Well-Posedness Beyond Known Noisy Euler Regimes

Definitions and mathematical statement

Let $\mathbb T^2$ denote the two-dimensional torus, and let $\omega$ denote the scalar vorticity in the stochastic two-dimensional incompressible Euler model intended by the supplied source. Write $L^p(\mathbb T^2)$ for the Lebesgue spaces with respect to area measure and $L^\infty(\mathbb T^2)$ for the essentially bounded functions. The record does not specify the stochastic equation, the noise, or the range of exponents in the existence results it describes.

Source-qualified research question. For that intended stochastic Euler model, determine whether one can extend existence to initial vorticities beyond the reportedly treated $L^p$ classes, extend uniqueness from the reportedly treated $L^\infty$ class to appropriate finite $L^p$ classes, and prove a suitable noncollapse property of an associated stochastic flow. The original formulation also asks about further regularization results; this additional target must be retained, although it is absent from the cleaned formulation.

Unresolved qualifications. This is a research programme rather than a fully specified conjecture. The stochastic forcing, stochastic integration convention, initial-data assumptions, treated exponent range, intended larger existence class, solution concept, time interval, and meaning of uniqueness are not supplied. The record also does not define the stochastic flow, noncollapse, or the requested regularization property. Consequently, no definite quantified existence, uniqueness, noncollapse, or regularization assertion can be extracted without additional source information. The descriptions of known existence and uniqueness results are supplied background claims, not independently verified theorems in this draft.

Short English statement

The supplied question asks whether results for noisy two-dimensional Euler vorticity on the torus extend to rougher initial data, whether uniqueness extends from $L^\infty$ to finite $L^p$, and whether the stochastic flow avoids collapse. The original wording also asks about regularization. The noise, solution classes, exponent ranges, and desired flow properties are unspecified, so a precise conjecture requires further source clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30003507$, source problem code catalogue entry, title: Well-Posedness Beyond Known Noisy Euler Regimes. The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Geophysical Fluid Dynamics (2018), Oberwolfach Reports, pp. 1421--1462, DOI 10.4171/owr/2017/23. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2017/23. The citation's parenthetical year is 2018, while the DOI contains 2017 and the registry records proposed year 2017; their relationship is not explained in the supplied record.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the extracted question as an unspecified request for results beyond partial stochastic Euler theorems, with no precise claim identified as settled. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked directly and the clean statement restored definitions. These are supplied verification claims; this draft has not visited or independently checked the report, and the supplied record still omits essential mathematical specifications.