Finite-time blow-up for Euler and Navier–Stokes equations

Determine whether smooth solutions to the incompressible Euler equations and to the incompressible Navier–Stokes equations admit finite-time singularities (blow-up) on a Riemannian manifold.

Background

The paper reviews universality phenomena in dynamical systems, with particular attention to Euler and Navier–Stokes flows. It recalls that the existence of finite-time blow-up for smooth solutions remains a central unresolved issue in PDEs and dynamics. This motivates Tao’s universality program and subsequent work embedding complex dynamics and computation into fluid flows.

The authors emphasize that computational universality might offer a route to constructing singular behaviors, yet the fundamental question of whether smooth Euler or Navier–Stokes solutions can blow up is still unresolved, and remains a benchmark open problem for the field.

References

One of the most important open problems in dynamical systems and partial differential equations is determining whether the Euler equations (and their viscous version, the Navier-Stokes equations) admit solutions that blow-up in finite time.

— Universality in computable dynamical systems: Old and new  (2507.10725 - González-Prieto et al., 14 Jul 2025) in Introduction

It remains unclear how to choose suitable constraints and ansatz for such a PINN formulation, and how to overcome the resulting optimization challenges.

— Stable Singularity of the Euler Equations on $\mathbb{R}^3$  (2609.10867 - Ganeshram et al., 9 Sep 2026) in Section 1, Introduction

Questions about the global-in-time well-posedness of equations such as the incompressible Navier--Stokes and Euler equations remain unsolved.

— Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing  (2609.16470 - Alpöge et al., 15 Sep 2026) in Section 1, Introduction

Whether blow-up can occur from data in the marginal space $H\alpha$, originally asked in , remains open.

— Cascade mechanisms for Navier-Stokes blow-up  (2609.26790 - Cheskidov et al., 22 Sep 2026) in Remark (iv) following Corollary \ref{cor:inviscid-forward-blowup}, Section 1, subsection “Inverse energy cascade and instantaneous blow-up”