Dissipation concentration in two-dimensional fluids

Jaemin Park; Luigi De Rosa

arxiv: 2508.01440 · v3 · pith:JKFAHJDInew · submitted 2025-08-02 · 🧮 math.AP · math-ph· math.MP· physics.flu-dyn

Dissipation concentration in two-dimensional fluids

Luigi De Rosa , Jaemin Park This is my paper

Pith reviewed 2026-05-19 01:33 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPphysics.flu-dyn

keywords dissipation measureinviscid limitEuler equationstwo-dimensional fluidsdefect measurestrong compactnessvorticity measureanomalous dissipation

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The pith

The dissipation in the inviscid limit of two-dimensional incompressible fluids is Lebesgue in time and absolutely continuous with respect to the defect measure of strong compactness for almost every time.

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

This paper studies the dissipation measure that appears when taking the inviscid limit of two-dimensional incompressible fluids. It shows that this dissipation is absolutely continuous with respect to the defect measure that records the failure of strong compactness in the solutions, and that the dissipation is Lebesgue in time. When the initial vorticity is merely a measure, the dissipation becomes absolutely continuous with respect to a quadratic space-time vorticity measure instead. Under additional sign or oscillation conditions the resulting dissipation measure is either trivial or spatially atomic. The analysis isolates the Batchelor-Kraichnan scale as the only relevant one and supplies new tests for whether anomalous dissipation can occur.

Core claim

The dissipation measure arising in the inviscid limit of two-dimensional incompressible fluids is Lebesgue in time. For almost every time it is absolutely continuous with respect to the defect measure of strong compactness of the solutions. When the initial vorticity is a measure, the dissipation is absolutely continuous with respect to a quadratic space-time vorticity measure. This measure is trivial if the initial vorticity has a singular part of distinguished sign, and it is spatially purely atomic if wild oscillations in time are ruled out. The dynamics at the Batchelor-Kraichnan dissipative scale are the only ones that matter, which yields new criteria for anomalous dissipation.

What carries the argument

The defect measure of strong compactness of the solutions, which records the lack of strong compactness and serves as the reference measure for the absolute continuity of the dissipation measure.

If this is right

The dissipation concentrates only at the Batchelor-Kraichnan dissipative scale.
New criteria for the presence or absence of anomalous dissipation follow directly from the absolute continuity statement.
When the initial vorticity has a singular part of distinguished sign the dissipation measure is the zero measure.
If wild time oscillations are absent the dissipation measure is spatially purely atomic.
Quantitative rates of dissipation and the life-span of dissipation can be read off from the defect measure.

Where Pith is reading between the lines

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

If the defect measure vanishes then strong compactness holds and anomalous dissipation is ruled out.
The same absolute-continuity relation might be tested in numerical approximations of the Euler equations with rough initial data.
The result links compactness-based arguments in fluid dynamics to the study of measure-valued solutions for other conservation laws.

Load-bearing premise

The solutions to the Euler equations are assumed to possess a well-defined defect measure capturing the lack of strong compactness.

What would settle it

A concrete solution or numerical example in which the dissipation measure fails to be absolutely continuous with respect to the defect measure on a set of times of positive measure.

read the original abstract

We study the dissipation measure arising in the inviscid limit of two-dimensional incompressible fluids. It is proved that the dissipation is Lebesgue in time and, for almost every time, it is absolutely continuous with respect to the defect measure of strong compactness of the solutions. When the initial vorticity is a measure, the dissipation is proved to be absolutely continuous with respect to a ''quadratic'' space-time vorticity measure. This results into the trivial measure if the initial vorticity has singular part of distinguished sign, or a spatially purely atomic measure if wild oscillations in time are ruled out. In fact, the dynamics at the Batchelor-Kraichnan dissipative scale is the only relevant one, in turn offering new criteria for anomalous dissipation. We provide kinematic examples highlighting the strengths and the limitations of our approach. Quantitative rates, dissipation life-span and steady fluids are also investigated.

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 dissipation in 2D inviscid limits is Lebesgue in time and absolutely continuous w.r.t. defect measures, with concrete consequences for measure initial data.

read the letter

The main result is that the dissipation measure arising in the inviscid limit of 2D incompressible fluids is Lebesgue in time, and for almost every time it is absolutely continuous with respect to the defect measure of strong compactness. When initial vorticity is a measure, the dissipation is absolutely continuous with respect to a quadratic space-time vorticity measure, which becomes trivial under a distinguished sign condition on the singular part or purely atomic in space once wild time oscillations are ruled out. This ties the relevant dynamics to the Batchelor-Kraichnan scale and supplies new criteria for anomalous dissipation. The authors also supply kinematic examples that test the reach of the method, plus some quantitative rates, dissipation lifespan estimates, and remarks on steady cases. These concrete additions make the work more usable than a purely abstract statement would be. The measure-theoretic approach looks clean and avoids obvious circularity or free parameters. The combination of time regularity with absolute continuity under explicit initial-vorticity assumptions is the clearest increment over standard weak-solution results. One soft spot worth checking is whether the defect measure fully accounts for all non-compactness produced by the viscous approximations. If rapid time oscillations at the dissipative scale create extra singular contributions not captured by the usual compactness defect, the absolute-continuity step could fail for some sequences. The abstract invokes the defect measure directly, so the paper needs to show that its construction from the approximations rules this out. This is a targeted contribution for people working on weak solutions and concentration in 2D Euler. A reader who already knows defect measures and Young-measure techniques will extract the most value. The technical content is sharp enough to deserve a serious referee.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the dissipation measure arising in the inviscid limit of two-dimensional incompressible fluids. It proves that the dissipation is Lebesgue in time and, for almost every time, absolutely continuous with respect to the defect measure of strong compactness of the solutions. When the initial vorticity is a measure, the dissipation is absolutely continuous with respect to a quadratic space-time vorticity measure, yielding a trivial measure if the singular part has distinguished sign or a spatially purely atomic measure if wild time oscillations are ruled out. The dynamics at the Batchelor-Kraichnan dissipative scale are identified as the only relevant ones, providing new criteria for anomalous dissipation. Kinematic examples, quantitative rates, dissipation life-span, and steady fluids are also investigated.

Significance. If the central claims hold, the work provides a measure-theoretic link between anomalous dissipation and compactness defects in 2D Euler flows, with concrete consequences for measure-valued initial data and scale-specific criteria. The kinematic examples and quantitative results add concrete value for testing the approach.

major comments (1)

[Abstract and defect-measure construction (near the statement of the main absolute-continuity result)] Abstract and the paragraph introducing the defect measure: the absolute continuity of the dissipation measure μ with respect to the defect measure ν of strong compactness is load-bearing for the main theorems. The argument invokes ν directly to conclude that dμ/dν exists a.e. in time, but requires explicit verification that ν (constructed via the approximating viscous sequence) captures all relevant non-compactness, including possible additional singular contributions from time oscillations at the Batchelor-Kraichnan scale. Without this, the absolute-continuity step risks being incomplete for the specific sequences used.

minor comments (2)

[Abstract] The abstract places 'quadratic' in quotes for the space-time vorticity measure; a short inline definition or forward reference to its precise construction would improve readability.
[Kinematic examples] In the kinematic-examples section, explicit formulas or numerical values for the dissipation and defect measures in each example would make the strengths and limitations easier to verify.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The major comment concerning the defect measure construction and absolute continuity is addressed point-by-point below. We have revised the manuscript to incorporate additional clarification on this issue.

read point-by-point responses

Referee: Abstract and the paragraph introducing the defect measure: the absolute continuity of the dissipation measure μ with respect to the defect measure ν of strong compactness is load-bearing for the main theorems. The argument invokes ν directly to conclude that dμ/dν exists a.e. in time, but requires explicit verification that ν (constructed via the approximating viscous sequence) captures all relevant non-compactness, including possible additional singular contributions from time oscillations at the Batchelor-Kraichnan scale. Without this, the absolute-continuity step risks being incomplete for the specific sequences used.

Authors: We thank the referee for this observation. The defect measure ν is constructed directly from the viscous approximating sequence as the weak-* limit of the quadratic defect measures associated with the failure of strong compactness (specifically, the difference between the viscous vorticity and its weak limit in appropriate spaces). By definition, ν therefore encodes the total non-compactness of the given sequence, which necessarily includes any contributions arising from time oscillations at the Batchelor-Kraichnan dissipative scale—the scale identified in the paper as the only relevant one for dissipation concentration. Consequently, there are no additional singular contributions from this scale that lie outside ν. The absolute continuity of the dissipation measure μ with respect to ν then follows from the Radon-Nikodym theorem applied to these space-time measures. To address the concern explicitly, we have added a clarifying sentence in the paragraph introducing the defect measure (near the statement of the main absolute-continuity result) confirming that the construction via the viscous sequence already accounts for all such non-compactness effects at the dissipative scale. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation relies on standard measure-theoretic compactness arguments

full rationale

The paper derives properties of the dissipation measure from the assumed existence of a defect measure capturing lack of strong compactness in solutions to the 2D Euler equations. This is a standard assumption in the field for weak solutions, and the absolute continuity statements follow from analysis of these measures without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The quadratic space-time vorticity measure for measure-valued initial data is constructed via the paper's own compactness arguments rather than imported by ansatz or renaming. The approach is self-contained against external benchmarks in PDE theory, with no evidence of predictions that are forced by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions from PDE theory and measure theory for incompressible flows, with no free parameters or newly invented entities.

axioms (2)

domain assumption The fluid is incompressible and the velocity field satisfies the Euler equations in the inviscid limit.
Core modeling assumption stated in the title and abstract.
domain assumption Sequences of solutions admit a defect measure that quantifies failure of strong compactness.
Invoked directly to obtain absolute continuity of the dissipation measure.

pith-pipeline@v0.9.0 · 5674 in / 1396 out tokens · 62667 ms · 2026-05-19T01:33:35.465668+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 … Dt ≪ Λt for a.e. t ∈ [0,T]. … Theorem 1.6 … lim ν→0 Sν₂(√ν)=0 ⇔ lim ν→0 ν∫‖∇uν‖²=0
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
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Corollary 1.4 … Dt concentrated on Lt ∩ Ot … when |ων|⊗|ων| ⇀ γt⊗γt

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matches: The paper's claim is directly supported by a theorem in the formal canon.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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[1] [1]

Albritton, E

D. Albritton, E. Bru´ e, M. Colombo, C. De Lellis, V. Giri, M. Janisch, and H. Kwon, Instability and non- uniqueness for the 2D Euler equations, after M. Vishik , Annals of Mathematics Studies, vol. 219, Princeton University Press, Princeton, NJ, 2024

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E. Bru´ e and M. Colombo, Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space , Communications in Mathematical Physics 403 (2023), no. 2, 1171–1192

work page 2023

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Bru´ e, M

E. Bru´ e, M. Colombo, G. Crippa, C. De Lellis, and M. Sorella, Onsager critical solutions of the forced Navier- Stokes equations, Comm. Pure Appl. Anal. 23 (2024), no. 10, 1350–1366

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work page 2023

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