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on Breakdown of periodic Navier–Stokes solutions in three dimensions

The statement

Let $\nu > 0$ be a positive real number. Does there exist a smooth, divergence-free, 1-periodic initial velocity field $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ and a smooth external force $f : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}^3$ that is 1-periodic in space and whose derivatives decay faster than any polynomial in time, such that there is no smooth solution pair $(v,p)$ to the three-dimensional periodic Navier–Stokes equations with viscosity $\nu$ , initial data $u_0$ , and forcing $f$ ? Specifically, no pair $(v,p)$ exists where $v : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}^3$ and $p : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}$ are both smooth, 1-periodic in space for all $t \geq 0$ , satisfy the Navier–Stokes equation

\frac{\partial v}{\partial t} + (v \cdot \nabla)v = \nu \Delta v - \nabla p + f

for all $x \in \mathbb{R}^3$ and $t \geq 0$ , maintain $\nabla \cdot v = 0$ for all $x$ and $t \geq 0$ , and satisfy the initial condition $v(x,0) = u_0(x)$ for all $x \in \mathbb{R}^3$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.