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 R3\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.

References

Despite its classical form, the global behaviour of solutions to eq:navier_stokes_classic in three space dimensions remains unresolved and constitutes one of the Clay Millennium Problems .

eq:navier_stokes_classic:

{∂tu+(u ⁣⋅ ⁣∇)u  =  − ∇p+νΔu,∇ ⁣⋅u  =  0,(x,t)∈Ω×(0,∞),u(x,0)=u0(x),\begin{cases} \partial_t u + (u \!\cdot\! \nabla)u \;=\; -\,\nabla p + \nu \Delta u, \\[2pt] \nabla\!\cdot u \;=\; 0, \end{cases} \qquad (x,t)\in\Omega\times(0,\infty), \qquad u(x,0)=u_0(x),

— Unified Frequency-Domain Reconstruction and Boundary Adaptation for Incompressible Navier-Stokes Equations  (2507.13356 - Nikitaeva, 16 Jun 2025) in Abstract; Section 1 (Introduction)

Whether an axisymmetric finite energy solution of the unforced 3D Navier-Stokes equations, arising from smooth and decaying initial data, can develop a finite time singularity remains an open problem.

— Regularity for axisymmetric Navier-Stokes with an Euler length  (2609.20762 - Constantin et al., 17 Sep 2026) in Section 1, Introduction

With swirl, the global regularity question is open and only conditional results are available.

— Regularity of asymptotically axisymmetric solutions to the 3D Navier-Stokes equations with analytic forcing  (2609.20803 - Constantin et al., 17 Sep 2026) in Section 1, subsection “Prior results on axisymmetric Navier-Stokes”

The global regularity problem for smooth finite-energy data remains open.

— Flexibility for the Three-Dimensional Navier-Stokes Equations via Moving Hill Vortices  (2608.20068 - Nguyen et al., 20 Aug 2026) in Section 1, subsection “Background and related work”

From a mathematical standpoint, the question of global existence and uniqueness of smooth solutions remains one of the major open problems in analysis.

— Energy Decay Induced by Localized Coercivity in the 3D Navier--Stokes Equations  (2609.03164 - Cissé, 2 Sep 2026) in Section 1, Introduction

Global well-posedness for the three-dimensional Navier-Stokes equations remains a major open problem.

— Global well-posedness of the 3D Navier-Stokes equations with helical $L^1$ vorticity  (2609.05193 - Gancedo et al., 4 Sep 2026) in Section 1, Introduction