Navier–Stokes global regularity conjecture

Let $u_0: \mathbb{R}^3 \to \mathbb{R}^3$ be a divergence-free vector field in the Schwartz class. A smooth vector field $u: [0,+\infty) \times \mathbb{R}^3 \to \mathbb{R}^3$ and a smooth function $p: \mathbb{R}^3 \to \mathbb{R}$ should exist such that

$$\partial_t u + (u \cdot \nabla)u = \nu \Delta u - \nabla p, \qquad \nabla \cdot u = 0, \qquad u(0,\cdot)=u_0,$$

for every $\nu>0$, together with $u \in L^\infty_t L^2_x([0,T] \times \mathbb{R}^3)$ for every $0<T<\infty$.

Navier–Stokes global regularity conjecture. For every $\nu>0$ and every such initial data $u_0$, such a global smooth finite-energy solution exists.

This is the three-dimensional homogeneous global regularity problem for the Navier–Stokes equations. The paper uses finite-time blowup for an averaged equation to identify a supercriticality barrier; the conjecture itself remains unresolved.