Breakdown scenario for the Navier–Stokes equations on \(\mathbb{R}^3\)

Let $\nu > 0$ be a fixed viscosity constant. A vector field $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ satisfies the decaying initial velocity condition if it is smooth ($C^\infty$), divergence-free ($\nabla \cdot u_0 = 0$), and all derivatives of $u_0$ decay faster than any polynomial at infinity: for every derivative order $m \in \mathbb{N}$ and every decay rate $K \in \mathbb{R}$, there exists a constant $C \in \mathbb{R}$ such that $\|\partial^m u_0(x)\| \leq C/(1 + \|x\|)^K$ for all $x \in \mathbb{R}^3$. A function $f : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}^3$ satisfies the decaying force condition if it is smooth on $\mathbb{R}^3 \times [0,\infty)$ and all derivatives of $f$ decay faster than any polynomial in both space and time: for every $m \in \mathbb{N}$ and $K \in \mathbb{R}$, there exists $C \in \mathbb{R}$ such that $\|\partial^m_{x,t} f(x,t)\| \leq C/(1 + \|x\| + t)^K$ for all $x \in \mathbb{R}^3$ and $t \geq 0$. A pair of functions $(v,p)$ with $v : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}^3$ and $p : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}$ is a global smooth solution with finite energy for initial data $u_0$ and force $f$ if: (i) $v$ and $p$ are smooth on $\mathbb{R}^3 \times [0,\infty)$; (ii) $v(\cdot, t)$ is square-integrable for each $t \geq 0$ and the kinetic energy $\int_{\mathbb{R}^3} \|v(x,t)\|^2 \, dx$ remains uniformly bounded for all $t \geq 0$; (iii) the incompressibility condition $\nabla \cdot v(\cdot, t) = 0$ holds for all $t \geq 0$; (iv) the initial condition $v(x,0) = u_0(x)$ holds for all $x \in \mathbb{R}^3$; and (v) the Navier–Stokes equation $\partial_t v + (v \cdot \nabla)v = \nu \Delta v - \nabla p + f$ holds for all $x \in \mathbb{R}^3$ and $t \geq 0$. Does there exist an initial velocity field $u_0$ satisfying the decaying initial velocity condition and a force $f$ satisfying the decaying force condition such that no pair $(v,p)$ is a global smooth solution with finite energy for this data?