openmath.dev
Esc

Loading…

↑ ↓ to move↵ to open

Appearance
Appearance

Theorem · version 1

Finite-time smooth breakdown requires divergent integrated vorticity

Statement

For an unforced three-dimensional Euler solution in the classical Sobolev class Hs(R3)H^s(\mathbb R^3), s>5/2s>5/2, a finite maximal smooth lifespan T* requires ∫0T∗∥curl⁡u(t)∥∞ dt=∞\int_0^{T_*}\|\operatorname{curl}u(t)\|_{\infty}\,dt=\infty. In particular, a finite value of this integral allows smooth continuation past that time.

Assumptions

  • Classical divergence-free Euler solution on ℝ³ in H^s with s > 5/2.
  • The solution is smooth up to each earlier time and the continuation criterion applies.

Scope

Beale–Kato–Majda is a conditional criterion; it does not itself construct a singularity.

Equation
Unforced inviscid Euler
Initial data
The cited regularity class of the continuation criterion
Conclusion
A vorticity-integral continuation criterion; not a blowup construction
Equation
Unforced inviscid Euler
Initial data
Classical divergence-free solution on ℝ³ in H^s, s > 5/2
Conclusion
A vorticity-integral continuation criterion; not a blowup construction

Evidence

Connections

In this problem · publication v7 · Selected context

Three-dimensional Euler regularity →

Conditional implication · Proved

Relationship not independently reviewed. This concerns its use in this problem.

Sources

Finite Time Blowup for the Euler Equation →

  • Theorem 1.1 and Section 1.1 → · Original passage ↗
    Recorded wording

    The manuscript states smooth compactly supported unforced Euler data with finite maximal lifespan.

    paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.

OpenAI · Preprint · Publication date Sep 8, 2026

Revision & attribution

Source revision: 2026-09-08. Observed Oct 13, 2026.

OpenAI attributes the argument to its internal research model. Bibliographic authorship is printed as OpenAI; individual human preparation and checking roles are not inferred.

Remarks on the Breakdown of Smooth Solutions for the 3-D Euler Equations →

J. Thomas Beale · Tosio Kato · Andrew Majda · Published · Publication date 1984

Revision & attribution

Source revision: 1984. Observed Oct 13, 2026.

No specific AI contribution is attributed by this source record.

History

Statement versions

Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.

Assessment history · 1

Assessments retained in these publication records. Earlier statement versions retain their own source histories.

Record details
Claim ID
claim-euler-vorticity-breakdown-criterion
Statement version
1
SHA-256
668831a689005f97b6f10092ac681272c1e9c430ab315d728d264c724078f94f

Publication records