Long-Time Navier–Stokes Flow in Exterior Domains

Definitions and mathematical statement

An exterior domain in two dimensions is an unbounded connected open set $\Omega\subset{\mathbb{R}}^2$ whose complement is compact. No boundary regularity or obstacle geometry is specified in the supplied record. Write $u(t,x)\in{\mathbb{R}}^2$ for the fluid velocity at time $t>0$ and position $x\in\Omega$.

Source-level problem. Determine the large-time behavior, as $t\to\infty$, of solutions of the Navier--Stokes initial-value problem in a two-dimensional exterior domain, including the associated Oseen evolution. Here the Oseen evolution refers to the corresponding linear incompressible viscous evolution with a constant transport term; the record does not specify its precise operator or its relation to the nonlinear flow under consideration.

This is a research question rather than a fully quantified mathematical assertion. The supplied record does not specify the governing equations in their intended normalization, forcing, boundary conditions, behavior at spatial infinity, admissible initial data, solution class, or viscosity. It also does not specify whether the requested large-time behavior means decay, convergence to a stationary flow, an asymptotic profile, or another property, nor the topology or rate in which such a conclusion should hold. These choices must be recovered from the source before a precise theorem or conjecture can be formulated.

The supplied dated literature assessment describes the question as open even for a sufficiently small nonzero Oseen parameter. Its normalization, the meaning of sufficiently small, and the accompanying hypotheses are not supplied. This qualification is retained as a report of the supplied assessment, not as an independently verified mathematical or literature claim.

Short English statement

Describe the long-time behavior of two-dimensional Navier--Stokes flow outside a bounded obstacle region, together with the associated Oseen evolution. The record leaves the data, boundary conditions, solution class, and intended asymptotic conclusion unspecified, so it does not determine a unique precise conjecture. The supplied open-status assessment, including its qualification about a sufficiently small nonzero Oseen parameter, requires independent review.

Sources

• Local UnsolvedMath registry, permanent ID $30002293$, source code catalogue entry, title: Long-Time Navier--Stokes Flow in Exterior Domains. The supplied record belongs to the Oberwolfach Reports Open Problems collection and records the proposed year as 2013; no individual proposer is identified.
• Geophysical Fluid Dynamics (2013), Oberwolfach Reports, pp. 521--577, DOI 10.4171/owr/2013/10. Supplied report citation and literature source: https://doi.org/10.4171/owr/2013/10. No problem-specific page or passage is supplied.
• Hishida, Decay estimates of the Oseen flow. Supplied literature source: https://eprints.lib.hokudai.ac.jp/repo/huscap/all/68077/tech153.pdf. The record provides no further bibliographic details or supporting passage.
• Literature assessment dated 2026-08-21: the record labels the problem open and describes the two-dimensional exterior-domain Oseen/Navier--Stokes large-time problem as still explicitly open, including for a sufficiently small nonzero Oseen parameter. This is supplied dated curation derived from the OWR material, not independently verified here; time-sensitive status and cited claims require independent verification.
• The supplied statement-status label is corrected_verified. Its accompanying verification note reports that the mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. This is supplied provenance, not verification performed for this draft.