The residual trick

Choose the flow first. The force is whatever the equation leaves over. The whole proof is making the leftover smooth.

For any divergence-free $u$ and any $p$, define

$$\mathcal{R}(u,p) := \partial_t u + (u\cdot\nabla)u - \nu\Delta u + \nabla p .$$

Set $f := \mathcal{R}(u,p)$. Then $(u,p,f)$ solves the forced Navier–Stokes equations, by definition. There is nothing to prove about existence. What must be proved is that a $u$ exists which becomes unbounded at time one while $\mathcal{R}(u,p)$, together with every space-time derivative of it, extends smoothly through time one. The paper calls $\mathcal{R}$ the momentum residual, and the entire construction is an exercise in cancellation inside it.

Why the obvious flows fail

Take any self-similar profile that concentrates, say $u(x,t) = \tau^{-1/2}U(x/\tau^{1/2})$ with $\tau=1-t$. Each term of the residual scales like $\tau^{-3/2}$. The residual of a generic profile therefore diverges like $\tau^{-3/2}$ at the singular point, and the force needed to sustain the flow is as singular as the flow. Nothing is gained.

The way out is to make the terms cancel. Individual terms may diverge; the sum must not. In the inner core the construction imposes the leading momentum balance exactly, so the residual there is small by design. In the exterior the flow is an exact solution of a heat equation, so the residual is zero. The trouble is in between, in the annulus where the concentrating core is joined to the smooth exterior. There the residual of the background is unbounded, of the same order as the terms it is built from, and no choice of axisymmetric profile removes it.

Momentum that the fluid supplies to itself

The paper's central move is to let the fluid supply the missing force. Add to the background $u_B$ a divergence-free increment $w$ with pressure increment $\pi$. The residual changes by an exact identity:

$$\mathcal{R}(u_B + w,\ p_B + \pi) = \mathcal{R}(u_B,p_B) + \mathcal{L}_{u_B}(w,\pi) + \nabla\cdot(w\otimes w),$$ $$\mathcal{L}_{u_B}(w,\pi) = \partial_t w + (u_B\cdot\nabla)w + (w\cdot\nabla)u_B - \nu\Delta w + \nabla\pi .$$

The linear term is how the increment evolves in the background. The quadratic term is the divergence of a momentum flux, and its angular average does not vanish even when $w$ does. In English: oscillations that average to zero still transport momentum, because the product of two zero-mean quantities has a mean. This is the Reynolds stress of turbulence theory, used here on purpose. Choose $w$ so that $\nabla\cdot\langle w\otimes w\rangle$ cancels the singular part of $\mathcal{R}(u_B,p_B)$, and the leading obstruction is gone.

Two conditions make this possible. The increments $w$ must be approximate solutions of the linearized equation, $\mathcal{L}_{u_B}(w,\pi)\approx 0$, otherwise the linear term reintroduces a singular residual. And their averaged flux must be able to point in the direction the background needs. The pulses page explains how both are arranged.

Cancellation to every order

Cancelling the leading term leaves smaller singular terms: interactions between pulses, the errors of the linearization, cutoffs. The construction removes these in a fixed cycle, recomputing the full residual after every operation so that nothing is dropped. Each cycle improves the rate at which the residual vanishes by a fixed amount, $\sigma_{j+1}=\sigma_j+\tfrac1{10}$. The corrections are then summed with cutoffs that shrink toward the singular point. The outcome is a residual that is flat: every Cartesian space-time derivative is $O(q^N)$ for every $N$ as the concentration scale $q\downarrow 0$. A flat function extends by zero smoothly. That is where the smooth force comes from.

Note what this does not require. The force need not be small, and it need not be simple. It is $C^\infty$ and compactly supported, and that is all the theorem asks. It is not real analytic, and it cannot be: Constantin, Ignatova and Vicol prove that a force analytic in space near the singular point forces regularity for flows of this type. The construction lives in the difference between smooth and analytic.

Ancestry

The trick is not new. Córdoba and Martínez-Zoroa built finite-time singularities for forced three-dimensional Euler in 2023 by amplifying increasingly concentrated vortex layers while keeping the force in a fixed Hölder class, and with Zheng extended the method to hypodissipative Navier–Stokes in 2024. Alpöge and Buckmaster pushed the method to smooth forcing for the porous medium, Boussinesq and Euler equations in August 2026. Tao's 2016 blow-up for an averaged Navier–Stokes equation is a different move: it modifies the nonlinearity rather than adding a force. The map draws the lineage.

What the 2026 construction adds is viscosity at full strength. Earlier forced constructions had none, or a small fractional power of it. Here the radial Reynolds number of the core is $O(1)$: viscous diffusion competes with inflow at every stage, and the pulses that supply the stress are damped by the same viscosity that they must outrun. The scales page makes those competitions quantitative.

Next: scales and exponents.