Navier–Stokes existence and smoothness

Statement

Let $\nu > 0$. Let $u_0 \colon \R^3 \to \R^3$ be a smooth vector field with divergence $\nabla \cdot u_0 = 0$. Assume that for every multi-index $\alpha$ and every $K > 0$ there is a constant $C_{\alpha K}$ with $\abs{\partial^\alpha u_0(x)} \le C_{\alpha K} (1 + \abs{x})^{-K}$ for all $x \in \R^3$. Then there exist smooth functions $u \colon \R^3 \times [0, \infty) \to \R^3$ and $p \colon \R^3 \times [0, \infty) \to \R$ such that:

1. $u$ and $p$ satisfy the Navier–Stokes equations $\partial_t u + (u \cdot \nabla) u = \nu \Delta u - \nabla p$ and $\nabla \cdot u = 0$ on $\R^3 \times [0, \infty)$, with no external force.
2. $u(x, 0) = u_0(x)$ for all $x \in \R^3$.
3. There is a constant $E$ with $\int_{\R^3} \abs{u(x, t)}^2 \, dx \le E$ for all $t \ge 0$.