Navier–Stokes: a new proof claim, and a dispute over credit

Here $u(x, t) \in \mathbb{R}^3$ is the velocity of the fluid at the point $x$ and time $t$, $p$ is the pressure, $\nu > 0$ the viscosity and $f$ an external force. The left side is the acceleration of a fluid particle: $\partial_t u$ is the change at a fixed point, and $(u \cdot \nabla) u$ the change from being carried along by the flow. This term is quadratic in $u$, and it is the source of all the difficulty.

The condition $\nabla \cdot u = 0$ says the fluid is incompressible. The pressure is then not a separate unknown: taking the divergence of the equation gives

For a smooth solution, multiplying the equation by $u$ and integrating over space gives the energy balance

Viscosity removes energy, and the force can add it. In 1934 Leray used this bound to construct weak solutions that exist for all time, but nobody knows whether they stay smooth or are unique.

Without a force, if $u$ solves the equation then so does the rescaled flow

In three dimensions its energy is $\|u_\lambda(t)\|_{L^2}^2 = \lambda^{-1} \|u(\lambda^2 t)\|_{L^2}^2$. Zooming in to small scales (large $\lambda$) makes the energy small, so the one quantity we control says almost nothing about the tiny scales where a singularity would form. This is called supercriticality. In two dimensions the energy does not change under this scaling, and smooth solutions are known to exist for all time.

The official problem is solved by proving any one of four statements: (A) smooth solutions exist for all time on $\mathbb{R}^3$ for every smooth, rapidly decaying initial velocity, with $f = 0$; (B) the same on the periodic box $\mathbb{T}^3$; (C) on $\mathbb{R}^3$ there are a smooth initial velocity and a smooth, decaying force $f$ for which no smooth solution with finite energy exists for all time; (D) the same breakdown on $\mathbb{T}^3$.

Alternatives C and D allow a force. That is why a construction with a smooth force can address the Millennium Problem while saying nothing about whether an unforced fluid can blow up.