Stochastic Regularization of Two-Dimensional Euler Flows

Definitions and mathematical statement

Let $\mathbb T^2$ denote the flat two-dimensional torus, with periodic spatial conditions. The deterministic incompressible Euler equations for velocity $u(t,x)\in{\mathbb{R}}^2$ and pressure $p(t,x)\in{\mathbb{R}}$ are \[ \partial_t u+(u\cdot\nabla)u+\nabla p=0, \qquad \nabla\cdot u=0, \qquad u(0,\cdot)=u_0. \] The initial velocity must satisfy the incompressibility constraint. The scalar vorticity is $\omega=\partial_1u_2-\partial_2u_1$; for sufficiently regular deterministic solutions it satisfies \[ \partial_t\omega+u\cdot\nabla\omega=0. \] For a spatial vector field $\sigma$, the transport operator on scalar vorticity is $L_\sigma\omega=\sigma\cdot\nabla\omega$. This defines the directional transport operation, but does not specify a stochastic Euler model.

Source question. Can a suitable transport-type stochastic perturbation of two-dimensional incompressible Euler flow on $\mathbb T^2$ restore uniqueness, prevent singular behavior, or otherwise improve well-posedness? The intended comparison is with the corresponding deterministic evolution for the relevant initial data.

Unresolved qualifications. The supplied record specifies neither an initial-data or vorticity class nor a solution concept, time interval, stochastic integration convention, covariance, spatial regularity, or admissible class of noise coefficients. It also leaves the meaning of uniqueness, singular behavior, and improved well-posedness unspecified. Consequently, the existential qualification "suitable" has no specified mathematical range, and the record does not determine a single fully quantified conjecture. The displayed deterministic equations provide context only; they do not select an unprovided stochastic equation or regularity hypothesis. A precise theorem-sized target requires these choices and a defined deterministic comparison.

Short English statement

The question asks whether suitable random transport can improve the behavior of incompressible Euler flow on the two-dimensional torus, especially uniqueness or avoidance of singular behavior. The supplied record leaves the noise, initial-data class, solution concept, and desired improvement unspecified, so this remains a qualified research question requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $30003506$, title "Stochastic Regularization of Two-Dimensional Euler Flows," source problem code catalogue entry. The collection is "Oberwolfach Reports Open Problems." No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Geophysical Fluid Dynamics (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/23, pp. 1421--1462. https://doi.org/10.4171/owr/2017/23. The background attributes the question to workshop open-problem discussions documented in Oberwolfach Reports. The supplied citation's 2018 title date and its 2017 DOI component and proposed-year field require bibliographic reconciliation.
• Supplied literature reference: Brzezniak, Flandoli, and Maurelli. https://doi.org/10.1007/s00205-015-0957-8.
• Supplied literature reference: Coghi and Maurelli, Kraichnan noise. https://doi.org/10.1088/1361-6544/ae621b.
• Supplied literature reference labeled "original OWR report." https://doi.org/10.4171/owr/2017/23.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and reports existence, uniqueness, and stronger regularity results for transport and Kraichnan noise in several vorticity classes, while describing a general regularization theorem for two-dimensional Euler as unresolved. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified. Its accompanying explanation says that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This supplied editorial label does not establish mathematical correctness or independent source verification here.