Navier–Stokes finite-time blow-up with smooth initial conditions

Determine whether solutions to the Navier–Stokes equations of fluid mechanics with smooth initial conditions develop finite-time blow-up (form singularities in finite time) or remain smooth for all time. Establish the conditions under which global regularity fails or holds for these smooth initial data, to clarify the long-time behavior of such flows.

Background

In discussing the Cauchy–Kovalevskaya theorem, the paper notes that while analytic data yield local existence and uniqueness, solutions can blow up in finite time, limiting global existence. The authors highlight the Navier–Stokes equations as a paradigmatic case where the question of blow-up versus global regularity for smooth initial conditions remains unresolved.

This open problem serves as a key example of how even classical PDEs central to physics may lack global solutions, emphasizing that local well-posedness does not guarantee long-time existence, and that such phenomena impact the empirical and conceptual adequacy of modeling assumptions.

References

For an interesting example where it is a famous open problem whether blow-up happens (with smooth initial conditions), consider the Navier--Stokes equations of fluid mechanics~\citep{constantin}.

— Hyperdeterminism? Spacetime 'Analyzed'  (2502.11004 - Chen et al., 16 Feb 2025) in Footnote in Section 3 (Analytic field theories), following Equation (1)

A significant open question when $d\geq3$ is whether the maximum lifespan $T$ can be finite, a scenario known as finite-time blow-up which is characterized by certain norms of the solution becoming infinite.

— Cascade mechanisms for Navier-Stokes blow-up  (2609.26790 - Cheskidov et al., 22 Sep 2026) in Section 1, Introduction

The non-density statement is proved for the zero-data slice. It does not classify every nonzero-data slice at or above the critical exponent.

— Distribution of Singular Data Generated by Compact Forced Navier-Stokes Blowup  (2609.10262 - Cao et al., 9 Sep 2026) in Section 6, immediately after the proof of Theorem 1.1

The extended-data projection reaches every smooth initial velocity because the force is free to depend on that velocity. This does not determine the density, dimension or stability of singular initial states under a single prescribed force.

— Distribution of Singular Data Generated by Compact Forced Navier-Stokes Blowup  (2609.10262 - Cao et al., 9 Sep 2026) in Section 7, Discussion