Breakdown of periodic Navier–Stokes solutions in three dimensions

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

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 $x \in \mathbb{R}^3$.