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arxiv: 2605.16963 · v1 · pith:ZBPSPEMDnew · submitted 2026-05-16 · 🧮 math.PR · math.AP

Stochastic Euler Equations with Pseudo-differential Noise: Continuous and Discontinuous Perturbations in Compressible and Incompressible Flows

Kenneth. H. Karlsen , Hao Tang , Feng-Yu Wang This is my paper

Pith reviewed 2026-05-19 18:54 UTC · model grok-4.3

classification 🧮 math.PR math.AP
keywords stochastic Euler equationspseudo-differential operatorsMarcus noiseMakino transforminvariant probability measuresShirikyan open problemcompressible and incompressible flowsmixed multiplicative noise
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The pith

Mixed continuous and discontinuous pseudo-differential noise yields local classical solutions to stochastic Euler equations and invariant measures for the damped incompressible case.

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

This paper establishes a local existence theory for classical solutions to stochastic Euler equations in both compressible and incompressible forms, driven by mixed multiplicative noise that includes continuous Stratonovich or Ito terms and discontinuous Marcus terms with pseudo-differential amplitudes. The authors introduce a generalized transformation that extends the Makino approach to a wide variety of pressure laws for the compressible barotropic equations. For the incompressible damped equations, they create an abstract criterion for invariant probability measures that works with mismatched topologies and use it to prove existence in all dimensions greater than or equal to two on both the torus and Euclidean space, thereby providing the first positive resolution to Shirikyan's open problem in a strengthened form. A reader would care about this work because it supplies mathematical tools for studying fluid dynamics under noise that features both smooth variations and sudden jumps, which better models certain physical phenomena.

Core claim

We study stochastic Euler equations in both compressible and incompressible regimes, on the whole space and on the torus, driven by genuinely mixed multiplicative noise: continuous Stratonovich/Itô components and a discontinuous Marcus component. The Stratonovich and Marcus noise amplitudes are pseudo-differential operators. We develop a local-in-time theory of classical solutions for both regimes, with new tools to control interactions between jump discontinuities and nonlocal operators. We establish a transformation principle for the compressible barotropic case that generalizes the Makino transform beyond the polytropic setting to a broad class of pressure laws. For the incompressible dam

What carries the argument

The generalized Makino-type transformation for non-polytropic pressure laws and the abstract Krylov-Bogoliubov-type criterion for invariant measures in systems with mismatched topologies under singular noise.

If this is right

  • Local classical solutions exist for compressible stochastic Euler equations with the specified class of pressure laws and pseudo-differential mixed noise.
  • Invariant probability measures exist for the damped stochastic incompressible Euler equations on T^d and R^d for d ≥ 2.
  • The new analytical tools handle the interaction of jump discontinuities with nonlocal pseudo-differential operators.
  • The framework applies to both periodic and non-periodic domains for the fluid equations.

Where Pith is reading between the lines

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

  • This approach may extend to global-in-time results or ergodic properties if additional dissipation is present.
  • Similar mixed noise models could be studied in other hyperbolic or parabolic fluid systems beyond Euler equations.
  • The resolution suggests that discontinuous noise components play a key role in establishing long-time statistical behavior where continuous noise alone may not suffice.

Load-bearing premise

The pressure laws belong to a broad class that permits a generalized Makino-type transformation while preserving the structure needed for classical solutions under the pseudo-differential noise.

What would settle it

Observation of finite-time blowup in a classical solution for a pressure law inside the claimed class under the mixed pseudo-differential noise, or non-existence of an invariant measure for the damped Euler system on the two-dimensional torus with this noise type.

Figures

Figures reproduced from arXiv: 2605.16963 by Feng-Yu Wang, Hao Tang, Kenneth. H. Karlsen.

Figure 1
Figure 1. Figure 1: Any smooth 𝑃 (𝜌) such that 𝑃 ′ (𝜌) > 0, 𝑃 (𝜌) = 2𝜌 5∕3 on [0, 1], 𝑃 (𝜌) = 3𝜌 on [2, 3] and 𝑃 (𝜌) = 𝜌 3∕2 on [4, ∞) can be included in A(5.2). 42 [PITH_FULL_IMAGE:figures/full_fig_p042_1.png] view at source ↗
read the original abstract

We study stochastic Euler equations in both compressible and incompressible regimes, on the whole space and on the torus, driven by genuinely mixed multiplicative noise: continuous Stratonovich/It\^o components and a discontinuous Marcus component. The Stratonovich and Marcus noise amplitudes are pseudo-differential operators. We develop a local-in-time theory of classical solutions for both regimes. The presence of pseudo-differential Marcus noise necessitates new analytical tools, which we develop to control the delicate interaction between jump discontinuities and nonlocal operators. We establish a transformation principle for the compressible barotropic case that generalizes the Makino transform beyond the polytropic setting and covers a broad class of physically relevant pressure laws outside the standard polytropic $\gamma$-law. This class includes (piecewise-defined) Chaplygin-type laws, the pressure law for white dwarf stars, etc. Most of these equations of state have not been analyzed in the stochastic compressible setting. For the incompressible damped case, we obtain further long-time behavior. We develop a criterion for the existence of invariant probability measures for a general Markov process accommodating mismatched topologies, extending the classical Krylov--Bogoliubov approach. This abstract criterion allows us to study invariant probability measures for a broad class of singular stochastic evolution systems in Hilbert spaces. Notably, this application gives what appears to be the first positive answer to Shirikyan's open problem on the damped Euler equations on $\mathbb T^2$ under genuinely mixed multiplicative noise. Furthermore, our framework goes beyond the original formulation of the problem: it resolves a substantially strengthened version in every dimension $d\ge 2$, on both $\mathbb T^d$ and $\mathbb R^d$.

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

1 major / 2 minor

Summary. The manuscript develops a local-in-time existence theory for classical solutions to stochastic Euler equations in compressible and incompressible settings, driven by mixed multiplicative noise consisting of continuous Stratonovich/Itô components and discontinuous Marcus components, where the noise amplitudes are pseudo-differential operators. For the compressible barotropic case, it establishes a generalized Makino-type transformation applicable to a broad class of pressure laws including Chaplygin-type and white-dwarf equations of state. For the incompressible damped case, it introduces an abstract criterion for invariant probability measures extending Krylov-Bogoliubov methods to accommodate mismatched topologies, and applies it to resolve a strengthened version of Shirikyan's open problem on the damped Euler equations under genuinely mixed multiplicative noise in dimensions d ≥ 2 on both the torus and Euclidean space.

Significance. If the central results hold, this work would be significant as it appears to provide the first positive resolution to Shirikyan's open problem for the damped Euler equations on T^2 under mixed multiplicative noise, and extends it substantially to higher dimensions and domains. The development of new tools to control interactions between jump discontinuities and nonlocal pseudo-differential operators, along with the abstract invariant measure criterion for singular stochastic evolution systems, could advance the field of stochastic fluid dynamics. The generalization of the Makino transform to non-polytropic pressure laws in the stochastic setting is also noteworthy, as these equations of state have not been previously analyzed in this context.

major comments (1)
  1. [Compressible barotropic case and generalized Makino transform] The assertion that the generalized Makino-type transformation preserves the structure needed for classical solutions under pseudo-differential Stratonovich and Marcus noise requires detailed verification. Since the transformation is pointwise nonlinear, post-application of non-local operators introduces commutator terms that may affect regularity. The energy estimates used for local or Fourier-multiplier noise may not suffice here. Please provide the specific commutator estimates or regularity controls in the relevant section to confirm closure of the local existence theorem for the broad class of pressure laws.
minor comments (2)
  1. [Notation and definitions] Ensure consistent use of notation for the pseudo-differential operators across sections, particularly distinguishing between Stratonovich and Marcus interpretations.
  2. [Abstract and introduction] The abstract mentions 'new analytical tools' for controlling jump discontinuities and nonlocal operators; a brief outline of these tools in the introduction would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the single major comment below and have incorporated additional details into the revised version to strengthen the presentation of the generalized Makino transform.

read point-by-point responses

  1. Referee: [Compressible barotropic case and generalized Makino transform] The assertion that the generalized Makino-type transformation preserves the structure needed for classical solutions under pseudo-differential Stratonovich and Marcus noise requires detailed verification. Since the transformation is pointwise nonlinear, post-application of non-local operators introduces commutator terms that may affect regularity. The energy estimates used for local or Fourier-multiplier noise may not suffice here. Please provide the specific commutator estimates or regularity controls in the relevant section to confirm closure of the local existence theorem for the broad class of pressure laws.

    Authors: We thank the referee for this observation. The generalized Makino-type transformation is introduced in Section 3 of the manuscript, where it is applied pointwise to the compressible barotropic system before the action of the pseudo-differential noise operators. To control the resulting commutators, we employ paradifferential calculus together with symbol estimates for the pseudo-differential amplitudes; these yield bounds that absorb the regularity loss into the existing Sobolev energy estimates, which are already adapted to the mixed Stratonovich/Marcus structure. The local existence theorem then closes for the indicated class of pressure laws. Nevertheless, we agree that the commutator calculations can be made more explicit. In the revision we add a dedicated lemma (Lemma 3.4) that isolates the commutator terms arising from the nonlinear transformation and the nonlocal operators, together with the precise regularity controls used to close the a priori estimates. This addition does not change the statements or proofs but improves readability and addresses the referee's request directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper develops new tools to control interactions between jump discontinuities and nonlocal pseudo-differential operators, then asserts a generalized Makino-type transformation that converts the barotropic system into symmetric hyperbolic form for a stated class of pressure laws (Chaplygin, white-dwarf, etc.). The long-time invariant-measure result follows from an abstract extension of the Krylov-Bogoliubov criterion that accommodates mismatched topologies. Neither step reduces by the paper's own equations to a quantity defined only in terms of itself, a fitted parameter renamed as prediction, or a load-bearing self-citation whose content is unverified outside the present work. The application to Shirikyan's problem is presented as a consequence of the developed framework rather than an input that is assumed.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results from stochastic analysis and PDE theory plus new technical estimates for pseudo-differential Marcus noise; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption The pressure laws satisfy structural conditions allowing a generalized Makino-type transformation that preserves classical solution regularity under the given noise.
    Invoked to extend the compressible barotropic theory beyond polytropic gases.
  • domain assumption The pseudo-differential operators satisfy symbol conditions that permit control of the interaction between jump discontinuities and nonlocal terms.
    Required for the new analytical tools mentioned for the Marcus component.

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