Math Conjectures

Global regularity for three-dimensional incompressible Euler

OPENLandmarkOpen problemProposed c. 1930 · Standard version

Canonical statement

For every divergence-free Schwartz vector field u0∈S(R3;R3)u_0\in\mathcal S(\mathbb R^3;\mathbb R^3), the maximal classical solution of
∂tu+(u⋅∇)u=−∇p,∇⋅u=0,u(⋅,0)=u0 \partial_tu+(u\cdot\nabla)u=-\nabla p,\qquad \nabla\cdot u=0,\qquad u(\cdot,0)=u_0
exists for all t≥0t\ge0 and belongs to C∞(R3×[0,∞))C^\infty(\mathbb R^3\times[0,\infty)).
View source LaTeX
For every divergence-free Schwartz vector field \(u_0\in\mathcal S(\mathbb R^3;\mathbb R^3)\), the maximal classical solution of \[ \partial_tu+(u\cdot\nabla)u=-\nabla p,\qquad \nabla\cdot u=0,\qquad u(\cdot,0)=u_0 \] exists for all \(t\ge0\) and belongs to \(C^\infty(\mathbb R^3\times[0,\infty))\).
PDE-002:[BealeKatoMajda1984Euler][Chae2007EulerBlowupSurvey][ChenHou2025EulerSingularities]
Partial differential equationsDynamical systemsMathematical physics

Notes

The problem asks whether every divergence-free Schwartz initial velocity field on R3\mathbb R^3 gives rise to a classical solution of the incompressible Euler equations that stays smooth for all t≥0t\ge 0. Unlike Navier–Stokes there is no viscous term, and the mechanism of concern is vortex stretching. The question has no single point of origin: it crystallized gradually from the early PDE theory of the Euler equations, roughly in the 1930s.

The classical Beale–Kato–Majda criterion shows that blow-up can occur only if the time integral of ∥ω(t)∥L∞\|\omega(t)\|_{L^\infty} diverges, so any singularity must be accompanied by unbounded vorticity growth [BealeKatoMajda1984Euler]; the surrounding blow-up literature is surveyed in [Chae2007EulerBlowupSurvey]. More recently, rigorous finite-time singularities have been constructed for solutions below the classical smoothness threshold, including C1,αC^{1,\alpha} regimes [ChenHou2025EulerSingularities], sharpening the sense in which the smooth problem is critical.

For smooth finite-energy data, however, neither blow-up nor global regularity has been proved, and the problem is open in both directions.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.