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Inviscid Incompressible Limit for Degenerate Compressible Navier-Stokes Equations on Expanding Domains

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Pith's one-line read Degenerate compressible Navier-Stokes on expanding domains converge, under simultaneous vanishing viscosity and Mach number, to incompressible Euler from ill-prepared data.

desk verdict Solid triple-limit extension for density-dependent viscosity; the math is standard relative-energy work with one real scope caveat on existence. read the letter →

arxiv 2607.25028 v1 pith:2OT6X5MR submitted 2026-07-27 math.AP

Tong Tang , Emil Wiedemann , Lu Zhu This is my paper · ORCID
classification math.AP MSC 35Q3035Q3135Q8676N10
keywords degeneratecompressibleNavier-StokessingularlimitslowMachnumberinviscidlimitexpandingdomainsrelativeenergydensity-dependentviscosityincompressibleEuler
topics Navier-Stokes Regularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that three singular limits can be taken at once for three-dimensional compressible flow whose viscosity vanishes with density: viscosity to zero, Mach number to zero, and the spatial domain expanding to fill all of space. Starting from rough, ill-prepared initial data, weak solutions of the degenerate Navier-Stokes system converge locally to a smooth solution of the incompressible Euler equations for as long as that Euler solution exists. The result extends earlier work that treated only constant viscosity, and it is the first simultaneous inviscid-plus-incompressible limit for density-dependent viscosity. A sympathetic reader cares because real fluids often have viscosity that depends on density and sit in large domains; the theorem says the ideal incompressible Euler model still captures the local interior dynamics once those parameters are small.

What carries the argument

A relative energy inequality comparing the weak Navier-Stokes solution to a corrected test field built from the Euler solution, the acoustic potential, and a boundary cut-off. The inequality absorbs acoustic oscillations, residual vacuum regions, and the degenerate viscous term so that Gronwall closes the convergence.

What would settle it

Produce a family of weak solutions satisfying the paper’s structural hypotheses on expanding domains whose density or momentum fails to converge strongly to the Euler solution on some fixed compact set inside a time interval where the smooth Euler solution still exists.

Watch

Extended reading notes

Core claim

For weak solutions of the three-dimensional degenerate compressible Navier-Stokes equations with density-dependent viscosity on a family of expanding domains, the simultaneous inviscid and low-Mach limits yield strong local convergence of density to 1 and of momentum to a smooth incompressible Euler velocity, even from ill-prepared compactly supported initial data, on any time interval short of the Euler lifespan.

Load-bearing premise

The argument takes as given the existence of the weak solutions it starts from; that existence is imported from earlier theory that needs a specific viscosity law and a drag term that itself must vanish in the limit.

Editorial extensions

If this is right

  • Local interior dynamics of slightly viscous, slightly compressible fluids with density-dependent viscosity are independent of distant boundaries once the domain is large enough.
  • The incompressible Euler system remains the correct target even when viscosity degenerates at vacuum, so the same ideal model covers a wider class of constitutive laws.
  • Ill-prepared data are admissible: acoustic waves disperse and do not prevent the low-Mach limit on expanding domains.
  • Any future existence theory for degenerate Navier-Stokes without drag immediately upgrades, under the same relative-energy argument, to the same triple limit.

Reading between the lines

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

  • The same corrector-plus-relative-energy pattern should apply verbatim on the whole space once existence without drag is available, removing the artificial friction term entirely.
  • Because the viscous remainder is controlled without Bresch–Desjardins entropy, the method is likely portable to other degenerate or non-Newtonian stress tensors that still admit a relative-energy inequality.
  • Quantitative rates could be read off from the explicit decay of the acoustic Strichartz norms and the cut-off errors once the Gronwall constant is tracked.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proves a simultaneous triple singular limit — vanishing viscosity (ε^α), low Mach number (ε), and expanding domains Ω_M exhausting ℝ³ — for the 3D isentropic compressible Navier–Stokes system with degenerate shear viscosity μ(ρ)=ρ, zero bulk viscosity, and a quadratic drag term. Starting from ill-prepared, compactly supported initial data, the authors show that any weak solution of (1.1) converges locally strongly (√ρu→v in L²_tL²_x(K), ρ→1) to the smooth solution of the incompressible Euler system on ℝ³, on any time interval where the latter exists. The proof is a relative-energy/weak-to-strong argument: a relative entropy inequality (Lemma 3.1) is derived from the weak formulation, the acoustic component is filtered by Strichartz estimates (2.12), the boundary corrector w_M of Feireisl–Nečasová–Sun [15] handles the expanding-domain geometry, and the remainder terms R1–R7 are absorbed via energy bounds (2.5)–(2.7) and a Cauchy–Schwarz treatment of the viscous term that avoids the Bresch–Desjardins structure, following Bisconti–Caggio–Dell'Oro [2]. The derivation is standard in strategy and, on my reading, essentially correct in execution; I found no structural gap in the convergence argument.

Significance. If the result stands, it is the first simultaneous inviscid + low-Mach limit for density-dependent (Bresch–Desjardins) viscosity, extending Feireisl–Nečasová–Sun (Nonlinearity 2014) from constant to degenerate viscosity, and it does so by a purely energetic argument avoiding the BD structure — a methodological simplification of independent interest. The strengths are concrete: the convergence statement is quantitative and conditional only on the existence of the target Euler solution, the proof is self-contained given the cited Strichartz estimates, and the treatment of the viscous remainder (§4, R6) is the genuinely new technical content. The main caveat is that the theorem's hypotheses quantify over weak solutions whose existence is imported, unverified, from [4]; if that import fails for the stated boundary conditions and γ-range, the theorem has no objects. This does not affect the convergence argument itself, which is sound.

major comments (4)
  1. [Remark 2.1 / Theorem 2.1] Theorem 2.1 quantifies over 'any weak solutions' of (1.1), but the only source of existence is Remark 2.1, which asserts that [4] (Bresch–Desjardins–Gérard-Varet, JMPA 2007) provides weak solutions in the sense of Definition 2.1 'for any data that satisfy (2.1) initially'. This is load-bearing: if [4]'s hypotheses do not literally cover the present system, the theorem quantifies over an empty class. The manuscript does not verify the match on any of the following points: (i) the boundary conditions — [4] treats Navier-type conditions on the velocity together with a condition on ∇ρ, whereas Definition 2.1 requires ρu=0 on ∂Ω_M (a much stronger condition than Navier slip) jointly with ∇ρ×n=0; (ii) the quadratic drag r₁(ε)ρ|u|u with arbitrary coefficient r₁(ε)>0; (iii) the full range γ>1, including large γ, where [4]'s construction has documented restrictions; (iv) the pressure law p(ρ)=ρ^γ
  2. [§4, final paragraph] The Gronwall conclusion at the end of §4 requires E(0)→0 as ε→0, but this is never verified in the manuscript. It follows standardly: u_{0,ε}→u_0, ρ^{(1)}_{0,ε}→ρ^{(1)}_0=s(0), and U(0)=u_0+w_M(0) with w_M(0) small by (2.13), so E(0)→0; and the pressure part of E(0) is controlled by (2.7) at t=0. However, since the whole proof is one Gronwall step on E(τ)−E(0), the initial-layer argument should be written out in a few lines rather than left implicit.
  3. [§4, Eqs. (4.19)–(4.20)] The decay rates asserted for ∥∇²_xΦ∥ in L², L⁴ and L^{2γ/(γ−1)} as (1+t/ε)^{−1} do not follow from (2.12): for the L² norm (2.12) gives no decay (exponent 0), for L^{2γ/(γ−1)} it gives exponent −1/γ, and for L⁴ it gives −1/2. The final conclusion R₆₂₁≤ε^{m+1}ln(1+τ/ε)→0 is nevertheless correct, since ∫₀^τ(1+t/ε)^{−β}dt≤ε/(β−1) for β>1 and the worst case (L² part, exponent 0) is directly bounded by ε^α·T via the conservation law (2.11). But the displayed claim is wrong as written and should be corrected, with the three pieces of (4.19) estimated separately according to their actual exponents.
  4. [§3, Eq. (3.13)] In the second equality, ρ∇_x(H''(1)s)·(U−u) is replaced by ∇_x(p'(1)s)·(U−u), silently dropping the factor ρ; the factor reappears in the final term −ε²ρ∂_t∇_xΦ·(U−u). The net identity is correct (H''(1)=p'(1)=γ and the acoustic equation (2.10)), but the intermediate line as printed is a scalar identity that does not hold; the ρ should be carried through or the step reordered.
minor comments (6)
  1. [§4, Eq. (4.9)] '(m=1−1/p>2)' is an evident typo; presumably m=1−1/p with p>3, or the intended exponent should be restated.
  2. [§4, Eq. (4.16)] The bound |R₇|≤η(ε)→0 is asserted in one line. It follows from r₁(ε)→0 together with sup_t∥√ρu∥_{L²}≤C (2.5) and U∈L∞_tL∞_x, but the estimate should be displayed since the drag is a nonstandard feature of the model.
  3. [§2.4, Remark 2.2] 'From to the compact support assumption on the initial data of (2.15)' — typo ('From to'), and the reference to (2.15) should be to (2.14)–(2.15) or to the initial data of (2.10).
  4. [§3, Eq. (3.3)/(3.8)] In (3.8) and elsewhere, time integrals appear inside spatial integrals written with dx only (e.g. ε^α∫∫√ρS_μ:∇_xU dxdt nested inside a dx expression); the notation should be cleaned up. Also 'We say that (ρ,u) is a weak solution ... if the following conditions are satisfied' — the second bullet (boundary conditions) lacks a bullet marker in the definition and (2.1) conflates regularity and trace conditions.
  5. [§4, Eq. (4.1)] The summation runs over R₁,…,R₇ but the viscous term is labelled R₆ᵢ (i=1,2,3) in (4.17); there is no explicitly named R₆ in (4.1). Please make the numbering consistent.
  6. [References] Reference [2] is listed as 'J. Math. Fluid Mech. 28:46 (2026), .' with a trailing comma and no DOI; reference [23] has a typo ('applicatons'); the 2010 MSC should be updated to 2020 MSC.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: standard weak-to-strong relative-energy limit; existence is imported externally, not self-defined.

full rationale

The paper proves a triple singular limit (vanishing viscosity, low Mach, expanding domains) for degenerate compressible Navier–Stokes by the relative-energy method. The target is an independently given strong solution v of incompressible Euler; the test functions are Euler plus acoustic waves plus a boundary corrector w_M. The relative entropy inequality (3.2)–(3.3) is derived from the weak formulation and energy inequality, then remainder terms R_i are estimated via Strichartz decay, energy bounds (2.5)–(2.7), and cut-off estimates (2.13), and absorbed by Gronwall. Nothing is fitted to data; no quantity is defined in terms of the claimed limit and then re-derived; citations (Feireisl et al., Bresch–Desjardins–Gérard-Varet, Bisconti et al.) supply method and existence background from other authors, not a self-citation uniqueness theorem that forces the triple-limit statement. Concerns about whether [4] literally covers the exact boundary conditions, drag, and γ-range are correctness/applicability issues, not circularity. The derivation chain is self-contained as a conditional weak-to-strong convergence argument.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central convergence theorem rests on standard PDE toolkit plus domain and constitutive hypotheses that guarantee weak solutions exist and acoustic waves do not hit the boundary on fixed time intervals. No numerical free parameters are fitted. The main external loads are existence of degenerate NS weak solutions with drag, local smooth Euler existence, and Strichartz/energy bounds for the acoustic system.

assumptions (6)
  • domain assumption Global weak solutions of the degenerate compressible NS system (1.1) exist on each fixed Ω_M for the chosen density-dependent viscosity and drag, in the sense of Definition 2.1 (imported from Bresch–Desjardins–Gérard-Varet).
    Stated in the Introduction and Remark 2.1 as the reason for including drag and the BD viscosity relation; without it the relative-energy argument has no sequence of solutions.
  • standard math Local-in-time smooth solutions of 3D incompressible Euler exist for compactly supported C^m initial data with m>4 (Kato–Lai).
    Section 2.3; the weak-to-strong limit is only claimed on [0,T] inside the Euler lifespan T_max.
  • standard math Acoustic system (2.10) satisfies energy conservation (2.11) and Strichartz decay (2.12).
    Section 2.4, citing Strichartz; used to kill acoustic remainders as ε→0.
  • domain assumption Expanding domains satisfy (H1)–(H4), in particular εM(ε)→∞, so acoustic waves from compactly supported data do not reach ∂Ω_M on fixed time intervals.
    Hypotheses after (1.1) and Remark 2.2; needed so the whole-space acoustic solution is admissible inside Ω_M and boundary layers are avoided.
  • domain assumption Pressure law p(ρ)=ρ^γ with γ>1, shear viscosity proportional to ρ, bulk viscosity zero, and drag coefficient r₁(ε)→0.
    Constitutive setup of (1.1); chosen as the prototype BD relation so existence applies.
  • ad hoc to paper Relative energy inequality (Lemma 3.1) holds for any finite weak solution against smooth compactly supported test functions.
    Derived in Section 3 from the weak form and energy inequality; load-bearing identity for all subsequent estimates.

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Pith. "Pith review of Inviscid Incompressible Limit for Degenerate Compressible Navier-Stokes Equations on Expanding Domains." pith.science (2026). https://pith.science/paper/2OT6X5MR

@misc{pith2026260725028,
  author       = {Pith},
  title        = {Pith review of: Inviscid Incompressible Limit for Degenerate Compressible Navier-Stokes Equations on Expanding Domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2OT6X5MR}},
  note         = {Machine review of arXiv:2607.25028}
}
abstract

We simultaneously investigate the inviscid and Low Mach number limits on expanding domains for the $3D$ degenerate compressible Navier-Stokes equations, whose viscosity depends on density. Starting from ill-prepared data, we show the limit system is the incompressible Euler system. Our result extends a previous result of Feireisl et al. concerning the constant viscosity Navier-Stokes equations, and is the first to establish the inviscid and incompressible limits at the same time for density-dependent viscosity.

Figures

Figures reproduced from arXiv: 2607.25028 by the authors.

Figure 1
Figure 1. K(t) ⊂⊂ ΩM Following [15], we introduce the cut-off function wM = −ηMv − ηM∇xΦ, where ηM satisfies ηM ∈ C ∞ c (R 3 ), 0 ≤ ηM ≤ 1, ηM|∂ΩM = 1, ηM(x) = 0, dist(x, ∂ΩM) > 1, and we have the estimate ∥∂twM(τ, ·)∥Lp(ΩM) + ∥wM(τ, ·)∥W2,p(ΩM) ≤ CM2( 1 p −1) (2.13) for 1 ≤ p ≤ ∞, τ ∈ (0, T). 2.6 Main result Theorem 2.1. Let α > 0 and suppose the initial data (1.3) are chosen in such a way that v0 ∈ C m(R 3 ), ρ (1) 0 ∈ C m(… view at source ↗

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