Finite Time Blowup for Navier–Stokes
2026 proposed solution with Lean formalization; independent validation ongoing, 2026 · paper · Mathematics and statistics
Constructs, for every positive viscosity, a solution of the three-dimensional incompressible Navier–Stokes equations that starts from rest, with a smooth compactly supported force. Its velocity becomes unbounded in finite time while kinetic energy stays bounded.
Checked: PDF from cdn.openai.com, fetched 5 Oct 2026. Page numbers are those of that PDF.
0 findings
None that passed a second check.
3 minor (typos and the like; most checked by the judge only)
Wording Theorem 1.1 binds u, p to the constructed fields, then says "there is no smooth solution (u, P) on R3 × [0, ∞)" and writes its energy as ∫|u(x,t)|^2. So u names two different fields, while the proof on p.124 calls the competitor "v, P".
- p. 1 Theorem 1.1, final sentence
Consequently, there is no smooth solution (u, P) on R3 × [0, ∞)
- p. 124 Proof of Theorem 1.1, third paragraph
Suppose that a global smooth solution v, P with the same data had uniformly bounded
Judge's note: I looked at the crops of p.1 and p.124. On p.1 the theorem statement uses u for the constructed velocity and then for the hypothetical global competitor (u, P), including inside its energy bound. The proof on p.124 names the competitor v, P and shows v = u on [0,1). The meaning can be recovered, but the notation clashes within the statement. This is minor.
Second check: another model agreed
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Typo p.45, Section 5.1: "We first solve the coefficient equations Equations (5.2) to (5.6)". The word 'equations' is printed twice in a row.
- p. 45 Section 5.1, first sentence
We first solve the coefficient equations Equations (5.2)
Judge's note: The page-image crop shows the duplication as printed. The second 'Equations' is the hyperlinked cross-reference label. This is cosmetic.
Second check: none (judge only)
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Typo p.54, Section 5.3: "Lemma 5.1 solves the inner equations Equations (5.2) to (5.6)". The word 'equations' is printed twice in a row.
- p. 54 Section 5.3, first paragraph
Lemma 5.1 solves the inner equations Equations (5.2) to (5.6)
Judge's note: The page-image crop confirms the duplication. It is a separate instance (different page) of the same cross-reference style issue as C006. This is cosmetic.
Second check: none (judge only)