Global Existence and Smoothness of Navier–Stokes Solutions on ℝ³

Let $\nu > 0$ be a real number (the viscosity), and let $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth vector field satisfying the following conditions: $u_0$ is divergence-free, meaning $\nabla \cdot u_0 = 0$ everywhere; and $u_0$ and all of its derivatives decay faster than any polynomial at infinity, meaning for every non-negative integer $m$ and every real number $K$, there exists a constant $C$ such that $\|\partial^m u_0(x)\| \leq C/(1 + \|x\|)^K$ for all $x \in \mathbb{R}^3$. Prove that there exist a velocity field $v : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}^3$ and a pressure field $p : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}$ such that:

for all $x \in \mathbb{R}^3$ and all $t \geq 0$; the velocity remains divergence-free, $\nabla \cdot v(\cdot, t) = 0$, for all $t \geq 0$; the initial condition $v(x,0) = u_0(x)$ holds for all $x \in \mathbb{R}^3$; both $v$ and $p$ are smooth ($C^\infty$) on $\mathbb{R}^3 \times [0,\infty)$; for each $t \geq 0$, the kinetic energy $\int_{\mathbb{R}^3} \|v(x,t)\|^2 \, dx$ is finite; and there exists a uniform bound $E$ such that $\int_{\mathbb{R}^3} \|v(x,t)\|^2 \, dx < E$ for all $t \geq 0$.