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arxiv: 2604.26547 · v1 · submitted 2026-04-29 · 🧮 math.PR

Non-uniqueness for the stochastic incompressible Euler equations with a passive tracer

Ashish Bawalia , Zdzis{\l}aw Brze\'zniak , Manil T. Mohan This is my paper

Pith reviewed 2026-05-07 11:29 UTC · model grok-4.3

classification 🧮 math.PR
keywords stochastic Euler equationspassive tracerpathwise non-uniquenessweak solutionsBaire category methodrandom coefficientsideal MHD
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The pith

The stochastic incompressible Euler equations with a passive tracer admit pathwise non-uniqueness of weak solutions in every spatial dimension at least two.

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

The paper establishes pathwise non-uniqueness for the stochastic incompressible Euler equations with a passive tracer under transport noise and linear multiplicative Stratonovich noise. The authors first apply classical transformations to recast the stochastic system as a deterministic Euler equation whose coefficients are random but pathwise fixed. They then invoke the Baire category method to produce infinitely many global weak solutions to this random-coefficient PDE, each satisfying an arbitrary positive bounded continuous time-dependent energy profile, in any dimension two or higher. Reversing the transformations yields the desired pathwise non-uniqueness for the original stochastic equations. The same argument supplies non-uniqueness for the three-dimensional stochastic ideal MHD system and extends earlier deterministic results that were limited to the constant energy profile equal to one.

Core claim

Via classical transformations, the stochastic incompressible Euler equations with a passive tracer are converted into PDEs with random coefficients. The Baire category method is then applied to construct infinitely many global-in-time weak solutions to these random PDEs in spatial dimensions greater than or equal to two, for arbitrary positive bounded continuous time-dependent energy profiles. Applying the inverse transformations produces pathwise non-uniqueness for the original stochastic partial differential equations. This extends earlier non-uniqueness results from constant energy profiles to general time-dependent ones and includes an application to the stochastic ideal MHD system.

What carries the argument

The Baire category method applied to the space of weak solutions of the random-coefficient incompressible Euler equations, together with the classical transformations that convert Stratonovich transport and multiplicative noise into random coefficients.

Load-bearing premise

The classical transformations preserve the class of weak solutions and the Baire category construction applies to the space of solutions for arbitrary positive bounded continuous time-dependent energy profiles in the random PDEs.

What would settle it

A proof that any two weak solutions of the stochastic Euler system must coincide pathwise for some admissible initial data and noise, or a numerical experiment in which all solution trajectories converge to the same path for a fixed noise realization, would falsify the non-uniqueness claim.

read the original abstract

In this work we investigate the phenomenon of pathwise non-uniqueness for the stochastic incompressible Euler equations with a passive tracer on the whole Euclidean space. The stochastic perturbations are interpreted as a transport noise and a linear multiplicative noise in the Stratonovich sense. In both cases, via classical transformations, we convert the SPDEs into PDEs with random coefficients. Using the Baire category method developed by De Lellis and Sz\'ekelyhidi Jr., we then construct infinitely many global-in-time weak solutions to the random PDEs in any spatial dimension greater than or equal to two. By applying the inverse transformations, we obtain pathwise non-uniqueness for the original SPDEs. Finally, we present an application of our result to the three-dimensional stochastic ideal MHD equations. This study can be regarded as a stochastic counterpart of Bronzi et al.~[\textit{Commun.~Math.~Sci.},~2015]. In particular, our non-uniqueness result in the random setting extends theirs from a constant energy profile equal to one to arbitrary positive, bounded, and continuous time-dependent profiles, originally established in two dimensions via the convex integration technique.

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.

Referee Report

2 major / 2 minor

Summary. The paper proves pathwise non-uniqueness for the stochastic incompressible Euler equations with a passive tracer on R^d (d≥2), with Stratonovich transport noise or linear multiplicative noise. Classical transformations convert the SPDEs to PDEs with random coefficients; the Baire category method of De Lellis–Székelyhidi is then applied to produce infinitely many global weak solutions for arbitrary positive bounded continuous time-dependent energy profiles. Inverse transformations recover pathwise non-uniqueness for the original SPDEs. An application to the 3D stochastic ideal MHD equations is given, extending the deterministic result of Bronzi et al. (2015) from constant energy to general profiles.

Significance. If the details hold, the result supplies a stochastic counterpart to the deterministic non-uniqueness theory for the Euler equations. It shows that the Baire-category construction remains viable under random coefficients and for general energy profiles, thereby broadening the scope of flexibility methods in stochastic fluid dynamics. The pathwise character of the non-uniqueness and the MHD application are of particular interest to the field.

major comments (2)
  1. [Transformation and random PDE sections (likely §§2–4)] The manuscript asserts that the classical transformations preserve the class of weak solutions and that the Baire-category construction applies directly to the random-coefficient PDEs for arbitrary positive bounded continuous energy profiles. Explicit verification of the a-priori bounds on the random coefficients (uniform in the probability space) is needed to guarantee that the iterative scheme converges pathwise; this step is load-bearing for the transfer of non-uniqueness.
  2. [Baire-category construction for random PDEs] The extension from constant energy (Bronzi et al.) to arbitrary continuous profiles is claimed to follow from the known flexibility of the Baire method, yet the random-coefficient setting may introduce additional integrability requirements on the noise terms. A precise statement of the admissible class of energy profiles and the corresponding solution space should be given to confirm the construction works without further restrictions.
minor comments (2)
  1. [Abstract and main theorem statement] The abstract states the result holds 'in any spatial dimension greater than or equal to two'; the proof should clarify whether the constants or the iteration scheme depend on dimension in a way that affects uniformity.
  2. [Introduction] The reference to Bronzi et al. (Commun. Math. Sci. 2015) should be given in full bibliographic form at first citation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance, and recommendation of minor revision. The comments correctly identify two points where additional explicit verification and clarification will strengthen the manuscript. We address each major comment below and will incorporate the requested details in the revised version.

read point-by-point responses

  1. Referee: [Transformation and random PDE sections (likely §§2–4)] The manuscript asserts that the classical transformations preserve the class of weak solutions and that the Baire-category construction applies directly to the random-coefficient PDEs for arbitrary positive bounded continuous energy profiles. Explicit verification of the a-priori bounds on the random coefficients (uniform in the probability space) is needed to guarantee that the iterative scheme converges pathwise; this step is load-bearing for the transfer of non-uniqueness.

    Authors: We agree that an explicit verification of the uniform bounds is necessary for a fully rigorous transfer of the pathwise non-uniqueness. In the revised manuscript we will add a dedicated lemma (placed after the transformation statements in §2 or §3) that derives the a-priori bounds on the random coefficients directly from the assumed regularity of the driving Brownian motion. The bounds are shown to hold uniformly in the probability space on a set of full measure, which is sufficient for the Baire-category iteration to converge pathwise. This step uses only the standard properties of Stratonovich transport and linear multiplicative noise already employed in the paper and does not alter any of the main statements. revision: yes

  2. Referee: [Baire-category construction for random PDEs] The extension from constant energy (Bronzi et al.) to arbitrary continuous profiles is claimed to follow from the known flexibility of the Baire method, yet the random-coefficient setting may introduce additional integrability requirements on the noise terms. A precise statement of the admissible class of energy profiles and the corresponding solution space should be given to confirm the construction works without further restrictions.

    Authors: We will add a short subsection (or expanded remark) at the beginning of the Baire-category construction section that precisely states the admissible class: energy profiles e(t) that are positive, bounded, and continuous on [0,∞). The solution space is the standard one for weak solutions of the random incompressible Euler system with prescribed energy profile, namely divergence-free vector fields in L^∞_t L²_x ∩ L²_t Ḣ¹_x whose kinetic energy equals e(t) almost surely. We explicitly verify that the random coefficients arising from the noise satisfy the same integrability and boundedness requirements used in the deterministic Baire-category arguments of De Lellis–Székelyhidi and Bronzi et al.; no additional restrictions are imposed by the randomness. This makes the extension to arbitrary continuous profiles fully rigorous while preserving the pathwise character of the construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by applying standard classical transformations (Stratonovich-to-Itô or flow-based) that map the original SPDEs to random-coefficient PDEs, followed by an invocation of the external Baire-category construction of De Lellis–Székelyhidi Jr. on those PDEs for arbitrary positive bounded continuous energy profiles, and finally inverting the maps. This chain relies on independently established external techniques and does not reduce any central claim to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation. The extension from the deterministic Bronzi et al. result is achieved by verifying the requisite a-priori bounds on the random coefficients, keeping the argument self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical tools from deterministic convex integration/Baire category theory and stochastic analysis; no new free parameters, invented entities, or ad-hoc axioms are introduced beyond domain assumptions on the applicability of known methods.

axioms (2)
  • domain assumption Classical transformations convert the SPDEs to random PDEs while preserving weak solutions and the energy profile class.
    Invoked to reduce the stochastic non-uniqueness question to a deterministic one with random coefficients.
  • domain assumption The Baire category method applies to the space of weak solutions of the random PDEs with arbitrary positive bounded continuous time-dependent energy profiles.
    Central to constructing infinitely many solutions in d >= 2.

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