BMO Blowup Criteria for Incompressible Flow Potentials

Definitions and mathematical statement

For an incompressible velocity field $u$, a three-dimensional vector potential is a vector field $A$ satisfying $u=\nabla\times A$. In two dimensions, a stream function is a scalar field $\psi$ satisfying $u=\nabla^\perp\psi$, where we use the sign convention $\nabla^\perp\psi=(-\partial_2\psi,\partial_1\psi)$. Existence and global representation by these potentials depend on the spatial domain and any boundary or topological conditions, which the supplied record does not specify. Moreover, $A$ is not uniquely determined: replacing $A$ by $A+\nabla\phi$ preserves its curl whenever these expressions are defined. The source supplies no gauge or normalization.

For reference, on ${\mathbb{R}}^d$, the bounded mean oscillation seminorm of a locally integrable scalar or vector field $f$ is \[ [f]_{\mathrm{BMO}({\mathbb{R}}^d)} :=\sup_B\frac{1}{|B|}\int_B|f(x)-f_B|\,{\,\mathrm{d}} x, \qquad f_B:=\frac{1}{|B|}\int_B f(x)\,{\,\mathrm{d}} x, \] where the supremum is over all Euclidean balls $B\subset{\mathbb{R}}^d$, and the absolute value is the Euclidean norm in the vector case. This seminorm defines a norm after quotienting by constant fields. This definition explains the notation in the whole-space setting; the source does not specify that setting, the intended domain-dependent version of BMO, or whether its notation denotes a seminorm or a normalized norm.

Source question, with unresolved hypotheses retained. For every incompressible Navier--Stokes flow in the intended, unspecified solution class, does a finite blowup time $t_*$ imply the corresponding assertion \[ \begin{aligned} d=3:\quad &\lim_{t\uparrow t_*}\|A(\cdot,t)\|_{\mathrm{BMO}}=+\infty,\\ d=2:\quad &\lim_{t\uparrow t_*}\|\psi(\cdot,t)\|_{\mathrm{BMO}}=+\infty? \end{aligned} \] Here $t$ approaches $t_*$ through times at which the flow and its chosen potential are defined. The source does not define blowup or specify the solution regularity, initial data, forcing, viscosity assumptions, spatial domain, boundary conditions, potential gauge, or BMO convention. Consequently, these implications are a statement template requiring those hypotheses before they become a fully specified mathematical proposition. In particular, the quantifier over potential representatives is unresolved. The displayed limits retain the source's full divergence assertion; they have not been weakened to unboundedness or divergence of a limsup. The intended scope of the two-dimensional clause also requires clarification, since the hypotheses governing the possibility of finite-time blowup are absent.

Short English statement

The question asks whether finite-time blowup of an incompressible Navier--Stokes flow forces the BMO size of its vector potential in three dimensions, or its stream function in two dimensions, to tend to infinity. The domain, solution class, meaning of blowup, and potential normalization must be specified before this becomes a precise universal claim.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002942$, source problem code catalogue entry, titled BMO Blowup Criteria for Incompressible Flow Potentials, in the Oberwolfach Reports Open Problems collection. 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. https://doi.org/10.4171/owr/2015/37. The background attributes the question to mathematical workshop open-problem discussions documented in Oberwolfach Reports. The supplied citation says 2016, while the DOI and proposed-year metadata contain 2015; these details are retained without reconciliation.
• Supplied literature reference, labeled "Ohkitani 2017": https://eprints.whiterose.ac.uk/id/eprint/111101/. No fuller bibliographic description or specific supporting result is supplied.
• Supplied literature reference, labeled "original OWR report": https://doi.org/10.4171/owr/2015/37.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the general BMO blowup criterion for the three-dimensional vector potential or two-dimensional stream function as unproved. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.