Global regularity or blow-up for 3D incompressible Navier–Stokes

Establish whether smooth solutions of the three-dimensional incompressible Navier–Stokes equations (\(\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u,\; \nabla \cdot u = 0\)) posed on \(\mathbb{R}^3\) with smooth divergence-free initial data remain smooth for all time, or construct an explicit finite-time blow-up example; that is, resolve the Clay Millennium Problem by determining the global regularity or singularity formation for the 3D incompressible Navier–Stokes equations.

Background

The paper studies the three-dimensional incompressible Navier–Stokes equations and introduces a frequency-domain framework that unifies weak, mild, and strong formulations. As motivation, the authors highlight that the global behavior of solutions in three dimensions is a major unresolved problem, recognized as a Clay Millennium Problem.

Within the historical overview, the authors reference Fefferman’s statement, which specifies the challenge: either prove global smoothness for smooth initial data on $\mathbb{R}^3$ or produce an explicit blow-up example. The paper’s contribution is to unify solution frameworks locally, but it does not claim to settle the global regularity question, which it explicitly notes remains unresolved.