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

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.

• The paper defines regular co-moving volumes by removing null sets from Borel regions, making their DiPerna–Lions flow images measurable despite the flow’s almost-everywhere group property.
• The paper develops bounded-domain transport theory through commutator estimates, strong convergence of mollified flows, and control of boundary leakage under the condition v∈W^{1,1}_0.
• The paper proves image- and inverse-image Reynolds transport and Liouville formulas, showing that both agree with limits of classical smooth-flow identities for weak Sobolev velocities.

Background and motivation

In classical fluid mechanics, balance laws are derived by integrating over material volumes advected by the flow map $X(s,t,\xi)$ of the ODE $\gamma'(s)=v(s,\gamma(s))$, and the Reynolds transport theorem supplies the interchange between the time derivative and the moving-volume integral. When $v$ is merely a Sobolev vector field—as arises for weak solutions of the Navier-Stokes equations—the classical ODE theory fails, and one must work within DiPerna-Lions theory [DL], which constructs generalized (a.e.) flow maps via uniqueness of weak solutions to the linear transport equation. In that setting, the inverse image $X(s,t,\cdot)^{-1}(A)$ of a Borel set is well behaved and its measure evolution is controlled by $\nabla\cdot v$, but the forward image $X(s,t,A)$—the natural candidate for a "co-moving volume"—need not be Lebesgue measurable at all. Consequently, the classical formulation of the Reynolds transport theorem over images does not even make sense in the Sobolev framework.

The paper under review addresses precisely this gap. Its central contribution is a notion of regular co-moving volume: for a Borel set $A\subset\Omega$ and each pair $(s,t)$, one removes a suitable null set $\dot N_A^{s,t}$ from $A$ so that the trimmed image $\gamma'(s)=v(s,\gamma(s))$0 is measurable and avoids the exceptional set

$\gamma'(s)=v(s,\gamma(s))$1

on which the group property fails. The author then proves a full Reynolds transport theorem for these trimmed volumes, together with an inverse-image formulation, and shows both coincide with the limits obtained from mollified flows.

Setting: DiPerna-Lions theory on a bounded domain

The working hypothesis is that $\gamma'(s)=v(s,\gamma(s))$2 is a bounded connected open set with $\gamma'(s)=v(s,\gamma(s))$3 and $\gamma'(s)=v(s,\gamma(s))$4 with weak spatial derivatives in the same class and $\gamma'(s)=v(s,\gamma(s))$5 a.e. in time; the $\gamma'(s)=v(s,\gamma(s))$6 condition encodes no-slip boundary behavior. The author first restates the DiPerna-Lions theorem in this bounded-domain form (Theorem 1): there exists a measurable map $\gamma'(s)=v(s,\gamma(s))$7 satisfying, for all Borel $\gamma'(s)=v(s,\gamma(s))$8,

$\gamma'(s)=v(s,\gamma(s))$9

together with the a.e. group identities and an integral representation $v$0 valid off a null set $v$1.

The author emphasizes that while the original whole-space treatment was sketched as doable for bounded domains, a fully detailed account appears to be missing from the literature; the paper supplies it. The proof strategy follows the DiPerna-Lions blueprint: mollify the zero-extension of $v$2 to obtain smooth fields $v$3 supported in $v$4 ($v$5 a ball), solve the transport equation with initial data $v$6 to recover the components of the approximate flow $v$7, and pass to the limit using compactness. A notable technical point is the handling of the extension outside $v$8: mollification causes a "leak" of trajectories into $v$9, which must be quantified; the assumption $X(s,t,\cdot)^{-1}(A)$0 is used here and is stated to be essential.

Analytic core: commutator estimates and strong convergence

The heart of the argument is a self-contained development of the linear transport equation

$X(s,t,\cdot)^{-1}(A)$1

in $X(s,t,\cdot)^{-1}(A)$2 with parameter $X(s,t,\cdot)^{-1}(A)$3. Existence is obtained by weak compactness of the approximate solutions $X(s,t,\cdot)^{-1}(A)$4, preserving the $X(s,t,\cdot)^{-1}(A)$5 bounds through elementary $X(s,t,\cdot)^{-1}(A)$6 arguments rather than an $X(s,t,\cdot)^{-1}(A)$7 calculus. The key lemma is the standard DiPerna-Lions commutator estimate: after truncating in time and mollifying, the defect $X(s,t,\cdot)^{-1}(A)$8 satisfies $X(s,t,\cdot)^{-1}(A)$9 as $\nabla\cdot v$0. The proof carefully exploits the freedom to use test functions not compactly supported inside $\nabla\cdot v$1—a choice forced by the shifted test functions $\nabla\cdot v$2 appearing in the commutator computation—and the geometric condition ensuring that mollification of data supported in $\nabla\cdot v$3 vanishes outside $\nabla\cdot v$4.

From the commutator estimate the author derives the $\nabla\cdot v$5-norm evolution identity

$\nabla\cdot v$6

which yields uniqueness in $\nabla\cdot v$7, continuity of the solution representative in $\nabla\cdot v$8, and—crucially—strong convergence $\nabla\cdot v$9 for every fixed $X(s,t,A)$0. A further diagonal/subsequence argument (Proposition on joint convergence) upgrades this to strong convergence in $X(s,t,A)$1 jointly in $X(s,t,A)$2, which is what later guarantees measurability properties of the limit flow map. The proof of strong convergence requires delicate control near $X(s,t,A)$3 and near the boundary strip $X(s,t,A)$4, handled via equicontinuity of the norms and Gronwall-type estimates.

Passing to the limit in the integral identity $X(s,t,A)$5 (justified by a stability lemma showing $X(s,t,A)$6 in $X(s,t,A)$7 whenever $X(s,t,A)$8 in $X(s,t,A)$9) yields the weak ODE identity and the a.e. group property of $A\subset\Omega$0.

Measurability of trimmed images

The measurability analysis rests on Egorov's theorem combined with measure regularity. Since $A\subset\Omega$1 strongly in $A\subset\Omega$2, one extracts sets where convergence is uniform; on closed subsets the limit map is continuous, so its image is compact and hence measurable. The main lemma establishes six assertions, of which the essential ones are:

• For any bounded measurable $A\subset\Omega$3 there is a null set $A\subset\Omega$4 such that $A\subset\Omega$5 is measurable.
• Approximate images $A\subset\Omega$6 can be trapped in open neighborhoods $A\subset\Omega$7 of the limit image with $A\subset\Omega$8, giving quantitative control of the leak.
• Preimages of null sets under $A\subset\Omega$9 are null (via the classical Liouville theorem applied to $(s,t)$0 and a contradiction argument), so trimming does not destroy mass.

A companion proposition then identifies the measure of regular co-moving volumes with the limit of the classical quantities:

$(s,t)$1

and analogously for integrals of $(s,t)$2 over images and preimages. The inclusion $(s,t)$3 plays a role in controlling boundary leakage. Importantly, the resulting formulas are independent of the particular choice of trimming null set, since each equals the same limit of smooth-flow quantities.

The Reynolds transport theorem

The main theorem states that for Borel $(s,t)$4, a regular co-moving volume, and $(s,t)$5,

$(s,t)$6

with the parallel inverse-image identity

$(s,t)$7

and the Liouville-type area formulas obtained by taking $(s,t)$8. These are exactly the limits of the classical identities for the mollified flows, and a.e.-differential forms are also available. The proof is short once the preceding propositions are in place: apply the classical theorem to $(s,t)$9 and pass to the limit using the identification of integrals over trimmed images with limits of integrals over $\dot N_A^{s,t}$0-images, noting measurability of $\dot N_A^{s,t}$1 from a.e. pointwise convergence.

Two structural remarks deserve emphasis. First, the entire argument uses only strong $\dot N_A^{s,t}$2 convergence of approximate flows—a property inherited from uniqueness of transport solutions—so the method is robust to refinements of the existence theory. Second, the result restores the classical picture without altering the vector field or the flow: the price is only the removal of a null set depending on $\dot N_A^{s,t}$3, and the measure evolution of the trimmed volume agrees with the untrimmed classical formula.

Limitations and open questions

Several restrictions are explicit. The theory is developed in three space dimensions on a bounded domain with $\dot N_A^{s,t}$4 regularity, corresponding to no-slip boundaries; the non-penetration condition $\dot N_A^{s,t}$5 on $\dot N_A^{s,t}$6 is explicitly excluded and left untreated. The trimming null set depends on $\dot N_A^{s,t}$7, and while the transport formula is trimming-independent, the question whether $\dot N_A^{s,t}$8 itself is measurable for general Borel $\dot N_A^{s,t}$9 remains negative in general and is not resolved. The a.e. nature of the group property means the construction cannot avoid the exceptional sets $A$0, and the paper does not address pointwise uniqueness questions—it cites the known dichotomy between a.e. uniqueness results (Robinson-Sadowski, Galeati) and positive-measure nonuniqueness constructions (Brué-Colombo-De Lellis, Kumar) but contributes nothing new there. Finally, the relation to two-phase flows with discontinuous velocity fields, where Bothe-Köhne retain surface-integral terms at interfaces, is noted as a distinct generalization not covered here.

Conclusion

This paper completes a missing piece of the DiPerna-Lions framework: it gives a detailed bounded-domain version of the flow-map theorem and, more substantively, defines regular co-moving volumes via controlled trimming of images, proving that their measures and the associated transport integrals coincide with the limits of the classical smooth-flow formulas. The Reynolds transport theorem and Liouville theorem thereby become available for Sobolev vector fields in the image formulation, restoring the standard tool for deriving balance laws in fluid mechanics at the level of weak solutions. The dependence of the construction on the no-slip class $A$1 and on $A$2 marks the precise boundary of the current result.