Circulation Budgets, Exponent Windows and a Dyadic Laboratory for Finite-Time Singularities of the Navier–Stokes Equations with and Without Forcing

We study structural constraints on finite-time singularities of the three-dimensional incompressible Navier–Stokes equations, motivated by the recently announced construction of blow-up with a smooth, compactly supported force, which addresses alternatives (C) and (D) of the Clay Millennium formulation. First, we prove an angular-momentum budget for the azimuthal mean of an arbitrary smooth solution: the supremum of the mean circulation ru‾θ is bounded by its initial value, by the time integral of the azimuthal moment of the force and by the time integral of a flux produced by non-axisymmetric fluctuations. Consequently, a Type II singularity whose mean swirl satisfies |u‾θ |≳(T-t)-1/2-h at distance ≃(T-t)1/2 from the axis, as in the announced leading profile, is incompatible with axial symmetry near the singular time for any force satisfying the Clay decay conditions, and it requires fluctuations that deliver a divergent angular-momentum flux; a scaling argument further indicates that these fluctuations remain comparable to the mean swirl. Second, we derive necessary exponent windows for singular structures of amplitude (T-t)-a concentrated on the length scale (T-t)b, and we show that the exponents of the announced profile lie in the range in which the energy equality persists through the singular time, so that the singularity carries no anomalous dissipation provided that the full solution obeys the same scaling. Third, we use the dyadic model of Katz–Pavlović and Cheskidov as a laboratory: we compute the inviscid blow-up time and the front-propagation law, prove that a prescribed self-similar front requires a force that is time-integrable but unbounded unless its residual cancels at leading order in every shell, and report numerical evidence on the interval of dissipation exponents between the known blow-up and regularity thresholds. Finally, we assess the published objections to the forced construction and list the requirements that any construction for the unforced problem must satisfy. No claim is made concerning the unforced problem.