Can smooth three-dimensional incompressible Navier–Stokes flow develop a finite-time singularity?

details For viscosity $\nu>0$, consider the three-dimensional incompressible Navier–Stokes equations

on $\mathbb R^3$ or the periodic three-torus, with smooth divergence-free initial velocity $u_0$ of finite energy. Prove that every such initial datum generates a global smooth solution, or construct admissible smooth initial data for which a singularity forms in finite time. A numerical simulation of apparent blow-up, or a proof for a restricted symmetry class, is not a resolution of the full problem.