For the incompressible Euler equations on boundary-free three-dimensional space, partial_t u+(u·nabla)u=-nabla p, nabla· u=0 on R^3×[0,∞), with smooth divergence-free initial data u(0,x)=u_0(x) of finite kinetic energy int_R^3|u_0(x)|^2,dx<∞, determine whether the corresponding solution remains smooth for every t≥ 0. Equivalently, determine whether there exists such initial data and a finite time T<∞ at which the solution develops a singularity and cannot be continued smoothly beyond T.

3D Euler regularity and singularity problem

For the incompressible Euler equations on boundary-free three-dimensional space, $\partial_t u+(u\cdot\nabla)u=-\nabla p$ , $\nabla\cdot u=0$ on $\mathbb{R}^3\times[0,\infty)$ , with smooth divergence-free initial data $u(0,x)=u_0(x)$ of finite kinetic energy $\int_{\mathbb{R}^3}|u_0(x)|^2\,dx<\infty$ , determine whether the corresponding solution remains smooth for every $t\ge 0$ . Equivalently, determine whether there exists such initial data and a finite time $T<\infty$ at which the solution develops a singularity and cannot be continued smoothly beyond $T$ .

Progress summary

Open

A technical clarification in August 2026 addressed an objection, but it did not settle whether smooth three-dimensional fluid flow can always remain regular.

The problem asks whether smooth, finite-energy solutions of the boundary-free $3\mathrm{D}$ Euler equations remain regular for all time or develop a finite-time singularity. The Hou–Luo scenario remains a proposed route, not a resolution of this general question.

Known results

Elgindi (2019): finite-time singularity under qualified nonsmooth conditions, not smooth-data blowup.
Hou and Chen (2019–2020): rigorous results for a related scenario, not the unrestricted problem.
Hou and Chen (2022): a computer-assisted singularity proof for a cylindrical-boundary setting, not boundary-free $3\mathrm{D}$ Euler.
Hou (2021): numerical evidence for a potential interior singularity, not a proof.

August 2026 clarification and January 2026 candidates

Chen and Hou distinguish nonlocal error from profile-residual error and argue that grid data do not establish the alleged failure of their residual estimates. The note neither verifies the exact-profile assumption nor proves singularity. Separately, Google DeepMind collaborators reported four neural-network-generated unstable candidates; these remain unproved.

Current status (as of August 2026): The full boundary-free smooth $3\mathrm{D}$ Euler regularity-versus-singularity problem remains open; special-domain, nonsmooth, model, numerical, and candidate results do not settle it.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

A clarification on the distinction between nonlocal error and profile residual error in a computer-assisted proof of 3D Euler singularity — arXiv — Jiajie Chen, Thomas Y. Hou

computer-assisted-proof fluid-dynamics partial-differential-equations

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