrache and S. Serfaty. Next order asymptotics and renormalized energy for Riesz interactions.J. Inst. Math. Jussieu, 16(3):501–569, 2017

    work page 2017

  53. [53]

    F. Poupaud. Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory.Z. Angew. Math. Mech., 72(8):359–372, 1992

    work page 1992

  54. [54]

    Poupaud and J

    F. Poupaud and J. Soler. Parabolic limit and stability of the Vlasov-Fokker-Planck system.Math. Models Methods Appl. Sci., 10(7):1027–1045, 2000

    work page 2000

  55. [55]

    Saint-Raymond

    L. Saint-Raymond. The gyrokinetic approximation for the Vlasov-Poisson system.Math. Models Methods Appl. Sci., 10(9):1305–1332, 2000

    work page 2000

  56. [56]

    Saint-Raymond

    L. Saint-Raymond. Convergence of solutions to the Boltzmann equation in the incompressible Euler limit.Arch. Ration. Mech. Anal., 166(1):47–80, 2003

    work page 2003

  57. [57]

    S. Serfaty. Mean field limit for Coulomb-type flows.Duke Math. J., 169(15):2887–2935, 2020. With an appendix by Mitia Duerinckx and Serfaty

    work page 2020

  58. [58]

    Unterreiter, A

    A. Unterreiter, A. Arnold, P. Markowich, and G. Toscani. On generalized Csisz´ ar-Kullback inequalities.Monatsh. Math., 131(3):235–253, 2000

    work page 2000

  59. [59]

    A. F. Vasseur. Recent results on hydrodynamic limits. InHandbook of differential equations: evolutionary equations. Vol. IV, Handb. Differ. Equ., pages 323–376. Elsevier/North-Holland, Amsterdam, 2008

    work page 2008

  60. [60]

    Wu, T.-C

    H. Wu, T.-C. Lin, and C. Liu. Diffusion limit of kinetic equations for multiple species charged particles.Arch. Ration. Mech. Anal., 215(2):419–441, 2015

    work page 2015

  61. [61]

    M. Zhong. Diffusion limit and the optimal convergence rate of the Vlasov-Poisson-Fokker-Planck system.Kinet. Relat. Models, 15(1):1–26, 2022. (Young-Pil Choi) Department of Mathematics Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03722, Republic of Korea Email address:ypchoi@yonsei.ac.kr (Jinwook Jung) Department of Mathematics and Research Instit...

    work page 2022