Finite‑time blowup for 3D incompressible Euler in free space

Determine whether smooth solutions of the three-dimensional incompressible Euler equations on R^3 (without boundary) can develop a finite-time singularity starting from smooth initial data.

Background

The paper reviews singularity formation results for Euler-type models and highlights that the Hou–Luo boundary scenario has been rigorously established in settings with a solid boundary. However, despite extensive numerical and analytical efforts, the question of whether blowup can occur in free space R3 without boundaries remains unresolved.

This open problem is central to mathematical fluid dynamics and motivates the numerical investigations in simplified models (the 1D Hou–Luo model and the 2D Boussinesq equations) reported in this paper.

References

Whether finite-time blowup can happen in the free space $R3$ still remains open.

— Novel Self-similar Finite-time Blowups with Singular Profiles of the 1D Hou-Luo Model and the 2D Boussinesq Equations: A Numerical Investigation  (2604.01868 - Chen et al., 2 Apr 2026) in Introduction (Section 1)