On the well-posedness of (nonlinear) rough continuity equations

James-Michael Leahy; Lucio Galeati; Torstein Nilssen

REVIEW 1 major objections 5 minor 14 references

On the well-posedness of (nonlinear) rough continuity equations

T0 review · 1 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper proves that the classical Yudovich theorem for 2D Euler extends to rough transport noise: initial vorticities in $L^1_x\cap L^\infty_x$ produce unique global solutions to the rough vorticity equation, with flow representation…

desk verdict A dense but solid paper that delivers the first rough-path Yudovich theorem on R^2 and a DiPerna–Lions theory for rough linear PDEs; deserves serious refereeing. read the letter →

arxiv 2502.04982 v2 pith:O3QEOYPA submitted 2025-02-07 math.AP math.PR

Lucio Galeati , James-Michael Leahy , Torstein Nilssen This is my paper · ORCID

classification math.APmath.PR MSC 60L2060L5060H1535R6035Q31

keywords roughpartialdifferentialequations2DEulerYudovichtheoremDiPerna-LionstheoryunboundeddriversflowrepresentationfractionalBrownianmotiongeometricpaths

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

The paper proves that the classical Yudovich theorem for 2D Euler equations extends to rough transport noise: initial vorticities in $L^1_x\cap L^\infty_x$ produce unique global solutions to the rough vorticity equation, with no Sobolev regularity required. The solutions are renormalized, propagate by the rough flow $\omega_t(x)=\omega_0(\Phi_t^{-1}(x))$, and conserve every $L^p_x$ norm. Well-posedness holds for geometric rough paths of finite $p$-variation with $p\in[2,3)$, which covers fractional Brownian motion with Hurst parameter $H\in(1/3,1)$ and yields a continuous random dynamical system. This matters because it gives a solution theory for stochastic fluid equations driven by non-Markovian, rough signals at the same level of regularity as in the deterministic theory.

What carries the argument

The argument runs on two coupled structures. On the Lagrangian side, a new solution theory for rough differential equations with Osgood non-Lipschitz drifts produces a continuous flow of homeomorphisms $\Phi_t$ with explicit modulus estimates, built in finite $p$-variation spaces rather than H\"older spaces. On the Eulerian side, the unbounded rough drivers framework gives an intrinsic distributional meaning to rough continuity and transport equations; the load-bearing identity is the product formula and conservativity $A^*_{st}=-A_{st}$ of the rough driver, which follows from $\nabla\cdot\xi=0$ and the geometricity of $Z$, yielding duality, uniqueness, and the flow representation. The quasi-incompressibility estimate for $\Phi_t$, expressed through the Jacobian determinant formula, is what converts the Lagrangian flow into $L^p$ conservation for the vorticity.

What would settle it

Compute the Jacobian determinant of the rough flow $\Phi_t$ generated by $\mathrm{d}y_t=(K*\omega_0)(y_t)\,\mathrm{d}t+\xi(y_t)\,\mathrm{d}Z_t$ for a smooth vortex $\omega_0$ and a fixed geometric rough path $Z$. The paper's quasi-incompressibility formula says the determinant equals $\exp(\int_0^t \nabla\cdot b_s\,\mathrm{d}s)$ with no noise term; if any numerical rough-path integrator shows dependence of $\det D\Phi_t$ on the rough area $Z_{st}$, the quasi-incompressibility identity fails and the theory collapses.

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Extended reading notes

Core claim

The central claim is that the Yudovich theory for 2D Euler survives rough transport noise. For a geometric rough path $Z$ of finite $p$-variation with $p\in[2,3)$, divergence-free $\xi\in C^3_b$, and the Biot\textendash Savart kernel $K$, every $\omega_0\in L^1_x\cap L^\infty_x$ gives a unique global solution to the rough vorticity equation, renormalized and of the form $\omega_t(x)=\omega_0(\Phi_t^{-1}(x))$, with $\|\omega_t\|_{L^p_x}=\|\omega_0\|_{L^p_x}$ for all $p\in[1,\infty]$. The solution map depends continuously on initial data and on the rough path in both weak and strong topologies, and if $Z$ is a random geometric rough path cocycle the dynamics generate a continuous random dynamical system on bounded subsets of $L^1_x\cap L^\infty_x$. The same framework also yields a DiPerna\textendash Lions type well-posedness theory for linear rough continuity and transport equations and weak existence for initial vorticities in $L^1_x\cap L^p_x$.

Load-bearing premise

The whole construction requires the noise to be a geometric rough path, meaning a Stratonovich-type lift; if the same signal is given an It\^o-type lift, the flow is no longer volume-preserving and the uniqueness argument collapses.

Editorial extensions

If this is right

The rough 2D Euler vorticity equation is globally well-posed in $L^1_x\cap L^\infty_x$ for geometric rough paths, with unique renormalized solutions and flow representation.
The solution map generates a continuous random dynamical system when the noise is a fractional Brownian motion with Hurst parameter $H\in(1/3,1)$, providing a pathwise framework for Wong\textendash Zakai and large-deviation results.
Linear rough continuity and transport equations are well-posed under DiPerna\textendash Lions regularity conditions, with product, duality, and renormalization formulas available.
For initial vorticities in $L^1_x\cap L^p_x$, weak solutions to the rough 2D Euler equations exist for every $p\in[1,\infty)$, and they are renormalized when $p\ge 2$.
The nonlinear continuity equation framework covers general convolutional kernels satisfying Osgood and integrability assumptions, going beyond the Biot\textendash Savart case.

Reading between the lines

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

Beyond the paper, the geometric requirement indicates that the theory targets Stratonovich-type or Wong\textendash Zakai limits; It\^o noise is explicitly excluded and would need a different duality argument.
The flow-representation and kernel estimates likely transfer to the torus and to bounded domains with slip boundary conditions, and to vanishing-viscosity limits of rough 2D Navier\textendash Stokes, as the paper lists as future perspectives.
The Osgood-drift RDE theorem may serve as a building block for rough Vlasov\textendash Poisson equations, where uniqueness relies on similar logarithmic moduli of continuity.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear [Colleague],

Punchline: this is the real thing. It proves a rough-path analogue of Yudovich's theorem for 2D Euler on the whole space, and along the way builds a DiPerna–Lions theory for linear rough continuity equations. If I had to point a skeptic at one paper to convince them the rough-programme works for low-regularity fluid noise, this would be it.

What is actually new is exactly what the abstract promises. The main theorems are as advertised: RDEs with Osgood drifts (even non-geometric rough paths, via a time-reversal lemma that fills a real gap), linear RPDEs under DiPerna–Lions conditions, and the nonlinear framework that covers the 2D Euler vorticity equation. The overlap with [RT24] is real but partial—they do torus Euler and log-Lipschitz RDEs; this paper does full space, general kernels, linear theory, and weak existence for L1∩Lp. The authors are honest about the relation.

The proofs are long and technical, but the architecture is sound. The product formula plus duality is used in place of renormalization as the central tool, which is a smart move and works. The stress test I ran—tracing from the Osgood flow to quasi-incompressibility to linear uniqueness to the nonlinear comparison—closes without an internal gap. The only soft spot of note is a display in Prop 5.8 that temporarily drops a |rho0| weight; the needed weighted estimate is available, so it's a typo not a gap.

The main scope limitation is real: everything requires geometric rough paths (Stratonovich-type noise) and divergence-free ξ. That's not a flaw—it's explicit and standard for the setting—but it does mean Itô-type noise is out. Similarly, ξ ∈ C^3_b is stronger than in the Brownian SPDE literature, a price of rough path theory. The paper leans heavily on the authors' own unbounded rough drivers framework, but that prior work is established and independently cited; self-citation here is not a red flag.

Who this is for: anyone working on rough PDEs, stochastic transport, or 2D Euler with transport noise. It deserves a serious referee; the length will make refereeing painful, but the content justifies it. My recommendation: send to peer review. I'd accept it after the usual minor fixes.

Referee Report

1 major / 5 minor

Summary. The paper develops a rough-path theory for (nonlinear) continuity and transport equations with non-Lipschitz drifts. It first proves well-posedness and the existence of a homeomorphic flow for RDEs with Osgood drifts (Theorem 3.9), then establishes a DiPerna–Lions-type well-posedness theory for linear rough continuity and transport equations on R^d with geometric p-rough-path noise for p in [2,3), including a product formula and duality (Theorem 4.20). These linear results are combined with the RDE flow theory to obtain flow representations and quasi-incompressibility. For nonlinear continuity equations with convolutional kernels, the paper proves existence, uniqueness, stability, and flow representation (Theorems 1.4, 5.9), and then specializes to the 2D Euler equations in vorticity form, obtaining a rough Yudovich theorem (Theorem 1.5) and a random dynamical system when the noise is a geometric rough path cocycle such as fractional Brownian motion with H in (1/3,1). A weak-existence result for L^1 ∩ L^p vorticities is also given (Theorem 1.6).

Significance. If the results are correct, this is a substantial contribution to the rough-PDE literature. It extends classical DiPerna–Lions and Yudovich theory to geometric rough-path noise, handles the full space, and constructs a continuous random dynamical system for rough 2D Euler, going beyond the partial overlap in [RT24]. The proofs are detailed and largely self-contained, and the main theorems are stated with explicit regularity assumptions. The paper also provides useful new tools, notably the RDE flow theory for Osgood drifts on p-variation spaces and the product-formula-based uniqueness argument. The geometric rough-path and divergence-free assumptions are clearly stated and are standard for Stratonovich-type noise, so they represent a scope limitation rather than a hidden inconsistency. The main issue I found is a load-bearing gap in the uniqueness proof for the nonlinear equation, which appears repairable.

major comments (1)

[§5.2, Proposition 5.8 (estimate of I^2_s)] The displayed computation of I^2_s is incorrect as written. The first equality should read I^2_s = ∫ | ∫ [K_s(x,Φ^1_s(y)) - K_s(x,Φ^2_s(y))] ρ0(y) dy | |ρ2_s(x)| dx; the factor ρ0(y) in the inner integral is missing. Consequently, the next bound by ∫ h(|Φ^1_s(y)-Φ^2_s(y)|) dy is unjustified, and in fact cannot hold with the unweighted Lebesgue measure: for the Biot-Savart kernel the double integral with respect to Lebesgue measure is not controlled by the displayed expression, and the y-integral is not controlled by I_s unless the measure |ρ0|dy is carried through. This step is load-bearing because it produces the Osgood inequality I_t ≲ ∫ ... h(I_s) ds that yields uniqueness in Proposition 5.8 and hence in Theorems 1.4 and 1.5. The gap is repairable: one should write I^2_s ≤ ∫ |ρ0(y)| [∫ |K_s(x,Φ^1_s(y))-K_s(x,Φ^2_s(y))| |ρ2_s(x)| dx] dy, apply Assumption 5.1 with f = ρ2_s in the K̃ form to bound the inner integral by C h(|Φ^1_s(y)-Φ^2_s(y)|) ||ρ2_s||_{L1∩L∞}, and then use Jensen's inequality with the finite measure |ρ0|dy; the proof should be amended accordingly.

minor comments (5)

[§5.2, Proposition 5.8 (estimate of I^1_s)] In the estimate for I^1_s, the constant depending on ||ρ0||_{L1} is suppressed; please write it explicitly, since Jensen's inequality with the finite measure |ρ0|dy introduces such a constant and h is only subadditive/concave, not homogeneous.
[§4.2, Lemma 4.17, display (77)] In the final condition of (77), "\tilde R ≥ κ‖h‖_{L^1_x}" should read "\tilde R ≥ κ‖h‖_{L^1_t}" because h is a time-dependent function introduced in (72).
[Definition 5.5, item ii] The word "beloging" should be "belonging".
[§4.4, Corollary 4.36] The proof invokes Corollary 4.35, which is stated for the continuity equation; since the statement concerns the transport equation, either the transport analogue should be stated explicitly or the proof should mention that the same argument applies verbatim using Corollary 4.32 and condition (109).
[§1.1, Theorems 1.4 and 1.5] The notation "for all q∈{p}∪(1,∞)" is redundant when p>1 and slightly confusing when p=1; consider rewording to "for all q∈[1,∞)" in the strong-convergence statements.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Yudovich-type theorem is derived from explicitly stated geometric-rough-path and divergence-free assumptions via a self-contained proof chain.

full rationale

The paper's central result, Theorem 1.5, is a proof-based extension of deterministic Yudovich theory to rough 2D Euler, and the derivation chain is not circular. The rough-path flow for Osgood drifts is constructed in Section 3 from Davie-type solution concepts, a priori estimates (Lemmas 3.3 and 3.4), mollification, and time-reversal arguments; it does not presuppose the Euler well-posedness. The quasi-incompressibility estimate in Corollary 3.12 uses the explicit Jacobian determinant formula (40), which is valid because the approximating flows are diffeomorphisms and because div xi = 0 and the rough path is geometric; these are stated hypotheses, not conclusions of the theorem. The linear rough continuity/transport theory in Section 4 is built on the unbounded rough driver framework, but the needed a priori estimates, product formula, duality, uniqueness, and renormalizability are either proved in the paper or quoted from prior works whose assumptions do not include the target result. The proof of Theorem 4.20 is a self-contained doubling-of-variables and blow-up argument, and uniqueness in Theorem 4.29 follows by duality from that product formula, not from an imported uniqueness postulate. The nonlinear problem in Section 5 is solved by comparing two candidate flows: Proposition 5.8 reduces uniqueness to an Osgood inequality for the weighted L1 distance between the two flows, with the kernel Assumptions 5.1-5.3 providing exactly the bounds needed; no fitted parameter or data subset is involved. The cited prior works, including the authors' own contributions such as [HLN21] for the rough Gronwall lemma, are technical and independent; they are not invoked to forbid alternatives or to establish the main theorem by fiat. The manuscript itself flags limitations, e.g. Remark 4.28 on non-divergence-free xi and Remark 5.21 comparing with deterministic DiPerna-Majda theory, but these are scope restrictions, not circular reductions. A displayed estimate in Proposition 5.8 is compressed and drops a weight in one intermediate display, but the required weighted Osgood bounds are available from Assumption 5.1, so this is a presentation issue rather than a circular step. Overall, the claim is self-contained under its stated assumptions and no prediction reduces by construction to an input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper is a pure mathematics work: no free parameters are fitted to data, and no new physical entities are postulated. The central claims rest on standard rough path theory, DiPerna-Lions theory, and the unbounded rough drivers framework, plus modeling assumptions on the noise (geometric rough paths) and the coefficients (DiPerna-Lions/Osgood regularity, divergence-free C^3_b vector fields). The main novel content is theorems built on these premises, all proved in the text.

assumptions (5)

domain assumption Geometric rough path lifts exist for the noise signals, e.g., fractional Brownian motion with Hurst parameter H∈(1/3,1).
The whole RPDE theory uses Z∈C^p_g for p∈[2,3); the authors build the RDS on such cocycles (Section 5.3, citing [BRS17]). Non-geometric lifts such as Itô Brownian are excluded.
domain assumption The drift b satisfies DiPerna-Lions regularity: b/(1+|x|)∈L1_t L1_x + L1_t L∞_x, b∈L1_t W^{1,1}_{loc}, ∇·b∈L1_t L∞_x.
Theorem 1.2 and Theorem 4.29 take these as hypotheses; they are exactly the conditions needed for the product formula (Theorem 4.20) and the commutator estimates (Proposition 4.25).
domain assumption The vector fields ξ are C^3_b and divergence-free (∇·ξ=0).
C^3_b is used for the RDE flow estimates in Lemmas 3.3-3.4 and the unbounded rough driver bounds (Lemma 4.11); ∇·ξ=0 gives conservativity (59) and the Jacobian formula (40).
domain assumption The kernel K satisfies Assumptions 5.1-5.3, and the Biot-Savart kernel in R^2 satisfies them via its Fourier multiplier properties (Theorem 1.4 proof).
These assumptions give the Osgood continuity, divergence bound and W^{1,1}_{loc} regularity of the drift u=K*ω used in Proposition 5.8.
standard math Standard rough path and unbounded rough drivers toolkit: sewing lemma (Lemma 2.7), rough Grönwall (Lemma A.5), and smoothing operators on the scales F_{l,R} uniform in R (Lemma 4.4 and Appendix C).
These are imported from [FV10, BG17, DGHT19a] and are the technical backbone of all a priori estimates; the paper proves the smoothing construction in Appendix C.

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Pith. "Pith review of On the well-posedness of (nonlinear) rough continuity equations." pith.science (2026). https://pith.science/paper/O3QEOYPA

@misc{pith2026250204982,
  author       = {Pith},
  title        = {Pith review of: On the well-posedness of (nonlinear) rough continuity equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O3QEOYPA}},
  note         = {Machine review of arXiv:2502.04982}
}

abstract

Motivated by applications to fluid dynamics, we study rough differential equations (RDEs) and rough partial differential equations (RPDEs) with non-Lipschitz drifts. We prove well-posedness and existence of a flow for RDEs with Osgood drifts, as well as well-posedness of weak $L^p$-valued solutions to linear rough continuity and transport equations on $\mathbb{R}^d$ under DiPerna--Lions regularity conditions; a combination of the two then yields flow representation formula for linear RPDEs. We apply these results to obtain existence, uniqueness and continuous dependence for $L^1\cap L^\infty$-valued solutions to a general class of nonlinear continuity equations. In particular, our framework covers the $2$D Euler equations in vorticity form with rough transport noise, providing a rough analogue of Yudovich's theorem. As a consequence, we construct an associated continuous random dynamical system, when the driving noise is a fractional Brownian motion with Hurst parameter $H \in (1/3,1)$. We further prove weak existence of solutions for initial vorticities in $L^1\cap L^p$, for any $p\in [1,\infty)$.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [10]

Lagrangian stability for a system of non-local continuity equations under Osgood condition

[IS23] Marco Inversi and Giorgio Stefani. Lagrangian stabi lity for a system of non-local continuity equations under Osgood condition. arXiv:2301.11822,

work page Pith review arXiv

[2] [1956]

The Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck systems in stochastic electromagnetic fields: local well-posedness

[BP22] Jacob Bedrossian and Stavros Papathanasiou. The Vla sov-Poisson and Vlasov-Poisson-Fokker-Planck systems in stochastic electromagnetic ﬁelds: local well-p osedness. arXiv:2211.03336,

work page Pith review arXiv

[3] [1963]

[Yud95] V. I. Yudovich. Uniqueness theorem for the basic non stationary problem in the dynamics of an ideal incompressible ﬂuid. Math. Res. Lett. , 2(1):27–38, 1995

work page 1995

[4] [1992]

Flexibil ity of two-dimensional euler ﬂows with integrable vorticity

[BCK24] Elia Bru` e, Maria Colombo, and Anuj Kumar. Flexibil ity of two-dimensional euler ﬂows with integrable vorticity. arXiv:2408.07934,

arXiv

[5] [1997]

[Loe06] Gr´ egoire Loeper

Reprint of t he 1990 original. [Loe06] Gr´ egoire Loeper. Uniqueness of the solution to the Vlasov–Poisson system with bounded density.Journal de math´ ematiques pures et appliqu´ ees, 86(1):68–79,

work page 1990

[6] [1998]

Medvedev

[Med02] A.V. Medvedev. On a concave diﬀerentiable majorant of a modulus of continuity. Real Anal. Exchange , 27(1):123–129, 2001/02. [M´ em14] Etienne M´ emin. Fluid ﬂow dynamics under location uncertainty. Geophysical & Astrophysical Fluid Dynamics, 108(2):119–146,

work page 2001

[7] [2010]

Navier-Stokes Equations with Navier Boundary Conditions and Stochastic Lie Transport: Well-Posedness and Inviscid Limit

Theory and applications. [Goo23] Daniel Goodair. Navier–Stokes equations with Navi er boundary conditions and stochastic Lie transport: Well-posedness and inviscid limit. arXiv:2308.04290; to appear on J. Diﬀerential Equations ,

work page Pith review arXiv

[8] [2011]

[Saint-Flour Probability Summer School]

Lectures from the 40th Probab ility Summer School held in Saint-Flour, 2010, ´Ecole d’´Et´ e de Probabilit´ es de Saint-Flour. [Saint-Flour Probability Summer School]. [FNS20] Peter K. Friz, Torstein Nilssen, and Wilhelm Stanna t. Existence, uniqueness and stability of semi-linear rough partial diﬀerential equations. Journal of Diﬀerential Equations , 268...

work page 2010