Existence and smoothness of periodic solutions to the three-dimensional Navier–Stokes equations

Let $\nu > 0$ be a positive real number. Let $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth vector field that is divergence-free, meaning $\nabla \cdot u_0 = 0$ everywhere, and is $1$-periodic in each coordinate, meaning $u_0(x + e_i) = u_0(x)$ for all $x \in \mathbb{R}^3$ and each standard basis vector $e_i$. Prove that there exist smooth functions $v : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}^3$ and $p : \mathbb{R}^3 \times [0,\infty) \to \mathbb{R}$ such that:

• $v$ and $p$ are $1$-periodic in each spatial coordinate for all $t \geq 0$;
• $v$ is divergence-free for all $t \geq 0$, meaning $\nabla \cdot v(\cdot, t) = 0$;
• $v$ satisfies the initial condition $v(x,0) = u_0(x)$ for all $x \in \mathbb{R}^3$;
• $v$ and $p$ satisfy the Navier–Stokes equations with zero external force:

for all $x \in \mathbb{R}^3$ and all $t \geq 0$;

• both $v$ and $p$ are infinitely differentiable ($C^\infty$) on $\mathbb{R}^3 \times [0,\infty)$.