Singular limits for compressible inviscid rotating fluids

Nilasis Chaudhuri

arxiv: 1907.06784 · v1 · pith:VHY3PBOWnew · submitted 2019-07-15 · 🧮 math.AP

Singular limits for compressible inviscid rotating fluids

Nilasis Chaudhuri This is my paper

Pith reviewed 2026-05-24 21:02 UTC · model grok-4.3

classification 🧮 math.AP

keywords singular limitsbarotropic Euler systemrotating fluidscompressible fluidsinviscid fluidsdissipative solutionsincompressible limitMach number

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The pith

Scaled barotropic Euler system for rotating compressible inviscid fluid in an infinite slab converges to horizontal incompressible inviscid flow as Mach and Rossby numbers vanish proportionally.

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

The paper examines the singular limit of the scaled barotropic Euler equations modeling a rotating, compressible, inviscid fluid where both the Mach number and Rossby number are proportional to a small parameter ε. When the fluid is confined to an infinite slab, the solutions converge to the horizontal motion of an incompressible inviscid system analogous to the Euler equations. This convergence is shown for a broad class of dissipative solutions of the compressible system, which approach strong solutions of the limiting incompressible system. The result identifies the effective large-scale dynamics under simultaneous rapid rotation and low compressibility, provided the slab geometry holds.

Core claim

If the fluid is confined to an infinite slab, the limit behaviour is identified as a horizontal motion of an incompressible inviscid system that is analogous to the Euler system. Dissipative solutions for the scaled compressible Euler systems converge to a strong solution of that incompressible inviscid system.

What carries the argument

The singular limit process applied to dissipative solutions of the scaled barotropic Euler system, with Mach and Rossby numbers both proportional to ε, under the infinite slab domain restriction.

If this is right

The limiting motion is purely horizontal with vanishing vertical velocity.
The density becomes constant and the flow satisfies an incompressible inviscid system analogous to the Euler equations.
Convergence holds from the general class of dissipative solutions to strong solutions of the limit system.
The result applies specifically when Mach and Rossby numbers scale together with ε.

Where Pith is reading between the lines

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

The slab geometry enforces the reduction to two-dimensional horizontal dynamics, suggesting the limit may fail or require different scalings in domains with vertical extent.
This type of limit could be used to justify reduced models in geophysical flows where rotation dominates and domains are thin.
Numerical tests could check whether the convergence rate matches the ε scaling for chosen initial data.

Load-bearing premise

The fluid must be confined to an infinite slab so that the limit reduces to purely horizontal motion.

What would settle it

A dissipative solution in the infinite slab whose vertical velocity component remains order one or whose density fails to become constant in the limit would contradict the claimed convergence to horizontal incompressible motion.

read the original abstract

We study singular limit for scaled barotropic Euler system modelling a rotating, compressible and inviscid fluid, where Mach and Rossby numbers are proportional to a small parameter $\epsilon$. If the fluid is confined to an infinite slab, the limit behaviour is identified as a horizontal motion of an incompressible inviscid system that is analogous to the Euler system. We consider a very general class of solutions, named dissipative solution for the scaled compressible Euler systems and will show that it converges to a strong solution of that incompressible inviscid system.

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.

Desk Editor's Note private letter to a colleague

The paper shows dissipative solutions of the scaled compressible rotating Euler system converge in a slab to a strong solution of a horizontal incompressible Euler-like system under proportional Mach-Rossby scaling.

read the letter

The key takeaway is that under proportional Mach-Rossby scaling, dissipative solutions of the compressible rotating Euler equations in a slab converge to a strong solution of a horizontal incompressible Euler-type system. The work applies the dissipative solution framework to this singular limit problem, which fits the inviscid setting well since it does not require strong uniqueness. The geometry restriction to the slab is explicitly used to get purely horizontal motion in the limit, and the abstract presents this as the main result. It does well in stating a clear convergence statement for a general class of solutions. Prior work on singular limits often uses viscosity or different scalings, so this combination for the inviscid compressible case is a step forward in that direction. The main soft spot is that the abstract supplies no proof sketch or error estimates, making it impossible to verify the handling of the singular terms from the abstract alone. The result depends heavily on the slab geometry, which is a strong assumption that may not extend easily to other domains. If the full paper has the details, that would address this. This paper is for specialists in PDEs for fluid dynamics, particularly those studying asymptotic reductions for rotating flows. A reader working on similar singular limit problems would get value from seeing how the dissipative solutions are used here. It deserves a serious referee because the result is specific and could be useful if the proof is correct. I would recommend engaging with it through peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the singular limit as ε→0 for the scaled barotropic Euler system modeling a rotating compressible inviscid fluid, with both Mach and Rossby numbers proportional to ε. Under the geometric assumption that the fluid is confined to an infinite slab, dissipative solutions of the scaled system are shown to converge to a strong solution of a horizontal incompressible inviscid system analogous to the Euler equations.

Significance. If the convergence result holds, the paper supplies a rigorous justification for the reduction of rotating compressible inviscid dynamics to horizontal incompressible motion when the domain is an infinite slab. The framework of dissipative solutions converging to strong solutions of the target system is a standard and appropriate device in singular-limit analyses for inviscid fluids; the geometric restriction is stated explicitly and is load-bearing for the horizontal reduction.

minor comments (2)

The abstract states the convergence result but supplies no proof details, error estimates, or handling of the singular terms; while the full manuscript presumably contains these, a brief indication of the main technical steps (e.g., the form of the dissipative solution definition or the compactness argument) would improve readability.
Notation for the scaled equations and the precise definition of dissipative solutions should be introduced early (ideally in §1 or §2) with explicit reference to the infinite-slab geometry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recommendation for minor revision. The report raises no specific major comments requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

No circularity: standard singular-limit passage under explicit geometric hypothesis

full rationale

The derivation relies on the well-posedness framework of dissipative solutions for the scaled compressible Euler system converging to a strong solution of the target incompressible system. This is a standard device in singular-limit analyses and is not reduced by construction to any fitted parameter or self-referential definition. The infinite-slab confinement is stated explicitly as a necessary hypothesis that forces the limit to be horizontal; it is not smuggled in via citation or self-definition. No load-bearing step equates a claimed prediction to its own input, and no uniqueness theorem or ansatz is imported from overlapping prior work by the same author. The result is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; the ledger therefore records only the explicit assumptions stated in the abstract. Standard PDE existence and regularity assumptions from the broader literature are not listed because they cannot be verified.

axioms (2)

domain assumption The fluid is confined to an infinite slab
Stated in the abstract as the geometric condition under which the horizontal incompressible limit holds.
domain assumption Dissipative solutions exist for the scaled compressible Euler system
The paper works with this very general class of solutions and shows convergence for them.

pith-pipeline@v0.9.0 · 5598 in / 1308 out tokens · 30611 ms · 2026-05-24T21:02:05.391236+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 5 internal anchors

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D Basari´ c, Vanishing viscosity limit for the compressible navier-sto kes system via measure-valued solutions, arXiv e-prints (2019Mar), arXiv:1903.05886, available a t 1903.05886

work page arXiv 1903
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D Breit, E Feireisl, and M Hofmanova, Dissipative solutions and semiﬂow selection for the comple te euler system, arXiv e-prints (2019Apr), arXiv:1904.00622, available a t 1904.00622

work page arXiv 1904
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, Generalized solutions to models of inviscid ﬂuids , arXiv e-prints (2019Jul), arXiv:1907.00757, available at 1907.00757

work page internal anchor Pith review Pith/arXiv arXiv 1907
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Existence of measure-valued solutions to a complete Euler system for a perfect gas

J. Brezina, Existence of measure-valued solutions to a complete Euler s ystem for a perfect gas , arXiv e-prints (May 2018), arXiv:1805.05570, available at 1805.05570

work page internal anchor Pith review Pith/arXiv arXiv 2018
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Bˇ rezina and E

J. Bˇ rezina and E. Feireisl, Measure-valued solutions to the complete Euler system , J. Math. Soc. Japan 70 (2018), no. 4, 1227–1245. MR3868717

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work page internal anchor Pith review Pith/arXiv arXiv
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J. Bˇ rezina and V . Mácha,Low stratifucation of the complete euler system , arXiv e-prints (2018Dec), arXiv:1812.08465, available at 1812.08465

work page internal anchor Pith review Pith/arXiv arXiv
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Chemin, B

J.-Y . Chemin, B. Desjardins, I. Gallagher, and E. Greni er, Mathematical geophysics, Oxford Lec- ture Series in Mathematics and its Applications, vol. 32, Th e Clarendon Press, Oxford Univer- sity Press, Oxford, 2006. An introduction to rotating ﬂuids and the Navier-Stokes equations. MR2228849

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D. G. Ebin, The motion of slightly compressible ﬂuids viewed as a motion with strong constraining force, Ann. of Math. (2) 105 (1977), no. 1, 141–200. MR0431261

work page 1977
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, Viscous ﬂuids in a domain with frictionless boundary , Global analysis–analysis on mani- folds, 1983, pp. 93–110. MR730604

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E Feireisl, I Gallagher, and A Novotný, A singular limit for compressible rotating ﬂuids , SIAM J. Math. Anal. 44 (2012), no. 1, 192–205. MR2888285

work page 2012
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E Feireisl, C Klingenberg, and S Markfelder, On the low mach number limit for the compressible Euler system, SIAM J. Math. Anal. 51 (2019), no. 2, 1496–1513. MR3942857 22

work page 2019
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Feireisl and M

E. Feireisl and M. Lukáˇ cová-Medvidová, Convergence of a mixed ﬁnite element–ﬁnite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions, Found. Com- put. Math. 18 (2018), no. 3, 703–730. MR3807359

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E Feireisl, M Lukacova-Medvidova, and H Mizerova, K− convergence as a new tool in numerical analysis, arXiv e-prints (2019Mar), arXiv:1904.00297, available a t 1904.00297

work page internal anchor Pith review Pith/arXiv arXiv 1904
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MR2499296

E Feireisl and A Novotný, Singular limits in thermodynamics of viscous ﬂuids , Advances in Math- ematical Fluid Mechanics, Birkhäuser V erlag, Basel, 2009. MR2499296

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, Scale interactions in compressible rotating ﬂuids , Ann. Mat. Pura Appl. (4) 193 (2014), no. 6, 1703–1725. MR3275259

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[20]

Gwiazda, A

P . Gwiazda, A. ´Swierczewska-Gwiazda, and E. Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible ﬂuid models , Nonlinearity 28 (2015), no. 11, 3873–

work page 2015
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Klainerman and A

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible ﬂuids , Comm. Pure Appl. Math. 34 (1981), no. 4, 481–

work page 1981
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M Oliver, Classical solutions for a generalized Euler equation in two dimensions, J. Math. Anal. Appl. 215 (1997), no. 2, 471–484. MR1490763

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S Schochet, The compressible Euler equations in a bounded domain: exist ence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), no. 1, 49–75. MR834481 23

work page 1986

[1] [1]

J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures, J. Convex Anal. 4 (1997), no. 1, 129–147. MR1459885

work page 1997

[2] [2]

Babin, A

A. Babin, A. Mahalov, and B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana Univ . Math. J. 48 (1999), no. 3, 1133–1176. MR1736966

work page 1999

[3] [3]

, 3D Navier-Stokes and Euler equations with initial data char acterized by uniformly large vor- ticity, Indiana Univ . Math. J.50 (2001), no. Special Issue, 1–35. Dedicated to Professors Ci prian Foias and Roger T emam (Bloomington, IN, 2000). MR1855663

work page 2001

[4] [4]

D Basari´ c, Vanishing viscosity limit for the compressible navier-sto kes system via measure-valued solutions, arXiv e-prints (2019Mar), arXiv:1903.05886, available a t 1903.05886

work page arXiv 1903

[5] [5]

D Breit, E Feireisl, and M Hofmanova, Dissipative solutions and semiﬂow selection for the comple te euler system, arXiv e-prints (2019Apr), arXiv:1904.00622, available a t 1904.00622

work page arXiv 1904

[6] [6]

, Generalized solutions to models of inviscid ﬂuids , arXiv e-prints (2019Jul), arXiv:1907.00757, available at 1907.00757

work page internal anchor Pith review Pith/arXiv arXiv 1907

[7] [7]

Existence of measure-valued solutions to a complete Euler system for a perfect gas

J. Brezina, Existence of measure-valued solutions to a complete Euler s ystem for a perfect gas , arXiv e-prints (May 2018), arXiv:1805.05570, available at 1805.05570

work page internal anchor Pith review Pith/arXiv arXiv 2018

[8] [8]

Bˇ rezina and E

J. Bˇ rezina and E. Feireisl, Measure-valued solutions to the complete Euler system , J. Math. Soc. Japan 70 (2018), no. 4, 1227–1245. MR3868717

work page 2018

[9] [9]

G Bruell and E Feireisl, On a singular limit for stratiﬁed compressible ﬂuids , arXiv e-prints (2018Feb), arXiv:1802.10340, available at 1802.10340

work page internal anchor Pith review Pith/arXiv arXiv

[10] [10]

Low stratifucation of the complete Euler system

J. Bˇ rezina and V . Mácha,Low stratifucation of the complete euler system , arXiv e-prints (2018Dec), arXiv:1812.08465, available at 1812.08465

work page internal anchor Pith review Pith/arXiv arXiv

[11] [11]

Chemin, B

J.-Y . Chemin, B. Desjardins, I. Gallagher, and E. Greni er, Mathematical geophysics, Oxford Lec- ture Series in Mathematics and its Applications, vol. 32, Th e Clarendon Press, Oxford Univer- sity Press, Oxford, 2006. An introduction to rotating ﬂuids and the Navier-Stokes equations. MR2228849

work page 2006

[12] [12]

D. G. Ebin, The motion of slightly compressible ﬂuids viewed as a motion with strong constraining force, Ann. of Math. (2) 105 (1977), no. 1, 141–200. MR0431261

work page 1977

[13] [13]

, Viscous ﬂuids in a domain with frictionless boundary , Global analysis–analysis on mani- folds, 1983, pp. 93–110. MR730604

work page 1983

[14] [14]

E Feireisl, I Gallagher, and A Novotný, A singular limit for compressible rotating ﬂuids , SIAM J. Math. Anal. 44 (2012), no. 1, 192–205. MR2888285

work page 2012

[15] [15]

E Feireisl, C Klingenberg, and S Markfelder, On the low mach number limit for the compressible Euler system, SIAM J. Math. Anal. 51 (2019), no. 2, 1496–1513. MR3942857 22

work page 2019

[16] [16]

Feireisl and M

E. Feireisl and M. Lukáˇ cová-Medvidová, Convergence of a mixed ﬁnite element–ﬁnite volume scheme for the isentropic Navier-Stokes system via dissipative measure-valued solutions, Found. Com- put. Math. 18 (2018), no. 3, 703–730. MR3807359

work page 2018

[17] [17]

E Feireisl, M Lukacova-Medvidova, and H Mizerova, K− convergence as a new tool in numerical analysis, arXiv e-prints (2019Mar), arXiv:1904.00297, available a t 1904.00297

work page internal anchor Pith review Pith/arXiv arXiv 1904

[18] [18]

MR2499296

E Feireisl and A Novotný, Singular limits in thermodynamics of viscous ﬂuids , Advances in Math- ematical Fluid Mechanics, Birkhäuser V erlag, Basel, 2009. MR2499296

work page 2009

[19] [19]

, Scale interactions in compressible rotating ﬂuids , Ann. Mat. Pura Appl. (4) 193 (2014), no. 6, 1703–1725. MR3275259

work page 2014

[20] [20]

Gwiazda, A

P . Gwiazda, A. ´Swierczewska-Gwiazda, and E. Wiedemann, Weak-strong uniqueness for measure-valued solutions of some compressible ﬂuid models , Nonlinearity 28 (2015), no. 11, 3873–

work page 2015

[21] [21]

Klainerman and A

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible ﬂuids , Comm. Pure Appl. Math. 34 (1981), no. 4, 481–

work page 1981

[22] [22]

M Oliver, Classical solutions for a generalized Euler equation in two dimensions, J. Math. Anal. Appl. 215 (1997), no. 2, 471–484. MR1490763

work page 1997

[23] [23]

S Schochet, The compressible Euler equations in a bounded domain: exist ence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), no. 1, 49–75. MR834481 23

work page 1986