Determining Wavenumbers Near Three-Dimensional Turbulence

Definitions and mathematical statement

Consider three-dimensional Navier--Stokes dynamics. The supplied record uses $d$ for an intermittency dimension, $\Lambda$ for a determining wavenumber, $\langle\Lambda\rangle$ for its time average, and $\kappa_d$ for the Kolmogorov dissipation wavenumber associated with $d$. These descriptions identify the intended quantities but do not supply their mathematical definitions.

The proposed question is whether, for intermittency dimensions $d$ sufficiently close to $3$ in an unspecified sense, one has the bound \[ \langle\Lambda\rangle\leq C\kappa_d, \] where $C>0$ is a constant with unspecified permitted dependence. The supplied statement attributes this proposed bound to predictions of Kolmogorov turbulence theory.

This is a schematic formulation requiring further hypotheses. The record does not specify the spatial domain, boundary conditions, forcing, viscosity assumptions, solution class, definition of intermittency dimension, construction of the determining wavenumber, normalization of $\kappa_d$, or time-averaging operation. It also does not specify the admissible range of $d$, whether $d=3$ is included, or the parameters over which the bound must hold uniformly. Consequently, no fully quantified inequality can be extracted without additional source information. The background describes the target as extending an "anomalous diffusion bound" to dimensions near three, but does not define that bound or explain its precise relationship to the displayed inequality.

Short English statement

The question asks whether the average determining wavenumber for three-dimensional Navier--Stokes dynamics is bounded by a constant multiple of the Kolmogorov dissipation wavenumber when the intermittency dimension is near $3$. Definitions, hypotheses, the meaning of "near," and the allowed dependence of the constant remain unspecified.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002941$, exact source problem code catalogue entry; title: "Determining Wavenumbers Near Three-Dimensional Turbulence." No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Mathematical Aspects of Hydrodynamics (2016), Oberwolfach Reports, pp. 2125--2187, DOI 10.4171/owr/2015/37. Original OWR report: https://doi.org/10.4171/owr/2015/37. The citation's 2016 date and the record's 2015 year are retained without reconciliation.
• Supplied full report text: https://ems.press/content/serial-article-files/46590.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, states that the report identifies extension of the anomalous diffusion bound to dimensions near three as open, and reports finding no later definitive resolution. 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 registry labels the statement corrected_verified and says that notation and publication prose were repaired without changing its substance. These are supplied editorial claims, not independent mathematical verification of this draft.