Submit progress

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

The statement

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:

\frac{\partial v}{\partial t} + (v \cdot \nabla) v = \nu \Delta v - \nabla p

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

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

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.