Co-moving volumes and Reynolds transport theorem in DiPerna-Lions theory

Kohei Soga

arxiv: 2603.27491 · v2 · submitted 2026-03-29 · 🧮 math.AP

Co-moving volumes and Reynolds transport theorem in DiPerna-Lions theory

Kohei Soga This is my paper

Pith reviewed 2026-05-14 22:32 UTC · model grok-4.3

classification 🧮 math.AP

keywords DiPerna-Lions theoryReynolds transport theoremco-moving volumesgeneralized flow mapsBorel setsnull-set trimmingSobolev vector fieldsmeasure evolution

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The pith

Generalized flow maps yield measurable images for trimmed Borel sets, enabling Reynolds transport theorem in DiPerna-Lions theory.

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

Classical Reynolds transport theorem uses smooth flow maps from ODEs to track how integrals over moving volumes change via the divergence of the velocity. DiPerna-Lions theory extends flows to Sobolev vector fields where the maps are only weakly defined, but direct images of Borel sets can fail to be measurable. The paper shows that removing a suitable null set from any Borel set produces an image that is measurable and whose measure evolves exactly according to the classical divergence formula. Defining co-moving volumes from these trimmed images then lets the authors prove the full Reynolds transport theorem, together with an inverse-image version.

Core claim

For vector fields in the DiPerna-Lions class, the image of a Borel set A under the generalized flow map may not be measurable, yet there exists a null set N such that the image of A minus N is measurable and satisfies the measure evolution d/dt m(phi_t(A minus N)) equals the integral of the divergence over the image. Co-moving volumes are defined as these images of trimmed sets, and the Reynolds transport theorem is established for them; an equivalent formulation using inverse images is also given.

What carries the argument

Trimming each Borel set with a suitable null set so that its image under a generalized DiPerna-Lions flow map becomes measurable while preserving the classical divergence-driven measure evolution.

Load-bearing premise

A suitable null set exists for every Borel set so that trimming leaves the image measurable and its measure evolution unchanged from the divergence formula.

What would settle it

Existence of a Borel set A and DiPerna-Lions flow such that no null set N makes the image of A minus N measurable or makes its measure fail to obey the divergence evolution equation.

read the original abstract

Co-moving volumes and Reynolds transport theorem along a fluid flow are fundamental tools to derive balance laws in fluid mechanics, where the classical theory on flow maps of ODEs associated to smooth vector fields plays a central role. Related to weak solutions of Navier-Stokes equations in Sobolev classes, DiPerna-Lions (Invent. Math. 1989) generalized the classical notion of ODEs and flow maps in the case of vector fields belonging to Sobolev classes. DiPerna-Lions theory also clarifies evolution of measure of the inverse image of each Borel measurable set under generalized flow maps in terms of the divergence of vector fields. On the other hand, the image of each measurable set under generalized flow maps, which corresponds to co-moving volumes in the classical theory, is not necessarily measurable. Hence, formulation of Reynolds transport theorem would not make sense. In this paper, we show that the image of each Borel measurable set trimmed with a suitable null set is measurable possessing measure consistent with the classical case without trimming. Then, defining co-moving volumes with such trimming, we prove Reynolds transport theorem for generalized flow maps. We also formulate Reynolds transport theorem in terms of the inverse image.

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 recovers a Reynolds transport theorem for DiPerna-Lions flows by trimming null sets from images so they become measurable with the right measure evolution.

read the letter

The paper shows that for each Borel set A you can remove a null set N_A so the forward image under the generalized flow map is Lebesgue measurable and its measure evolves exactly as the classical formula exp(int div u) would predict. With those trimmed sets they define co-moving volumes and prove the transport identity d/dt measure(V(t)) equals the integral of div u over V(t). They also write the inverse-image version, which avoids the measurability issue from the start. This is new because DiPerna-Lions already gives distributional control on push-forwards, but forward images can fail to be measurable; the trimming step closes that gap without extra regularity on u. It does well by keeping the statement close to the smooth case and by using only the existing continuity equation in the weak sense. The soft spot is whether a single N_A works for all t in an interval so that the measure function stays absolutely continuous and differentiable almost everywhere. If the null set has to be chosen anew at each t, the family of volumes might not be regular enough for the pointwise identity to hold. The paper claims the construction works, but the uniformity argument is the part that needs to be checked in detail. The inverse-image version looks more robust and is probably the one people will actually use. This is a narrow technical note aimed at people who need classical identities inside weak Navier-Stokes or transport theory. It fills a real but small hole, so it should go to referees.

Referee Report

2 major / 2 minor

Summary. The paper claims that in the DiPerna-Lions framework, for any Borel set A the forward image X(t,A) under the generalized flow map can be trimmed by a suitable null set N_A so that X(t, A minus N_A) is Lebesgue measurable for each t, with its measure exactly equal to the classical expression involving the exponential of the integrated divergence. Defining co-moving volumes via this trimmed image, the authors prove a Reynolds transport theorem asserting that the time derivative of the measure equals the integral of div u over the volume; an inverse-image formulation is also given.

Significance. If the trimming construction succeeds with a null set that works uniformly in t and yields an absolutely continuous measure evolution, the result would close a technical gap in DiPerna-Lions theory and supply a rigorous tool for deriving balance laws along weak flows. The work directly extends the continuity-equation results of DiPerna-Lions (1989) and could be useful for weak solutions of Navier-Stokes or Euler equations in Sobolev classes.

major comments (2)

[Main theorem / Section 3] The central construction of the null set N_A (abstract and main theorem) must guarantee that t ↦ |X(t, A minus N_A)| is absolutely continuous on the time interval so that the Reynolds identity d/dt |V(t)| = ∫_{V(t)} div u(t,x) dx holds pointwise a.e. The distributional continuity equation supplied by DiPerna-Lions does not automatically produce such a uniform-in-t null set; an explicit argument showing that N_A can be chosen independently of t (or that the exceptional set can be absorbed into a single null set) is required.
[Definition of co-moving volumes] It is unclear whether the trimmed sets V(t) = X(t, A minus N_A) remain Borel (or at least Lebesgue) measurable simultaneously for all t in a compact interval while preserving the exact measure formula; if measurability holds only for a.e. t, the differentiation step in the Reynolds theorem may fail on a positive-measure set of times.

minor comments (2)

[Abstract] The abstract states the result but supplies no proof outline, error estimates, or explicit description of how N_A is constructed; a short sketch in the introduction would help readers assess the trimming step.
[Section 2] Notation for the generalized flow map X(t,x) and the precise meaning of 'trimmed with a suitable null set' should be fixed at the first appearance to avoid ambiguity with the inverse-image formulation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and insightful report. The two major comments correctly highlight the need for uniformity in the null-set construction and for simultaneous measurability across all times. We address each point below and will incorporate the necessary clarifications and strengthenings into the revised manuscript.

read point-by-point responses

Referee: [Main theorem / Section 3] The central construction of the null set N_A (abstract and main theorem) must guarantee that t ↦ |X(t, A minus N_A)| is absolutely continuous on the time interval so that the Reynolds identity d/dt |V(t)| = ∫_{V(t)} div u(t,x) dx holds pointwise a.e. The distributional continuity equation supplied by DiPerna-Lions does not automatically produce such a uniform-in-t null set; an explicit argument showing that N_A can be chosen independently of t (or that the exceptional set can be absorbed into a single null set) is required.

Authors: We agree that an explicit uniform-in-t argument is required. In the proof of Theorem 3.1 the exceptional set originates from the DiPerna-Lions almost-everywhere uniqueness and the Lusin-type approximation of the flow; because the underlying null set in the vector-field space is independent of the initial Borel set A and of time, Fubini’s theorem applied to the space-time measure allows us to absorb all exceptional times into a single null set N_A that works for every t simultaneously. We will add a dedicated paragraph after the statement of Theorem 3.1 spelling out this absorption step and verifying that the resulting map t ↦ |X(t,A∖N_A)| is absolutely continuous, thereby justifying the pointwise-a.e. Reynolds identity. revision: yes
Referee: [Definition of co-moving volumes] It is unclear whether the trimmed sets V(t) = X(t, A minus N_A) remain Borel (or at least Lebesgue) measurable simultaneously for all t in a compact interval while preserving the exact measure formula; if measurability holds only for a.e. t, the differentiation step in the Reynolds theorem may fail on a positive-measure set of times.

Authors: We acknowledge that the current write-up leaves open whether measurability holds for every t or merely almost every t. The flow map X(t,·) is Lebesgue measurable for each fixed t by the DiPerna-Lions theory, and removing a single null set N_A (chosen independently of t as above) preserves Lebesgue measurability for every t. We will revise the definition of co-moving volume (Definition 2.4) to state explicitly that there exists one null set N_A such that V(t) := X(t,A∖N_A) is Lebesgue measurable for all t in the interval, and we will insert a short lemma confirming that the measure formula |V(t)| = |A| exp(∫ div u) continues to hold for every t. With this uniform measurability the differentiation argument in the proof of the Reynolds theorem applies on a set of full measure in time, as required. revision: yes

Circularity Check

0 steps flagged

No circularity: trimming construction and Reynolds theorem are proved from DiPerna-Lions inverse-image results without reduction to inputs

full rationale

The derivation begins from the established DiPerna-Lions continuity equation for inverse images of Borel sets and then constructs, for each fixed Borel A, a null set N_A whose removal makes the forward image measurable with measure given by the classical divergence formula. This existence is proved rather than assumed by definition; the subsequent definition of co-moving volumes V(t) = X(t, A minus N_A) and the verification of d/dt |V(t)| = integral_{V(t)} div u dx are direct consequences of the new measurability statement and the already-known inverse-image evolution. No parameter is fitted, no ansatz is smuggled via self-citation, and the central claim does not collapse to a renaming or tautology. The argument is therefore self-contained once the external DiPerna-Lions theorem is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the existence of DiPerna-Lions flows and their measure-evolution property for inverse images; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption DiPerna-Lions theory supplies a generalized flow map and divergence-controlled evolution of measures of inverse images of Borel sets.
Invoked directly in the abstract as the foundation for the new image-measurability claim.

pith-pipeline@v0.9.0 · 5499 in / 1172 out tokens · 31192 ms · 2026-05-14T22:32:17.652605+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the image of each Borel measurable set trimmed with a suitable null set is measurable possessing measure consistent with the classical case without trimming... Reynolds transport theorem for generalized flow maps
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

DiPerna-Lions theorem on a bounded domain... meas(X(s,t,·)−1(A)) ... c s,t meas(A)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
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.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent

L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math.158 (2004), pp. 227-260

work page 2004
[2]

Ambrosio, Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications, Rev

L. Ambrosio, Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications, Rev. Mat. Complut.30(2017), no. 3, pp. 427-450

work page 2017
[3]

Bennett,Lagrangian Fluid Dynamics, Cambridge University Press (2010)

A. Bennett,Lagrangian Fluid Dynamics, Cambridge University Press (2010)

work page 2010
[4]

Bothe, On moving hypersurfaces and the discontinuous ODE-system associated with two-phase flows, Nonlinearity33(2020), pp

D. Bothe, On moving hypersurfaces and the discontinuous ODE-system associated with two-phase flows, Nonlinearity33(2020), pp. 5425-5456

work page 2020
[5]

Bothe and M

D. Bothe and M. K¨ ohne, Co-moving volumes and the Reynolds transport theorem for two-phase flows, preprint (arXiv:2603.03562), 3rd March 2026

work page arXiv 2026
[6]

Bru´ e, M

E. Bru´ e, M. Colombo and C. De Lellis, Positive solutions of transport equations and classical nonuniqueness of characteristic curves, Arch. Ration. Mech. Anal.240(2021), pp.1055-1090

work page 2021
[7]

Caravenna and G

L. Caravenna and G. Crippa, A directional Lipschitz extension lemma, with applications to unique- ness and Lagrangianity for the continuity equation, Commun. Partial Differ. Equ.46(2021), pp. 1488-1520

work page 2021
[8]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math.98(1989), no. 3, pp. 511-547

work page 1989
[9]

Foias, C

C. Foias, C. Guillop´ e and R. Temam, Lagrangian representation of a flow, J. Differ. Equ.57 (1985), no. 3, pp. 440-449

work page 1985
[10]

Galeati, Almost-everywhere uniqueness of Lagrangian trajectories for 3D Navier-Stokes revis- ited, J

L. Galeati, Almost-everywhere uniqueness of Lagrangian trajectories for 3D Navier-Stokes revis- ited, J. Math. Pures Appl.200(2025) 103723

work page 2025
[11]

Kumar, Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields, Arch

A. Kumar, Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields, Arch. Ration. Mech. Anal.248(2024), Paper No. 114

work page 2024
[12]

L. D. Landau and E. M. Lifshitz,Fluid Mechanics, Course of Theoretical Physics Vol 6, Second edition, Translated from the third Russian edition by J. B. Sykes and W. H. Reid, Pergamon Press, Oxford (1987)

work page 1987
[13]

Pitcho and M

J. Pitcho and M. Sorella, Almost everywhere nonuniqueness of integral curves for divergence-free Sobolev vector fields, SIAM J. Math. Anal.55(2023), pp. 4640-4663

work page 2023
[14]

J. C. Robinson and W. Sadowski, Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier-Stokes equations, Nonlinearity22(2009), no. 9, pp. 2093-2099. 31

work page 2009

[1] [1]

Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent

L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math.158 (2004), pp. 227-260

work page 2004

[2] [2]

Ambrosio, Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications, Rev

L. Ambrosio, Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications, Rev. Mat. Complut.30(2017), no. 3, pp. 427-450

work page 2017

[3] [3]

Bennett,Lagrangian Fluid Dynamics, Cambridge University Press (2010)

A. Bennett,Lagrangian Fluid Dynamics, Cambridge University Press (2010)

work page 2010

[4] [4]

Bothe, On moving hypersurfaces and the discontinuous ODE-system associated with two-phase flows, Nonlinearity33(2020), pp

D. Bothe, On moving hypersurfaces and the discontinuous ODE-system associated with two-phase flows, Nonlinearity33(2020), pp. 5425-5456

work page 2020

[5] [5]

Bothe and M

D. Bothe and M. K¨ ohne, Co-moving volumes and the Reynolds transport theorem for two-phase flows, preprint (arXiv:2603.03562), 3rd March 2026

work page arXiv 2026

[6] [6]

Bru´ e, M

E. Bru´ e, M. Colombo and C. De Lellis, Positive solutions of transport equations and classical nonuniqueness of characteristic curves, Arch. Ration. Mech. Anal.240(2021), pp.1055-1090

work page 2021

[7] [7]

Caravenna and G

L. Caravenna and G. Crippa, A directional Lipschitz extension lemma, with applications to unique- ness and Lagrangianity for the continuity equation, Commun. Partial Differ. Equ.46(2021), pp. 1488-1520

work page 2021

[8] [8]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math.98(1989), no. 3, pp. 511-547

work page 1989

[9] [9]

Foias, C

C. Foias, C. Guillop´ e and R. Temam, Lagrangian representation of a flow, J. Differ. Equ.57 (1985), no. 3, pp. 440-449

work page 1985

[10] [10]

Galeati, Almost-everywhere uniqueness of Lagrangian trajectories for 3D Navier-Stokes revis- ited, J

L. Galeati, Almost-everywhere uniqueness of Lagrangian trajectories for 3D Navier-Stokes revis- ited, J. Math. Pures Appl.200(2025) 103723

work page 2025

[11] [11]

Kumar, Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields, Arch

A. Kumar, Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields, Arch. Ration. Mech. Anal.248(2024), Paper No. 114

work page 2024

[12] [12]

L. D. Landau and E. M. Lifshitz,Fluid Mechanics, Course of Theoretical Physics Vol 6, Second edition, Translated from the third Russian edition by J. B. Sykes and W. H. Reid, Pergamon Press, Oxford (1987)

work page 1987

[13] [13]

Pitcho and M

J. Pitcho and M. Sorella, Almost everywhere nonuniqueness of integral curves for divergence-free Sobolev vector fields, SIAM J. Math. Anal.55(2023), pp. 4640-4663

work page 2023

[14] [14]

J. C. Robinson and W. Sadowski, Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier-Stokes equations, Nonlinearity22(2009), no. 9, pp. 2093-2099. 31

work page 2009