Navier–Stokes existence and smoothness

Let $\nu>0$. The unknowns are the velocity field $u:\mathbb{R}^3\times[0,\infty)\to\mathbb{R}^3$ and the pressure $p:\mathbb{R}^3\times[0,\infty)\to\mathbb{R}$. Given an external force $f$, the incompressible Navier-Stokes equations are

$$\frac{\partial u}{\partial t}+(u\cdot\nabla)u =\nu\Delta u-\nabla p+f, \qquad \nabla\cdot u=0,$$

with initial condition

$$u(x,0)=u_0(x).$$

Here $\Delta=\sum_{j=1}^3\partial^2/\partial x_j^2$, and the initial velocity must be divergence-free:

$$\nabla\cdot u_0=0.$$

Admissible data and solutions

In the whole-space formulation, $u_0$ is smooth and rapidly decreasing: for every multi-index $\alpha$ and every $K\geq 0$ there is a constant $C_{\alpha,K}$ such that

$$\left|\partial_x^\alpha u_0(x)\right| \leq C_{\alpha,K}(1+|x|)^{-K}.$$

When a force is allowed, it is smooth and rapidly decreasing in space and time: for every $\alpha,m,K$,

$$\left|\partial_x^\alpha\partial_t^m f(x,t)\right| \leq C_{\alpha,m,K}(1+|x|+t)^{-K}.$$

A physically reasonable whole-space solution is globally smooth,

$$u,p\in C^\infty\!\left(\mathbb{R}^3\times[0,\infty)\right),$$

and has uniformly bounded kinetic energy:

$$\sup_{t\geq 0}\int_{\mathbb{R}^3}|u(x,t)|^2\,dx<\infty.$$

In the periodic formulation, space is the three-torus $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$. The fields are $1$-periodic in each coordinate,

$$u(x+e_j,t)=u(x,t),\qquad p(x+e_j,t)=p(x,t),\qquad f(x+e_j,t)=f(x,t)$$

for $j=1,2,3$. The initial velocity is smooth, periodic, and divergence-free. When a force is allowed, all its space-time derivatives decrease faster than any power of time:

$$\left|\partial_x^\alpha\partial_t^m f(x,t)\right| \leq C_{\alpha,m,K}(1+t)^{-K}.$$

The problem

The Clay problem is solved by proving at least one of the following four statements.

(A) Global regularity on $\mathbb{R}^3$. For every smooth, rapidly decreasing, divergence-free $u_0$, with $f\equiv 0$, there exist global smooth $u$ and $p$ satisfying the equations and the bounded-energy condition.

(B) Global regularity on $\mathbb{T}^3$. For every smooth, periodic, divergence-free $u_0$, with $f\equiv 0$, there exist global smooth periodic $u$ and $p$ satisfying the equations.

(C) Breakdown on $\mathbb{R}^3$. There exist smooth, rapidly decreasing, divergence-free initial data $u_0$ and a smooth rapidly decreasing force $f$ for which no global smooth bounded-energy solution exists.

(D) Breakdown on $\mathbb{T}^3$. There exist smooth periodic, divergence-free initial data $u_0$ and a smooth periodic force $f$ with the stated time decay for which no global smooth periodic solution exists.

Thus the question is whether arbitrary smooth three-dimensional incompressible data remain smooth for all time, or whether some admissible data develop a finite-time singularity.