Anomalous Dissipation in Stochastic Navier–Stokes Equations

Definitions and mathematical statement

Let $\mathbb T^3$ denote the three-dimensional torus, with periodic spatial boundary conditions, and let $\nu>0$ denote viscosity. For a random velocity field $u^\nu(t,x)$, write \[ \|u^\nu(t)\|_{L^2_x}^2=\int_{\mathbb T^3}|u^\nu(t,x)|^2\,{\,\mathrm{d}} x. \] Here $\mathbb E_\nu$ denotes expectation on the probability space supporting the solution at viscosity $\nu$; the probability spaces may depend on $\nu$. Stationarity means that the law of the velocity process is invariant under time translations, so that this expected squared norm is independent of $t$, whenever defined. A martingale solution is understood in the probabilistically weak sense, with a probability space and driving stochastic process supplied as part of the solution; its precise admissibility requirements depend on the stochastic equation under consideration.

Existence question, conditional on unspecified source hypotheses. For the three-dimensional stochastic Navier--Stokes equations on $\mathbb T^3$ with the forcing and solution class intended by the cited sources, does there exist a family of stationary martingale solutions $(u^\nu)_{0<\nu<\nu_0}$, for some $\nu_0>0$, satisfying the property called weak anomalous dissipation in the supplied record, \[ \lim_{\nu\downarrow0}\nu\,\mathbb E_\nu\|u^\nu(t)\|_{L^2_x}^2=0? \] The displayed family formulation makes explicit the viscosity dependence implicit in the source statement. The record does not specify the stochastic forcing, its covariance or nondegeneracy, whether its law is fixed as $\nu$ varies, spatial mean constraints, or the regularity and energy requirements imposed on martingale solutions. These are unresolved hypotheses, so this is a conditional transcription rather than a fully specified stochastic PDE problem. The supplied literature assessment also mentions solution sequences; it is unclear whether a family for all sufficiently small viscosities is required or whether a sequence $\nu_j\downarrow0$ with the corresponding limit suffices. The stated condition concerns the velocity $L^2$ norm; no replacement by a gradient norm is intended.

Short English statement

The question asks whether stationary martingale solutions of the three-dimensional stochastic Navier--Stokes equations on a periodic domain can be chosen as viscosity tends to zero so that viscosity times their expected squared $L^2$ velocity norm tends to zero. The forcing assumptions and exact solution class are missing, and the intended choice between a full viscosity family and a vanishing-viscosity sequence needs clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004148$, source problem code catalogue entry, title "Anomalous Dissipation in Stochastic Navier--Stokes Equations," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2019 and supplies no individual proposer attribution.
• Supplied report citation: Nonlinear Hyperbolic Problems: Modeling, Analysis, and Numerics (2020), Oberwolfach Reports, pp. 1419--1497, DOI 10.4171/owr/2019/24. https://doi.org/10.4171/owr/2019/24. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Bedrossian--Coti Zelati--Punshon-Smith--Weber, supplied literature reference: https://arxiv.org/abs/1803.09695.
• Oberwolfach problem report, supplied literature reference: https://ems.press/content/serial-article-files/46803.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that the four-fifths-law result is conditional on weak anomalous dissipation, while existence of suitable Navier--Stokes solutions or sequences satisfying that property is explicitly left open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.