pith. sign in

arxiv: 2605.24392 · v1 · pith:GKFQFRTFnew · submitted 2026-05-23 · 🧮 math.AP

Hydrodynamic Limit of the Boltzmann Equation toward Generic Riemann Solutions with Shocks

Mingi Choe , Moon-jin Kang , Chanwoo Kim This is my paper

Pith reviewed 2026-06-30 13:26 UTC · model grok-4.3

classification 🧮 math.AP
keywords hydrodynamic limitBoltzmann equationRiemann solutionscompressible Euler systemshock wavescontact discontinuitykinetic shock layermacro-micro decomposition
0
0 comments X

The pith

The one-dimensional Boltzmann equation converges to Riemann solutions of the compressible Euler system, including shocks and contacts, as the Knudsen number vanishes.

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

The paper proves that solutions of the Boltzmann equation with hard-sphere collisions in one space dimension approach the local Maxwellian distributions tied to Riemann solutions of the Euler equations. The covered Riemann solutions include superpositions of two shocks with a contact discontinuity or a rarefaction with a contact and a shock. Convergence occurs globally in time in the L2 norm without excising either the shock layers or the initial layer, provided the initial data are well-prepared and the wave strengths are small. Shock positions are tracked by time-dependent shifts that turn out to be functions of bounded variation. This supplies a rigorous passage from kinetic to fluid description even when discontinuities are present.

Core claim

For suitably well-prepared initial data and sufficiently small wave strength, the corresponding Boltzmann solution exists globally in time and converges, as the Knudsen number vanishes, to the local Maxwellian associated with the Riemann solution in L^2([0,T]×R_x×R^3_ξ) for any T>0. The shock locations are modulated by dynamical unknowns, the Shifts, which are obtained as BV functions on [0,T]. The convergence is proved without removing either the shock layer or the initial layer. In the special case of a single shock, the analysis gives a sharp quantitative description of the kinetic shock layer up to the dynamically selected Shift.

What carries the argument

Macro-micro decomposition combined with a kinetic adaptation of the a-contraction method for shocks, layer analysis, and compactness arguments for the shifts.

If this is right

  • The Boltzmann solution exists globally in time when the wave strength is small.
  • Convergence to the local Maxwellian holds in L2 for any finite time horizon without layer removal.
  • Shock locations are tracked by BV shifts obtained from the compactness argument.
  • A sharp quantitative description of the kinetic shock layer is available in the single-shock case.

Where Pith is reading between the lines

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

  • The same smallness condition that controls interactions may also be the threshold beyond which stronger wave interactions produce different kinetic structures.
  • The use of dynamical shifts to locate shocks suggests that the limit procedure preserves the locations predicted by the underlying hyperbolic system.
  • The layer-analysis techniques developed here could be tested on other collision kernels that admit similar contraction estimates.

Load-bearing premise

The wave strengths must be small enough that global existence and convergence can be obtained without excising the layers.

What would settle it

A concrete counter-example or numerical computation for wave strengths above the smallness threshold in which the Boltzmann solution fails to converge to the corresponding Euler Riemann solution in the stated L2 norm.

read the original abstract

We establish the hydrodynamic limit of the one-dimensional Boltzmann equation with hard-sphere collisions toward Riemann solutions of the compressible Euler system. The Riemann solutions covered by our result include generic superpositions of elementary waves: either two shock waves and a contact discontinuity, or a rarefaction wave, a contact discontinuity, and a shock wave. For suitably well-prepared initial data and sufficiently small wave strength, we prove that the corresponding Boltzmann solution exists globally in time and converges, as the Knudsen number vanishes, to the local Maxwellian associated with the Riemann solution in $L^2([0,T]\times\mathbb R_x\times\mathbb R^3_\xi)$ for any $T>0$. The shock locations are modulated by dynamical unknowns, the Shifts, which are obtained as BV functions on $[0,T]$. A distinctive point of our result is that the convergence is proved without removing either the shock layer or the initial layer. In the special case of a single shock, our analysis gives a sharp quantitative description of the kinetic shock layer, up to the dynamically selected Shift. The proof combines the macro-micro decomposition, a kinetic adaptation of the $a$-contraction method for shocks, layer analysis, and compactness arguments for the Shifts.

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

2 major / 0 minor

Summary. The paper establishes the hydrodynamic limit of the one-dimensional Boltzmann equation with hard-sphere collisions toward Riemann solutions of the compressible Euler system. For suitably well-prepared initial data and sufficiently small wave strength, the Boltzmann solution exists globally in time and converges in L²([0,T]×R_x×R³_ξ) to the local Maxwellian associated with the Riemann solution (including generic superpositions of two shocks plus contact, or rarefaction plus contact plus shock) as the Knudsen number vanishes, without removing shock or initial layers. Shock locations are modulated by BV shifts obtained via compactness. The proof combines macro-micro decomposition, a kinetic adaptation of the a-contraction method, layer analysis, and compactness arguments.

Significance. If the result holds, it is a notable advance in kinetic theory: it extends hydrodynamic limits to multi-wave Riemann solutions containing shocks, achieves convergence without layer removal, and supplies a sharp quantitative description of the kinetic shock layer in the single-shock case. The adaptation of a-contraction to the kinetic setting and the handling of modulated shifts via BV compactness are technically distinctive strengths that could influence subsequent work on non-perturbative limits.

major comments (2)
  1. [Abstract] Abstract: the title advertises convergence 'toward Generic Riemann Solutions with Shocks', yet the result is stated only for 'sufficiently small wave strength'. The smallness is used both to close the a-contraction estimates around the modulated Riemann solution and to control interactions among shocks, contacts, and rarefactions while keeping the shifts BV. No explicit threshold is supplied, nor is a necessity argument given showing that the smallness can be removed. This directly affects whether the central claim reaches the generic (finite-strength) case suggested by the title.
  2. [Abstract] Abstract and §1 (presumed introduction): the global-in-time existence and L² convergence without layer removal are asserted under the small-wave-strength hypothesis, but the manuscript does not indicate whether the a-contraction and compactness arguments contain a quantitative dependence on wave strength that would permit an explicit threshold or a continuation argument to finite strength. This is load-bearing for the advertised scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the positive evaluation of the significance of our results. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the title advertises convergence 'toward Generic Riemann Solutions with Shocks', yet the result is stated only for 'sufficiently small wave strength'. The smallness is used both to close the a-contraction estimates around the modulated Riemann solution and to control interactions among shocks, contacts, and rarefactions while keeping the shifts BV. No explicit threshold is supplied, nor is a necessity argument given showing that the smallness can be removed. This directly affects whether the central claim reaches the generic (finite-strength) case suggested by the title.

    Authors: We use 'generic' in the title to denote the general admissible combinations of elementary waves (two shocks plus contact, or rarefaction plus contact plus shock), as opposed to specially chosen or single-wave configurations; this is consistent with the abstract, which already states the small-wave-strength hypothesis. The smallness is indeed essential to close the kinetic a-contraction estimates and to keep the modulated shifts of bounded variation. We do not claim the result for arbitrary finite strength, nor do we supply an explicit threshold or a necessity argument for removing smallness, as these lie outside the perturbative framework of the paper. To prevent misinterpretation we will revise the abstract and the opening of §1 to state explicitly that 'generic' refers to wave configurations under the small-strength assumption. revision: yes

  2. Referee: [Abstract] Abstract and §1 (presumed introduction): the global-in-time existence and L² convergence without layer removal are asserted under the small-wave-strength hypothesis, but the manuscript does not indicate whether the a-contraction and compactness arguments contain a quantitative dependence on wave strength that would permit an explicit threshold or a continuation argument to finite strength. This is load-bearing for the advertised scope.

    Authors: The constants appearing in the a-contraction estimates and in the compactness argument for the BV shifts depend quantitatively on the wave strength; smallness is required to absorb interaction and layer error terms. We have not computed an explicit numerical threshold, which would demand a complete bookkeeping of all implicit constants, nor have we developed a continuation argument to finite strength, because the method is inherently perturbative. We will add a short clarifying paragraph in the introduction describing this dependence without altering the stated scope of the theorem. revision: partial

Circularity Check

0 steps flagged

No circularity; direct mathematical proof under explicit smallness assumption

full rationale

The derivation is a standard PDE analysis proof that combines macro-micro decomposition, kinetic a-contraction, layer analysis, and compactness for the shifts. The small wave strength is stated upfront as a hypothesis needed to close global estimates and control interactions; it is not obtained by fitting, self-definition, or reduction to a prior self-citation. No load-bearing step equates the target convergence statement to its own inputs by construction, and the argument remains independent of any fitted or renamed quantities. This is the normal case of a self-contained existence/convergence theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, invented entities, or non-standard axioms are stated. The result rests on standard background results in kinetic theory and hyperbolic PDEs.

axioms (1)
  • standard math Standard existence and regularity results for the Boltzmann equation and compressible Euler system are assumed as background.
    The proof combines macro-micro decomposition and a-contraction, which presuppose these classical tools.

pith-pipeline@v0.9.1-grok · 5750 in / 1131 out tokens · 30453 ms · 2026-06-30T13:26:17.215898+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

66 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    Dallas Albritton, Jacob Bedrossian, and Matthew Novack,Kinetic shock profiles for the Landau equation, Ars Inven. Anal. (2026), Paper No. 1, 87

    2026

  2. [2]

    John Anderson, Sanchit Chaturvedi, and Cole Graham,Vanishing viscosity and matched asymptotics near shock formation, arXiv preprint arXiv:2506.17156 (2025)

    work page arXiv 2025

  3. [3]

    Gi-Chan Bae and Chanwoo Kim,Quantitative closure analysis toward ideal fluids, arXiv preprint arXiv:2603.14671 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  4. [4]

    David Levermore,Fluid dynamic limits of kinetic equations

    Claude Bardos, Fran¸ cois Golse, and C. David Levermore,Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys.63(1991), no. 1-2, 323–344

    1991

  5. [5]

    ,Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46(1993), no. 5, 667–753

    1993

  6. [6]

    6, 1807–1835

    Florent Berthelin and Alexis F Vasseur,From kinetic equations to multidimensional isentropic gas dynamics before shocks, SIAM journal on mathematical analysis36(2005), no. 6, 1807–1835

    2005

  7. [7]

    ,From discrete velocity Boltzmann equations to gas dynamics before shocks, Journal of Statistical Physics (2009)

    2009

  8. [8]

    9, 2069–2120

    Tristan Buckmaster, Steve Shkoller, and Vlad Vicol,Formation of shocks for 2d isentropic compressible Euler, Communications on Pure and Applied Mathematics75(2022), no. 9, 2069–2120

    2022

  9. [9]

    9, 2073–2191

    ,Formation of point shocks for 3d compressible Euler, Communications on Pure and Applied Mathematics76(2023), no. 9, 2073–2191

    2023

  10. [10]

    Caflisch,The fluid dynamic limit of the nonlinear Boltzmann equation, Communications on Pure and Applied Mathe- matics33(1980), no

    Russel E. Caflisch,The fluid dynamic limit of the nonlinear Boltzmann equation, Communications on Pure and Applied Mathe- matics33(1980), no. 5, 651–666

    1980

  11. [11]

    Caflisch and Basil Nicolaenko,Shock profile solutions of the Boltzmann equation, Communications in Mathematical Physics86(1982), no

    Russel E. Caflisch and Basil Nicolaenko,Shock profile solutions of the Boltzmann equation, Communications in Mathematical Physics86(1982), no. 2, 161–194. 98 CHOE, KANG, AND KIM

    1982

  12. [12]

    PDE9 (2023), no

    Sanchit Chaturvedi and Cole Graham,The inviscid limit of viscous Burgers at nondegenerate shock formation, Ann. PDE9 (2023), no. 1, Paper No. 1, 80

    2023

  13. [13]

    Demetrios Christodoulou,The formation of shocks in 3-dimensional fluids, European Mathematical Society, 2007

    2007

  14. [14]

    Rational Mech

    Constantine M Dafermos,The second law of thermodynamics and stability, Arch. Rational Mech. Anal.70(1979), no. 2, 167–179

    1979

  15. [15]

    Ronald J DiPerna,Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J.28(1979), no. 1, 137–188

    1979

  16. [16]

    Saehoon Eo, Namhyun Eun, and Moon-Jin Kang,Stability of a Riemann shock in a physical class: from Brenner-Navier-Stokes- Fourier to Euler, arXiv preprint arXiv:2411.03613 (2024)

    work page arXiv 2024

  17. [17]

    4, Gauthier–Villars, Paris, 2000

    Fran¸ cois Golse,From Kinetic to Macroscopic Models, Kinetic Equations and Asymptotic Theory, Series in Applied Mathematics, vol. 4, Gauthier–Villars, Paris, 2000

    2000

  18. [18]

    1, 81–161

    Fran¸ cois Golse and Laure Saint-Raymond,The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels, Inventiones mathematicae155(2004), no. 1, 81–161

    2004

  19. [19]

    Gorban and Ilya V

    Alexander N. Gorban and Ilya V. Karlin,Hilbert’s 6th problem: Exact and approximate hydrodynamic manifolds for kinetic equations, Bulletin of the American Mathematical Society51(2014), no. 2, 186–246

    2014

  20. [20]

    Part II: Fine particles regime, Indiana University mathematics journal (2004), 1517–1536

    Thierry Goudon, Pierre-Emmanuel Jabin, and Alexis Vasseur,Hydrodynamic limit for the Vlasov-Navier-Stokes equations. Part II: Fine particles regime, Indiana University mathematics journal (2004), 1517–1536

    2004

  21. [21]

    Yan Guo,Smooth Irrotational Flows in the Large to the Euler–Poisson System inR 3+1, Communications in Mathematical Physics195(1998), 249–265

    1998

  22. [22]

    2, 469–501

    Yan Guo and Juhi Jang,Global Hilbert expansion for the Vlasov–Poisson–Boltzmann system, Communications in Mathematical Physics299(2010), no. 2, 469–501

    2010

  23. [23]

    1, 205–214

    Yan Guo, Juhi Jang, and Ning Jiang,Local Hilbert expansion for the Boltzmann equation, Kinetic and Related Models2(2009), no. 1, 205–214

    2009

  24. [24]

    5, 5526–5574

    Sungho Han, Moon-Jin Kang, and Jeongho Kim,Large-time behavior of composite waves of viscous shocks for the barotropic Navier–Stokes equations, SIAM Journal on Mathematical Analysis55(2023), no. 5, 5526–5574

    2023

  25. [25]

    1, 89–116

    Feimin Huang, Jing Li, and Akitaka Matsumura,Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier–Stokes system, Archive for rational mechanics and analysis197(2010), no. 1, 89–116

    2010

  26. [26]

    3, 1741–1811

    Feimin Huang, Yi Wang, Yong Wang, and Tong Yang,The limit of the Boltzmann equation to the Euler equations for Riemann problems, SIAM Journal on Mathematical Analysis45(2013), no. 3, 1741–1811

    2013

  27. [27]

    2, 293–326

    Feimin Huang, Yi Wang, and Tong Yang,Hydrodynamic limit of the Boltzmann equation with contact discontinuities, Commu- nications in Mathematical Physics295(2010), no. 2, 293–326

    2010

  28. [28]

    4, 1246–1297

    Feimin Huang, Zhouping Xin, and Tong Yang,Contact discontinuity with general perturbations for gas motions, Advances in Mathematics219(2008), no. 4, 1246–1297

    2008

  29. [29]

    Xushan Huang and Hobin Lee,Time-asymptotic stability of composite wave of viscous shocks and viscous contact wave for Navier-Stokes-Fourier equations, arXiv preprint arXiv:2504.04014 (2026)

    work page arXiv 2026

  30. [30]

    Moon-Jin Kang,L 2-type contraction for shocks of scalar viscous conservation laws with strictly convex flux, J. Math. Pures Appl. (9)145(2021), 1–43

    2021

  31. [31]

    Moon-Jin Kang and Hobin Lee,Long-time behavior toward composite wave of shocks for 3D barotropic Navier-Stokes system, Commun. Math. Sci.23(2025), 1459–1518

    2025

  32. [32]

    Moon-Jin Kang and HyeonSeop Oh,L 2 decay for large perturbations of viscous shocks for multi-D Burgers equation, Anal. Appl. (Singap.)23(2025), no. 3, 475–488

    2025

  33. [33]

    Moon-Jin Kang and Alexis F Vasseur,Asymptotic analysis of Vlasov-type equations under strong local alignment regime, Math. Mod. Meth. Appl. Sci.25(2015), 2153–2173

    2015

  34. [34]

    ,Criteria on contractions for entropic discontinuities of systems of conservation laws, Arch. Ration. Mech. Anal.222 (2016), no. 1, 343–391

    2016

  35. [35]

    ,L 2-contraction for shock waves of scalar viscous conservation laws, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire34 (2017), no. 1, 139–156

    2017

  36. [36]

    ,Contraction property for large perturbations of shocks of the barotropic Navier-Stokes system, J. Eur. Math. Soc. (JEMS) 23(2021), no. 2, 585–638

    2021

  37. [37]

    Math.224(2021), no

    ,Uniqueness and stability of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large family of Navier-Stokes systems, Invent. Math.224(2021), no. 1, 55–146

    2021

  38. [38]

    Moon-Jin Kang, Alexis F Vasseur, and Yi Wang,Uniqueness of a planar contact discontinuity for 3D compressible Euler system in a class of zero dissipation limits from Navier-Stokes-Fourier system, Comm. Math. Phys.384(2021), no. 3, 1751–1782

    2021

  39. [39]

    Math.419(2023), Paper No

    ,Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier–Stokes equations, Adv. Math.419(2023), Paper No. 108963, 66

    2023

  40. [40]

    ,Time-asymptotic stability of generic Riemann solutions for compressible Navier–Stokes–Fourier equations, Archive for Rational Mechanics and Analysis249(2025), no. 4, 42

    2025

  41. [41]

    2, 97–124

    Shuichi Kawashima, Akitaka Matsumura, and Takaaki Nishida,On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Communications in Mathematical Physics70(1979), no. 2, 97–124

    1979

  42. [42]

    3, 173–193

    P-L Lions and Nader Masmoudi,From the Boltzmann Equations to the Equations of Incompressible Fluid Mechanics, i, Archive for Rational Mechanics and Analysis158(2001), no. 3, 173–193

    2001

  43. [43]

    3-4, 178–192

    Tai-Ping Liu, Tong Yang, and Shih-Hsien Yu,Energy method for Boltzmann equation, Physica D: Nonlinear Phenomena188 (2004), no. 3-4, 178–192

    2004

  44. [44]

    2, 333–371

    Tai-Ping Liu, Tong Yang, Shih-Hsien Yu, and Hui-Jiang Zhao,Nonlinear stability of rarefaction waves for the Boltzmann equation, Archive for rational mechanics and analysis181(2006), no. 2, 333–371

    2006

  45. [45]

    1, 133–179

    Tai-Ping Liu and Shih-Hsien Yu,Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Communi- cations in Mathematical Physics246(2004), no. 1, 133–179. HYDRODYNAMIC LIMIT OF THE BOLTZMANN EQUATION 99

    2004

  46. [46]

    3, 869–997

    ,Invariant manifolds for steady Boltzmann flows and applications, Archive for Rational Mechanics and Analysis209 (2013), no. 3, 869–997

    2013

  47. [47]

    1, 1–169

    Jonathan Luk and Jared Speck,Shock formation in solutions to the 2d compressible Euler equations in the presence of non-zero vorticity, Inventiones Mathematicae214(2018), no. 1, 1–169

    2018

  48. [48]

    Pure Appl

    Nader Masmoudi and Laure Saint-Raymond,From the Boltzmann equation to the Stokes-Fourier system in a bounded domain, Comm. Pure Appl. Math.56(2003), no. 9, 1263–1293

    2003

  49. [49]

    Akitaka Matsumura and Kenji Nishihara,Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan Journal of Applied Mathematics3(1986), no. 1, 1–13

    1986

  50. [50]

    3, 573–596

    Antoine Mellet and Alexis Vasseur,Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equa- tions, Communications in mathematical physics281(2008), no. 3, 573–596

    2008

  51. [51]

    Takaaki Nishida,Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Communications in Mathematical Physics61(1978), 119–148

    1978

  52. [52]

    Stefano Olla, S. R. S. Varadhan, and Horng-Tzer Yau,Hydrodynamical limit for a Hamiltonian system with weak noise, Com- munications in Mathematical Physics155(1993), no. 3, 523–560

    1993

  53. [53]

    11, 6752–6808

    Alin Pogan and Kevin Zumbrun,Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks, Journal of Differential Equations264(2018), no. 11, 6752–6808

    2018

  54. [54]

    ,Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks, Kinetic and Related Models12(2019), no. 1, 1–36

    2019

  55. [55]

    1, 47–80

    Laure Saint-Raymond,Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Archive for rational mechanics and analysis166(2003), no. 1, 47–80

    2003

  56. [56]

    3, 705–744

    ,Hydrodynamic Limits: Some Improvements of the Relative Entropy Method, Annales de l’Institut Henri Poincar´ e C, Analyse Non Lin´ eaire26(2009), no. 3, 705–744

    2009

  57. [57]

    2005, 20120351

    Marshall Slemrod,Admissibility of weak solutions for the compressible Euler equations,n≥2, Philosophical Transactions of the Royal Society A371(2013), no. 2005, 20120351

    2013

  58. [58]

    10, 1497–1501

    ,From Boltzmann to Euler: Hilbert’s 6th problem revisited, Computers & Mathematics with Applications65(2013), no. 10, 1497–1501

    2013

  59. [59]

    2118, 20170222

    ,Hilbert’s sixth problem and the failure of the Boltzmann to Euler limit, Philosophical Transactions of the Royal Society A376(2018), no. 2118, 20170222

    2018

  60. [60]

    Seiji Ukai and Kiyoshi Asano,The Euler limit and the initial layer of the nonlinear Boltzmann equation, Hokkaido Mathematical Journal12(1983), 303–324

    1983

  61. [61]

    S. R. S. Varadhan,Entropy methods in hydrodynamic scaling, Nonequilibrium Problems in Many-Particle Systems (L. Arkeryd, ed.), Lecture Notes in Mathematics, vol. 1551, Springer, Berlin, 1993, pp. 112–145

    1993

  62. [62]

    Math., vol

    Alexis F Vasseur,Relative entropy and contraction for extremal shocks of conservation laws up to a shift, Recent advances in partial differential equations and applications, Contemp. Math., vol. 666, Amer. Math. Soc., Providence, RI, 2016, pp. 385–404

    2016

  63. [63]

    Yi Wang and Qiuyang Yu,Time-asymptotic stability of generic Riemann solutions for Boltzmann equation, arXiv preprint arXiv:2501.04399 (2025)

    work page arXiv 2025

  64. [64]

    5, 621–665

    Zhouping Xin,Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Communications on pure and applied mathematics46(1993), no. 5, 621–665

    1993

  65. [65]

    1, 63–80

    Horng-Tzer Yau,Relative entropy and hydrodynamics of Ginzburg–Landau models, Letters in Mathematical Physics22(1991), no. 1, 63–80

    1991

  66. [66]

    3, 409–443

    Shih-Hsien Yu,Hydrodynamic limits with shock waves of the Boltzmann equation, Communications on Pure and Applied Math- ematics58(2005), no. 3, 409–443. (Mingi Choe) Department of Mathematical Sciences Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea Email address:mgchoe@kaist.ac.kr (Moon-jin Kang) Department of Mathemat...

    2005