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Tiny holes of subcritical size leave the non-homogeneous heat-conducting fluid equations unchanged in the limit.

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2026-07-14 07:29 UTC pith:AW3AP6VK

load-bearing objection Clean subcritical completion for the non-homogeneous heat-conducting system; standard pipeline, solid, modest novelty.

arxiv 2607.11036 v1 pith:AW3AP6VK submitted 2026-07-13 math.AP

Homogenization of a non-homogeneous incompressible heat-conducting fluid in perforated domains

Yong Lu , Jiaojiao Pan , Luqi Wang This is my paper
classification math.AP MSC 35B2776M5080M40
keywords homogenizationperforated domainsnon-homogeneous incompressible fluidheat-conducting fluidsubcritical holestemperature-dependent viscosity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies a three-dimensional non-homogeneous incompressible fluid whose viscosity and heat conductivity depend on temperature, flowing through a domain perforated by many small solid holes. The holes have diameter much smaller than their mutual spacing (order ε^α with α>3). The authors prove that any sequence of finite-energy weak solutions converges, after suitable extensions, to a weak solution of the same system on the solid domain without holes. In other words the holes become invisible: they neither produce an effective friction term nor alter the macroscopic density, velocity or temperature equations. The result completes the picture for this model by treating the subcritical regime that earlier work had left open. A sympathetic reader cares because the same geometric threshold already governs Stokes, Navier–Stokes and compressible flows; confirming that the heat-conducting non-homogeneous case obeys the identical rule shows the threshold is robust across a wide class of fluid models.

Core claim

Under the stated assumptions on initial data and external force, every sequence of finite-energy weak solutions on the perforated domains Ω_ε converges (density strongly in C([0,T];L^p), velocity strongly in L^{2} and weakly in L^{2}W^{1},^{2}, temperature strongly in L^{2}W^{1},^{2}) to a weak solution of the original non-homogeneous heat-conducting system posed on the full domain Ω without holes.

What carries the argument

A family of cut-off functions g_ε that vanish on the holes, combined with a Bogovskii-type operator B_ε on the perforated domain whose operator-norm bound carries the factor (1+ε^((3-q)α-3)/q). The factor tends to zero precisely when α>3, so that all remainder terms generated by the cut-offs vanish and the momentum equation passes to the limit without extra friction.

Load-bearing premise

The operator-norm bound on the Bogovskii map on the perforated domain must improve with a positive power of ε; if that power vanished or became negative the remainder terms would not disappear and the holes would leave a macroscopic trace.

What would settle it

Construct a sequence of solutions for α=3 (the critical scaling) and check whether the limit still satisfies the original system or instead acquires a Brinkman friction term proportional to velocity; any nonzero friction term would show that the strict inequality α>3 is essential.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies homogenization of the 3D non-homogeneous incompressible heat-conducting Navier–Stokes–Boussinesq system in periodically perforated domains whose holes have diameter of order ε^α with α>3 (subcritical regime). Viscosity μ depends on temperature; the heat conductivity is piecewise constant (κ_f in the fluid, κ_s in the solid). For finite-energy weak solutions, the authors prove that the constantly extended density, zero-extended velocity and temperature converge (strongly in the topologies of Theorem 1.2) to a weak solution of the same system on the full domain without holes. The argument proceeds by uniform energy bounds, DiPerna–Lions compactness for density, a cut-off/Bogovskii extension of the momentum equation with remainder of order ε^σ (σ>0), Aubin–Lions strong convergence of velocity, passage to the limit in the quasi-static heat equation, and an elliptic identity yielding strong convergence of temperature.

Significance. The result completes the homogenization picture for this model in three dimensions by treating the subcritical regime, complementing the critical (Brinkman) case already obtained by Feireisl–Lu–Sun. The proof is a careful, self-contained application of the standard perforated-domain toolkit (energy estimates, DiPerna–Lions, Bogovskii operator with the known ε-dependent bound, Aubin–Lions, elliptic regularity for the heat equation). No free parameters or circular arguments appear; the only external black box is a published Bogovskii estimate that is correctly invoked for α>3. The contribution is solid and of clear interest to the mathematical fluid-mechanics and homogenization communities.

minor comments (6)
  1. Abstract (and the corresponding sentence in the introduction): the claim that “the viscosity and the heat conductivity coefficient are assumed to depend on the temperature” is inaccurate. Only μ=μ(Θ) depends on temperature; κ_ε is the piecewise-constant function (1.10) and the limit conductivity is the constant κ_f. Please correct the abstract and the introductory wording.
  2. Sign inconsistency between the original momentum equation (1.7)_2 (force −Θ∇_x F) and the limit system (1.22)_2 (force +Θ∇_x F). The weak formulations (1.17), (3.2) and (3.38) are internally consistent with the “+” convention of (1.22). The sign in (1.7) should be aligned with the rest of the paper.
  3. Page 3, display (1.7)_4 and the subsequent discussion: the quasi-static heat equation is presented as a high-Péclet approximation of an evolutionary equation that still contains the factor ϱ. A one-sentence clarification that the density factor is dropped under the same approximation (or a pointer to the precise derivation in [14]) would help the reader.
  4. Lemma 3.1 is cited from [9,27] for α≥1; the paper only needs α>3. It would be useful to state explicitly that the constant C is independent of ε and of the particular hole shape T_0 (as long as the geometric assumptions (1.4)–(1.5) hold).
  5. Typographical: “an an admissible” (p. 9, after the definition of φ_1); “e·to denote” (Notations, missing space); occasional missing spaces after commas in multi-line displays. A careful proof-reading pass is recommended.
  6. In the energy inequality (1.19) the work of the buoyancy force does not appear. While this is consistent with the definition of finite-energy weak solutions adopted in the paper, a brief remark explaining why the force term is absent (or absorbed) would improve readability for non-specialists.

Circularity Check

0 steps flagged

No significant circularity: standard weak-limit homogenization argument for the subcritical regime, with self-citations supplying independent published lemmas rather than tautological premises.

full rationale

The paper derives Theorem 1.2 by a direct, self-contained passage to the limit from the finite-energy weak formulations (1.16)–(1.19). Uniform bounds follow from the energy inequality (1.19) plus Korn/Poincaré (2.2)–(2.6); density compactness is the classical DiPerna–Lions result restated as Prop. 2.1; the extended momentum equation (Prop. 3.2) is obtained by cut-off + Bogovskii correction whose remainder vanishes precisely because α>3 makes the factor ε^σ positive; strong velocity convergence uses Aubin–Lions on the projected momentum (3.8)–(3.18); the heat equation passes by strong L^q convergence of κ_ε (3.19) and an elliptic identity (3.27)–(3.31) that yields strong temperature convergence. The only external black boxes are the already-published Bogovskii bound (Lemma 3.1, cited from Diening–Feireisl–Lu and Lu–Schwarzacher) and the critical-case companion paper [12]; neither statement contains the target subcritical theorem, so the citations are independent support rather than circular premises. No free parameters are fitted, no uniqueness theorem is imported to forbid alternatives, and no ansatz is smuggled in. Score 1 reflects only the ordinary presence of author-overlapping citations that are not load-bearing for the circularity patterns listed.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

Pure existence/uniqueness-of-limit theorem in mathematical fluid dynamics. No numerical fitting. All free parameters are geometric (α>3, domain regularity) or constitutive (bounds on μ, κ). Background analytic tools are standard or previously published. No new physical entities are postulated.

axioms (6)
  • standard math DiPerna-Lions renormalized continuity equation and associated strong compactness of density in C([0,T];L^p) for bounded divergence-free velocity in L^2(0,T;W^{1,2})
    Invoked as Proposition 2.1 (citing Lions' book) to obtain eϱ_ε → ϱ strongly; load-bearing for the convective term.
  • domain assumption Existence of a Bogovskii-type right inverse of divergence on the perforated domain Ω_ε with the precise operator-norm bound (1+ε^{((3-q)α-3)/q}) for 1<q<∞ (Lemma 3.1)
    Cited from Diening-Feireisl-Lu and Lu-Schwarzacher; the only tool that makes the remainder G_ε of order ε^σ with σ>0 when α>3.
  • domain assumption Finite-energy weak solutions exist for each fixed ε>0 (Definition 1.1)
    Stated as 'following the well-known argument in Lions' book'; not re-proved for temperature-dependent μ and piecewise κ.
  • domain assumption μ is Lipschitz continuous and bounded between positive constants; κ_ε equals κ_f on fluid and κ_s on solid; ∇F ∈ L^∞
    Assumptions (1.8), (1.10), (1.13); needed for energy bounds, strong convergence of μ(Θ_ε), and passage in the heat equation.
  • domain assumption Holes satisfy the geometric packing (1.3)-(1.6) with a_ε=ε^α, α>3, and T_0 of class C^{2,β}
    Defines the subcritical regime and supplies the cut-off estimates (3.4)-(3.5).
  • standard math Korn and Poincaré inequalities on Ω_ε and Ω; elliptic L^2 regularity for the limit heat equation
    Used for uniform bounds (2.3) and higher regularity of Θ (3.23) that feeds strong temperature convergence.

pith-pipeline@v1.1.0-grok45 · 23016 in / 3281 out tokens · 34932 ms · 2026-07-14T07:29:06.541123+00:00 · methodology

0 comments
read the original abstract

This paper provides the study of the homogenization of the 3D non-homogeneous incompressible heat-conducting fluid in perforated domains with holes of subcritical size, where the viscosity and the heat conductivity coefficient are assumed to depend on the temperature. The diameter of the holes is of order $\varepsilon^{\alpha} \ (\alpha>3)$, where $\varepsilon > 0$ is a small parameter that measures the mutual distance between the holes. We prove that as $\varepsilon\to 0$, the limit behavior of velocity, density and temperature is governed by the original system in homogeneous domain without holes.

discussion (0)

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Works this paper leans on

35 extracted references · 20 canonical work pages

  1. [1]

    Allaire,Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes, I

    G. Allaire,Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes, I. Abstract framework, a volume distribution of holes, Arch. Ration. Mech. Anal., 113 (3) (1990), pp. 209–259, https://doi:10.1007/BF00375065

    work page doi:10.1007/bf00375065 1990

  2. [2]

    Allaire,Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes, II

    G. Allaire,Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes, II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes, Arch. Ration. Mech. Anal., 113 (3) (1990), pp. 261–298, https://doi:10.1007/BF00375066

    doi:10.1007/bf00375066 1990

  3. [3]

    Basari´ c and N

    D. Basari´ c and N. Chaudhuri,Low Mach number limit on perforated domains for the evolutionary Navier-Stokes-Fourier system, Nonlinearity, 37 (6) (2024), 065008, https://doi:10.1088/1361-6544/ad3da9

    work page doi:10.1088/1361-6544/ad3da9 2024

  4. [4]

    Basari´ c, F

    D. Basari´ c, F. Oschmann and J. Pan,Qualitative derivation of a density dependent incom- pressible Darcy law, Sci. China Math., 69 (2026), pp. 1–24, https://doi.org/10.1007/s11425- 025-2540-2

    doi:10.1007/s11425- 2026

  5. [5]

    Bella and F

    P. Bella and F. Oschmann,Homogenization and low Mach number limit of compressible Navier-Stokes equations in critically perforated domains, J. Math. Fluid Mech., 24 (3) (2022), pp. 79–89, https://doi:10.1007/s00021-022-00707-1

    work page doi:10.1007/s00021-022-00707-1 2022

  6. [6]

    Bella and F

    P. Bella and F. Oschmann,Inverse of divergence and homogenization of compressible Navier- Stokes equations in randomly perforated domains, Arch. Ration. Mech. Anal., 247 (2) (2023), pp. 14–42, https://doi:10.1007/s00205-023-01847-y

    work page doi:10.1007/s00205-023-01847-y 2023

  7. [7]

    Chandrasekhar,Hydrodynamic and hydrodynamic stability, Clarendon Press, Oxford, 1961

    S. Chandrasekhar,Hydrodynamic and hydrodynamic stability, Clarendon Press, Oxford, 1961

    1961

  8. [8]

    Darcy,Les fontaines publiques de la ville de Dijon, Victor Dalmont, 1856

    H. Darcy,Les fontaines publiques de la ville de Dijon, Victor Dalmont, 1856

  9. [9]

    Diening, E

    L. Diening, E. Feireisl and Y. Lu,The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier-Stokes system, ESAIM Control Optim. Calc. Var., 23 (2017), pp. 851–868, https://doi:10.1051/cocv/2016016

    work page doi:10.1051/cocv/2016016 2017

  10. [10]

    DiPerna and P.-L

    R.-J. DiPerna and P.-L. Lions,Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), pp. 511–547, https://doi.org/10.1007/BF01393835

    doi:10.1007/bf01393835 1989

  11. [11]

    Feireisl and Y

    E. Feireisl and Y. Lu,Homogenization of stationary Navier-Stokes equations in domains with tiny holes, J. Math. Fluid Mech., 17 (2) (2015), pp. 381–392, https://doi:10.1007/s00021-015- 0200-2

    work page doi:10.1007/s00021-015- 2015

  12. [12]

    Feireisl, Y

    E. Feireisl, Y. Lu and Y. Sun,Homogenization of a non-homogeneous heat conducting fluid, Asymptot. Anal., 125 (3–4) (2021), pp. 327–346, https://doi:10.3233/ASY-201658

    work page doi:10.3233/asy-201658 2021

  13. [13]

    Feireisl, Y

    E. Feireisl, Y. Namlyeyeva and ˇS. Neˇ casov´ a,Homogenization of the evolutionary Navier-Stokes system, Manuscripta Math., 149 (1–2) (2016), pp. 251–274, https://doi:10.1007/s00229-015- 0778-y. 17

    work page doi:10.1007/s00229-015- 2016

  14. [14]

    Feireisl and A

    E. Feireisl and A. Novotn´ y,Singular Limits in thermodynamics of viscous fluids, Birkh¨ auser Verlag, Basel, 2009

    2009

  15. [15]

    Feireisl, A

    E. Feireisl, A. Novotn´ y and T. Takahashi,Homogenization and singular limits for the complete Navier-Stokes Fourier system, J. Math. Pures Appl., 94 (1) (2010), pp. 33–57, https://doi:10.1016/j.matpur.2009.11.006

    work page doi:10.1016/j.matpur.2009.11.006 2010

  16. [16]

    H¨ ofer, K

    R.-M. H¨ ofer, K. Kowalczyk and S. Schwarzacher,Darcy’s law as low Mach and homogenization limit of a compressible fluid in perforated domains, Math. Models Methods Appl. Sci., 31 (9) (2021), pp. 1787–1819, https://doi:10.1142/S0218202521500391

    work page doi:10.1142/s0218202521500391 2021

  17. [17]

    H¨ ofer, Y

    R.-M. H¨ ofer, Y. Lu and F. Oschmann,Qualitative/quantitative homogenization of some non-Newtonian flows in perforated domains, Math Ann., 395 (39) (2026), pp. 1–52, https://doi.org/10.1007/s00208-026-03390-z

    work page doi:10.1007/s00208-026-03390-z 2026

  18. [18]

    H¨ ofer,ˇS

    R.-M. H¨ ofer,ˇS. Neˇ casov´ a and F. Oschmann,Quantitative homogenization of the compressible Navier-Stokes equations towards Darcy’s law, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 43 (3) (2026), pp. 669–709, https://doi 10.4171/aihpc/156

    work page doi:10.4171/aihpc/156 2026

  19. [19]

    Ladyzhenskaya and N.-N

    O.-A. Ladyzhenskaya and N.-N. Uralceva.,Linear and quasilinear elliptic equations, Academic Press, New York and London, 1968

    1968

  20. [20]

    Ligni` eres,The small P´ eclet number approximation in stellar radiative zones, Astronomy and astrophysics, pp

    F. Ligni` eres,The small P´ eclet number approximation in stellar radiative zones, Astronomy and astrophysics, pp. 1–10, 1999, https://api.semanticscholar.org/CorpusID:17968719

    1999

  21. [21]

    Lions,Mathematical topics in fluid dynamics, Vol.1, Incompressible models, Oxford Science Publication, Oxford, 1998

    P.-L. Lions,Mathematical topics in fluid dynamics, Vol.1, Incompressible models, Oxford Science Publication, Oxford, 1998

    1998

  22. [22]

    Lions,Mathematical topics in fluid dynamics, Vol.2, Compressible models, Oxford Science Publication, Oxford, 1998

    P.-L. Lions,Mathematical topics in fluid dynamics, Vol.2, Compressible models, Oxford Science Publication, Oxford, 1998

    1998

  23. [23]

    Lu,Homogenization of Stokes equations in perforated domains: A unified approach, J

    Y. Lu,Homogenization of Stokes equations in perforated domains: A unified approach, J. Math. Fluid Mech., 22 (3) (2020), pp. 44–56, https://doi:10.1007/s00021-020-00506-6

    doi:10.1007/s00021-020-00506-6 2020

  24. [24]

    Y. Lu, J. Pan and P. Yang,Homogenization of Inhomogeneous Incompressible Navier-Stokes Equations in Domains with Very Tiny Holes, Acta Math. Sin. Engl. Ser., in press, 2026

    2026

  25. [25]

    Lu and M

    Y. Lu and M. Pokorn´ y,Homogenization of stationary Navier-Stokes-Fourier system in domains with tiny holes, J. Differential Equations, 278 (2021), pp. 463–492, https://doi:10.1016/j.jde.2020.10.032

    work page doi:10.1016/j.jde.2020.10.032 2021

  26. [26]

    Lu and Z

    Y. Lu and Z. Qian,Homogenization of some evolutionary non-Newtonian flows in porous me- dia, J. Differential Equations, 411 (2024), pp. 619–639, https://doi:10.1016/j.jde.2024.08.021

    work page doi:10.1016/j.jde.2024.08.021 2024

  27. [27]

    Lu and S

    Y. Lu and S. Schwarzacher,Homogenization of the compressible Navier-Stokes equations in domains with very tiny holes, J. Differential Equations, 265 (4) (2018), pp. 1371–1406, https://doi:10.1016/j.jde.2018.04.007

    work page doi:10.1016/j.jde.2018.04.007 2018

  28. [28]

    Lu and P

    Y. Lu and P. Yang,Homogenization of Evolutionary Incompressible Navier-Stokes System in Perforated Domains, J. Math. Fluid Mech., 25 (2023), pp. 4–23, https://doi:10.1007/s00021- 022-00745-9. 18

    work page doi:10.1007/s00021- 2023

  29. [29]

    Masmoudi,Homogenization of the compressible Navier-Stokes equations in a porous medium, ESAIM Control Optim

    N. Masmoudi,Homogenization of the compressible Navier-Stokes equations in a porous medium, ESAIM Control Optim. Calc. Var., 8 (2002), pp. 885–906, https://doi:10.1051/cocv:2002053

    doi:10.1051/cocv:2002053 2002

  30. [30]

    Mikeli´ c,Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary, Ann

    A. Mikeli´ c,Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary, Ann. Mat. Pura Appl., 158 (1991), pp. 167–179, https://doi:10.1007/BF01759303

    doi:10.1007/bf01759303 1991

  31. [31]

    Neˇ casov´ a and F

    ˇS. Neˇ casov´ a and F. Oschmann,Homogenization of the two-dimensional evolutionary compressible Navier-Stokes equations, Calc. Var. Partial Differential Equations, 62 (184) (2023) , pp. 1–13, https://doi:10.1007/s00526-023-02526-2

    work page doi:10.1007/s00526-023-02526-2 2023

  32. [32]

    Neˇ casov´ a and J

    ˇS. Neˇ casov´ a and J. Pan,Homogenization problems for the compressible Navier–Stokes system in 2D perforated domains, Math. Methods Appl. Sci., 45 (12) (2022), pp. 7859–7873, https://doi:10.1002/mma.8283

    work page doi:10.1002/mma.8283 2022

  33. [33]

    Oschmann and M

    F. Oschmann and M. Pokorn´ y,Homogenization of the unsteady compressible Navier-Stokes equations for adiabatic exponentγ >3, J. Differential Equations, 377 (2023), pp. 271–296, https://doi:10.1016/j.jde.2023.08.040

    work page doi:10.1016/j.jde.2023.08.040 2023

  34. [34]

    Pan,Homogenization of non-homogeneous incompressible Navier-Stokes system in critically perforated domains, J

    J. Pan,Homogenization of non-homogeneous incompressible Navier-Stokes system in critically perforated domains, J. Math. Fluid Mech., 27 (24) (2025), pp. 1–16, https://doi.org/10.1007/s00021-025-00931-5

    work page doi:10.1007/s00021-025-00931-5 2025

  35. [35]

    Tartar,Incompressible fluid flow in a porous medium: convergence of the homogenization process, in Nonhomogeneous media and vibration theory, edited by E

    L. Tartar,Incompressible fluid flow in a porous medium: convergence of the homogenization process, in Nonhomogeneous media and vibration theory, edited by E. S´ anchez-Palencia, pp. 368–377, 1980. 19

    1980