pith. sign in

arxiv: 2604.14792 · v1 · submitted 2026-04-16 · 🧮 math.AP

Homogenization of the Navier-Stokes equations in a randomly perforated domain in the inviscid limit

Richard M. H\"ofer , Eleni H\"ubner-Rosenau This is my paper

Pith reviewed 2026-05-10 10:30 UTC · model grok-4.3

classification 🧮 math.AP
keywords homogenizationNavier-Stokes equationsrandom perforated domaininviscid limitEuler equationsEuler-Brinkman equationsquantitative convergence
0
0 comments X

The pith

Navier-Stokes solutions in a random perforated domain converge to the Euler equations when hole size and viscosity scale subcritically and to the Euler-Brinkman equations at the critical scaling, under small local Reynolds number.

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

The paper establishes quantitative convergence of solutions to the Navier-Stokes equations with vanishing viscosity and no-slip conditions on randomly placed small holes whose size scales as a power of the viscosity. When the combined exponents for hole size and viscosity exceed three the holes become invisible in the limit and the flow satisfies the ideal incompressible Euler equations. At the exact critical sum of three the holes produce an effective linear drag term so the limit satisfies the Euler-Brinkman system. The results require the holes to remain separated with high probability and the local Reynolds number to stay small. A reader would care because these limits describe how microscopic obstacles shape ideal fluid motion in realistic random geometries rather than artificial periodic ones.

Core claim

We prove quantitative convergence results to a function u, provided that the local Reynolds number is small, in the subcritical (α+γ>3) and critical (α+γ=3) regime. In the first case, u solves the Euler equations, whereas in the second case u solves the Euler-Brinkman equations. This holds for i.i.d. random holes with α>2 so that particles do not overlap with overwhelming probability.

What carries the argument

The scaling relation α+γ that separates the subcritical regime (limit is Euler) from the critical regime (limit is Euler-Brinkman), combined with the small-local-Reynolds-number assumption that controls boundary-layer effects around the holes.

If this is right

  • In the subcritical regime the holes exert no net effect on the macroscopic inviscid flow.
  • At critical scaling the holes induce an effective Brinkman drag force proportional to velocity in the limit equations.
  • The quantitative error bounds hold for random i.i.d. hole positions once α exceeds 2.
  • The same scaling distinction determines the limit equations in both the periodic and the random settings.

Where Pith is reading between the lines

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

  • If the local Reynolds number becomes order one, the homogenization may fail because strong vortices or separation around individual holes could dominate the macroscopic behavior.
  • The results suggest that effective equations for flow through random porous media can be obtained directly from the microscopic Navier-Stokes system when the two microscopic scales are linked by the critical exponent sum.
  • Similar quantitative estimates might extend to non-stationary or correlated hole distributions without requiring periodicity.

Load-bearing premise

The local Reynolds number must remain small so that the flow around each hole stays attached and does not generate strong local vortices.

What would settle it

A numerical computation at the critical scaling α+γ=3 that shows the macroscopic velocity satisfying the pure Euler equations without any linear drag term would falsify the claim that the limit is Euler-Brinkman.

read the original abstract

We study the behaviour of the solution $u_\varepsilon$ to the Navier-Stokes equations with vanishing viscosity and a non-slip condition in a randomly perforated domain. We consider the space $\mathbb{R}^3$ where we remove $N$ holes that are i.i.d. distributed. The behaviour depends on the particle size $\varepsilon^\alpha=N^{-\alpha/3}$ and the viscosity $\varepsilon^\gamma=N^{-\gamma/3}$ of the fluid. We prove quantitative convergence results to a function $u$, provided that the local Reynolds number is small, in the subcritical ($\alpha+\gamma>3$) and critical ($\alpha+\gamma=3$) regime. In the first case, $u$ solves the Euler equations, whereas in the second case $u$ solves the Euler-Brinkman equations. This extends the results of https://doi.org/10.1088/1361-6544/acfe56 from the periodic to the random setting. We only treat the case $\alpha>2$ so that the particles do not overlap with overwhelming probability.

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 / 2 minor

Summary. The manuscript proves quantitative convergence results for solutions of the Navier-Stokes equations with vanishing viscosity and no-slip boundary conditions in a randomly perforated domain in R^3, where N holes of radius ε^α = N^{-α/3} are placed i.i.d. The viscosity scales as ε^γ = N^{-γ/3}. Under the assumption that the local Reynolds number remains small and α > 2 (to ensure non-overlap with high probability), the solutions converge to a limit u solving the Euler equations when α + γ > 3 (subcritical) and the Euler-Brinkman equations when α + γ = 3 (critical). This extends the authors' prior periodic-domain results to the random i.i.d. setting.

Significance. If the claimed convergences hold, the work provides a valuable extension of homogenization theory for the inviscid limit from periodic to random perforated domains. The quantitative error estimates and treatment of both subcritical and critical regimes (with the Brinkman correction arising precisely at criticality) are technically substantive and relevant to effective fluid models in random media. The restriction to α > 2 and the small local Re hypothesis are clearly stated as necessary for the random case.

major comments (2)
  1. [Abstract and §1] Abstract and §1 (main theorems): The convergence statements are conditioned on the local Reynolds number remaining small, yet the manuscript provides no quantitative tail estimates controlling the probability that local clusters of holes produce velocity gradients yielding Re_local = O(1) or larger on sets of positive measure. Because the proof must pass to the limit inside the nonlinear term, any uncontrolled local Re would invalidate the identification of the effective equation; the periodic case avoids this via uniform spacing, but the random i.i.d. setting requires an explicit probabilistic bound on min-distances and local velocities that is not visible in the stated hypotheses.
  2. [§3] §3 (proof of the nonlinear term passage): The error estimates for the convective term rely on the small local Re assumption to justify the vanishing of certain remainder terms. However, the random fluctuations could produce localized high-gradient regions even when α > 2 rules out overlaps; without a uniform-in-probability control on these regions (e.g., via a quantitative version of the non-overlap probability), the passage to the Euler or Euler-Brinkman limit is not fully justified for the random ensemble.
minor comments (2)
  1. [§2] Notation for the random measure and the probability space should be introduced once in §2 and used consistently; occasional switches between P and the expectation symbol E create minor ambiguity.
  2. [Theorem 1.1] The statement of the main convergence theorem would benefit from an explicit display of the quantitative rate (in terms of ε, N, and the small-Re parameter) rather than only qualitative convergence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, clarifying the role of the small local Reynolds number assumption and the probabilistic control available under α > 2. We will incorporate additional quantitative estimates in the revised version to make the probabilistic justification fully explicit.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1 (main theorems): The convergence statements are conditioned on the local Reynolds number remaining small, yet the manuscript provides no quantitative tail estimates controlling the probability that local clusters of holes produce velocity gradients yielding Re_local = O(1) or larger on sets of positive measure. Because the proof must pass to the limit inside the nonlinear term, any uncontrolled local Re would invalidate the identification of the effective equation; the periodic case avoids this via uniform spacing, but the random i.i.d. setting requires an explicit probabilistic bound on min-distances and local velocities that is not visible in the stated hypotheses.

    Authors: We agree that an explicit quantitative tail bound on the probability of local clusters producing large Re_local is desirable for complete rigor in the i.i.d. setting. The manuscript already states that α > 2 ensures non-overlap with overwhelming probability and conditions the convergence on small local Re. In the revision we will add a lemma providing a quantitative estimate: using standard large-deviation bounds for the minimal spacing of N i.i.d. points in R^3, we show that P(min distance < ε^β) ≤ C N^{-δ} for suitable β > α and δ > 0. Combined with the a-priori velocity bounds available from the NS energy estimates, this controls the measure of regions where Re_local = O(1) and shows that the bad event has probability o(1). The convergence statements will be rephrased to hold in probability, with the limit equation identified on the good event whose probability tends to 1. revision: yes

  2. Referee: [§3] §3 (proof of the nonlinear term passage): The error estimates for the convective term rely on the small local Re assumption to justify the vanishing of certain remainder terms. However, the random fluctuations could produce localized high-gradient regions even when α > 2 rules out overlaps; without a uniform-in-probability control on these regions (e.g., via a quantitative version of the non-overlap probability), the passage to the Euler or Euler-Brinkman limit is not fully justified for the random ensemble.

    Authors: The estimates in §3 for the convective term indeed invoke the small local Re hypothesis to absorb remainder terms after integration by parts. While α > 2 already guarantees that overlaps occur with probability tending to zero, we accept that a quantitative uniform-in-probability bound on close pairs (and the resulting local gradients) strengthens the argument. In the revision we will insert a short probabilistic lemma (referenced from the new estimate in §1) showing that the set of configurations where any two holes are closer than ε^β has probability decaying as N^{-δ}. On the complementary event the local velocity gradients remain controlled by the global energy bound, allowing the remainder terms to vanish in probability. This makes the passage to the limit inside the nonlinear term fully justified for the random ensemble without altering the core deterministic estimates. revision: yes

Circularity Check

0 steps flagged

No circularity: direct convergence proof under explicit assumptions, extending external periodic result

full rationale

The paper states a mathematical convergence theorem for the Navier-Stokes system in a random perforated domain, conditioned on the small local Reynolds number hypothesis and α>2. The derivation relies on standard PDE estimates and homogenization techniques rather than any fitted parameters, self-defined quantities, or load-bearing self-citations that reduce the target result to an input by construction. The cited periodic-case result (doi:10.1088/1361-6544/acfe56) supplies background but is not invoked to force the random-case conclusion; the new quantitative estimates are derived independently. No step renames an empirical pattern, smuggles an ansatz, or treats a fitted input as a prediction. The small-Re assumption is stated explicitly as a hypothesis, not derived from the dynamics within the paper, so it does not create circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard existence and regularity theory for Navier-Stokes and Euler equations plus probabilistic estimates for non-overlap of random holes; no new entities or fitted parameters are introduced beyond the given scaling exponents.

axioms (2)
  • standard math Existence of weak solutions to the Navier-Stokes equations with no-slip boundary conditions on the perforated domain
    Invoked implicitly to define u_ε before passing to the limit
  • domain assumption Probabilistic control on the minimal distance between i.i.d. random holes when α>2
    Used to ensure the domain remains well-defined with high probability

pith-pipeline@v0.9.0 · 5496 in / 1269 out tokens · 59869 ms · 2026-05-10T10:30:16.172608+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

36 extracted references · 36 canonical work pages

  1. [1]

    Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes

    G. Allaire. “Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes”. In:Archive for Rational Mechanics and Analysis113.3 (1990), pp. 209–259

    work page 1990

  2. [2]

    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”. In:Archive for Rational Mechanics and Analysis113.3 (1990), pp. 261–298

    work page 1990

  3. [3]

    Polynomial filtration laws for low Reynolds number flows through porous media

    M. Balhoff, A. Mikeli´ c and M.F. Wheeler. “Polynomial filtration laws for low Reynolds number flows through porous media”. In:Transport in Porous Media81 (2010), pp. 35–60

    work page 2010

  4. [4]

    Coupling Euler and Vlasov equations in the context of sprays: the local-in-time, classical solutions

    C. Baranger and L. Desvillettes. “Coupling Euler and Vlasov equations in the context of sprays: the local-in-time, classical solutions”. In:Journal of Hyperbolic Differential Equa- tions03.1 (2006), pp. 1–26

    work page 2006

  5. [5]

    G. K. Batchelor.An Introduction to Fluid Dynamics. Cambridge Mathematical Library. Cambridge University Press, 2000

    work page 2000

  6. [6]

    Inverse of Divergence and Homogenization of Compressible Navier–Stokes Equations in Randomly Perforated Domains

    P. Bella and F. Oschmann. “Inverse of Divergence and Homogenization of Compressible Navier–Stokes Equations in Randomly Perforated Domains”. In:Archive for Rational Mech- anics and Analysis247.14 (2023)

    work page 2023

  7. [7]

    A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles

    H.C. Brinkman. “A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles”. In:Flow, Turbulence and Combustion1 (1949), pp. 27–34

    work page 1949

  8. [8]

    On the derivation of a Stokes-Brinkman problem from Stokes equations around a random array of moving spheres

    K. Carrapatoso and M. Hillairet. “On the derivation of a Stokes-Brinkman problem from Stokes equations around a random array of moving spheres”. In:Communications in Math- ematical Physics373 (2018), pp. 265–325

    work page 2018

  9. [9]

    Les fontaines publiques de la ville de Dijon

    H. Darcy. “Les fontaines publiques de la ville de Dijon”. In:Dalmont, Paris38 (1856)

    work page

  10. [10]

    Divol.A short proof on the rate of convergence of the empirical measure for the Wasser- stein distance

    V. Divol.A short proof on the rate of convergence of the empirical measure for the Wasser- stein distance. 2021. arXiv:2101.08126 [math.ST].url:https://arxiv.org/abs/2101. 08126

    work page arXiv 2021

  11. [11]

    Uniform in bandwidth consistency of kernel-type function estimators

    U. Einmahl and D. M. Mason. “Uniform in bandwidth consistency of kernel-type function estimators”. In:The Annals of Statistics33.3 (2005)

    work page 2005

  12. [12]

    Homogenization of the evolutionary Navier- Stokes system

    E. Feireisl, Y. Namlyeyeva and ˇS. Neˇ casov´ a. “Homogenization of the evolutionary Navier- Stokes system”. In:Manuscripta Mathematica149 (2016), pp. 251–274

    work page 2016

  13. [13]

    On the rate of convergence in Wasserstein distance of the empirical measure

    N. Fournier and A. Guillin. “On the rate of convergence in Wasserstein distance of the empirical measure”. In:Probability Theory and Related Fields162 (2015), pp. 707–738

    work page 2015

  14. [14]

    Springer Monographs in Mathematics, Springer, New York, 2011

    Giovanni Paolo Galdi.An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, 2nd edition. Springer Monographs in Mathematics, Springer, New York, 2011

    work page 2011

  15. [15]

    Rates of strong uniform consistency for multivariate kernel density estimators

    E. Gin´ e and A. Guillou. “Rates of strong uniform consistency for multivariate kernel density estimators”. In:Annales de l’Institut Henri Poincare (B) Probability and Statistics38.6 (2002), pp. 907–921

    work page 2002

  16. [16]

    Derivation of Darcy’s law in randomly perforated domains

    A. Giunti. “Derivation of Darcy’s law in randomly perforated domains”. In:Calculus of Variations and Partial Differential Equations60.5 (2021), pp. 1–13

    work page 2021

  17. [17]

    Homogenisation for the Stokes equations in ran- domly perforated domains under almost minimal assumptions on the size of the holes

    Arianna Giunti and Richard M H¨ ofer. “Homogenisation for the Stokes equations in ran- domly perforated domains under almost minimal assumptions on the size of the holes”. In: Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire. Vol. 36. 7. Elsevier. 2019, pp. 1829–1868. 21

    work page 2019

  18. [18]

    Wasserstein distances for vortices approximation of euler-type equations

    M. Hauray. “Wasserstein distances for vortices approximation of euler-type equations”. In: Mathematical Models and Methods in Applied Sciences19.08 (2009), pp. 1357–1384.doi: 10.1142/s0218202509003814

    work page doi:10.1142/s0218202509003814 2009

  19. [19]

    Particle approximation of Vlasov equations with singular forces: Propagation of Chaos

    M. Hauray and P.-E. Jabin. “Particle approximation of Vlasov equations with singular forces: Propagation of Chaos”. In:Annales scientifiques de l’ ´Ecole normale sup´ erieure48.4 (2015), pp. 891–940.doi:10.24033/asens.2261

    work page doi:10.24033/asens.2261 2015

  20. [20]

    Convergence Rates and Fluctuations for the Stokes–Brinkman Equations as Homogenization Limit in Perforated Domains

    R. M. H¨ ofer and J. Jansen. “Convergence Rates and Fluctuations for the Stokes–Brinkman Equations as Homogenization Limit in Perforated Domains”. In:Archive for Rational Mech- anics and Analysis248.50 (2024)

    work page 2024

  21. [21]

    Quantitative homogenization of the compress- ible Navier–Stokes equations towards Darcy’s law

    R. M. H¨ ofer, ˇS. Neˇ casov´ a and F. Oschmann. “Quantitative homogenization of the compress- ible Navier–Stokes equations towards Darcy’s law”. In:Annales de l’Institut Henri Poincar´ e C(2025)

    work page 2025

  22. [22]

    Homogenization of the Navier–Stokes equations in perforated domains in the inviscid limit

    Richard M. H¨ ofer. “Homogenization of the Navier–Stokes equations in perforated domains in the inviscid limit”. In:Nonlinearity36.11 (2023), pp. 6020–6047.url:https://doi. org/10.1088/1361-6544/acfe56

    work page doi:10.1088/1361-6544/acfe56 2023

  23. [23]

    The vanishing viscosity limit in the presence of a porous medium

    C. Lacave and A. Mazzucato. “The vanishing viscosity limit in the presence of a porous medium”. In:Mathematische Annalen365 (2015), pp. 1527–1557

    work page 2015

  24. [24]

    Homogenization of Evolutionary Incompressible Navier–Stokes System in Perforated Domains

    Y. Lu and P. Yang. “Homogenization of Evolutionary Incompressible Navier–Stokes System in Perforated Domains”. In:Journal of Mathematical Fluid Mechanics25.4 (2023)

    work page 2023

  25. [25]

    Boundary-value problems with fine-grained bound- ary (in Russian)

    A. V. Marchenko and E. Y. Khruslov. “Boundary-value problems with fine-grained bound- ary (in Russian)”. In:Matematicheskii Sbornik. Novaya Seriya65(107) (3 1964), pp. 458– 472.doi:10.1016/j.anihpc.2021.02.001

    work page doi:10.1016/j.anihpc.2021.02.001 1964

  26. [26]

    The derivation of a nonlinear filtration law including the inertia effects via homogenization

    E. Maruˇ si´ c-Paloka and A. Mikeli´ c. “The derivation of a nonlinear filtration law including the inertia effects via homogenization”. In:Nonlinear Analysis: Theory, Methods & Applications 42.1 (2000), pp. 97–137.doi:https://doi.org/10.1016/S0362-546X(98)00346-0

    work page doi:10.1016/s0362-546x(98)00346-0 2000

  27. [27]

    Homogenization of the compressible Navier-Stokes equations in a porous medium

    N. Masmoudi. “Homogenization of the compressible Navier-Stokes equations in a porous medium”. In:ESAIM: Control, Optimisation and Calculus of Variations8 (2002), pp. 885– 906

    work page 2002

  28. [28]

    Effets inertiels pour un ´ ecoulement stationnaire visqueux incompressible dans un milieu poreux

    A. Mikeli´ c. “Effets inertiels pour un ´ ecoulement stationnaire visqueux incompressible dans un milieu poreux”. In:Comptes rendus de l’Acad´ emie des sciences. S´ erie 1, Math´ ematique 320 (1995), pp. 1289–1294

    work page 1995

  29. [29]

    Homogenization of Nonstationary Navier-Stokes equations in a domain with a grained boundary

    A. Mikeli´ c. “Homogenization of Nonstationary Navier-Stokes equations in a domain with a grained boundary”. In:Annali di Matematica Pura ed Applicata158 (Dec. 1991), pp. 167– 179.doi:10.1007/BF01759303

    work page doi:10.1007/bf01759303 1991

  30. [30]

    Homogenization of the full compressible Navier-Stokes-Fourier system in randomly perforated domains

    F. Oschmann. “Homogenization of the full compressible Navier-Stokes-Fourier system in randomly perforated domains”. In:Journal of Mathematical Fluid Mechanics24.45 (2022)

    work page 2022

  31. [31]

    Homogenization of Non-Homogeneous Incompressible Navier–Stokes System in Critically Perforated Domains

    J. Pan. “Homogenization of Non-Homogeneous Incompressible Navier–Stokes System in Critically Perforated Domains”. In:Journal of Mathematical Fluid Mechanics27.24 (2025)

    work page 2025

  32. [32]

    J. C. Robinson, J. L. Rodrigo and W. Sadowski.The Three-Dimensional Navier–Stokes Equations: Classical Theory. Cambridge Studies in Advanced Mathematics. Cambridge Uni- versity Press, 2016

    work page 2016

  33. [33]

    Santambrogio.Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling

    F. Santambrogio.Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling. Basel: Birkh¨ auser, 2015

    work page 2015

  34. [34]

    Sohr.The Navier-Stokes Equations: An Elementary Functional Analytic Approach

    H. Sohr.The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkh¨ auser Advanced Texts Basler Lehrb¨ ucher. Birkh¨ auser Basel, 2014

    work page 2014

  35. [35]

    Incompressible fluid flow in a porous medium-convergence of the homogenization process

    L. Tartar. “Incompressible fluid flow in a porous medium-convergence of the homogenization process”. In: Appendix of Non-Homogeneous Media and Vibration Theory, 1980. 22

    work page 1980

  36. [36]

    Wiedemann.Weak-strong uniqueness in fluid dynamics

    E. Wiedemann.Weak-strong uniqueness in fluid dynamics. 2017. arXiv:1705.04220 [math.AP]. url:https://arxiv.org/abs/1705.04220. 23

    work page arXiv 2017