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LIT-1255: A proof of Onsager's conjecture

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StatusActive — read 2026-10-20 (NOTE-974), skimmed, from arXiv:1608.08301v2: the introduction, the statement of the theorem, the Main Lemma and its three sublemmas, the gluing construction and the convex-integration ansatz, and the whole proof of the theorem from the Main Lemma; the long estimates of §§9–10 and 17 and the appendix were not checked. Worth reading for the negative half of Onsager's conjecture: for every α < 1/3 there is a nonzero weak solution of the 3D incompressible Euler equations on the periodic box, Hölder-continuous with exponent α in space and time, that vanishes outside a finite time interval and so does not conserve energy (THEORY-763). With Constantin, E and Titi (LIT-1248) it fixes 1/3 as the threshold. Read it knowing that its solutions gain energy from zero and lose it again, so they are not dissipative (admissible) solutions, which came later (Buckmaster, De Lellis, Székelyhidi and Vicol 2019); that they are built by convex integration from the zero solution and so also show non-uniqueness, not that turbulent or vanishing-viscosity limits are of this kind; that the endpoint 1/3, two dimensions and finite-energy solutions on R³ are left open; and that the arXiv v2 copy read is a 2024 revision, not compared with the Annals text.
Version2
Tagsmathematics, natural-sciences
Cited as “paper” byNOTE-974 — Isett — A proof of Onsager's conjecture
Cited as “source” byTHEORY-763 — For every α < 1/3 there is a nonzero weak solution of the 3D incompressible Euler equations on the periodic box, with velocity Hölder-continuous of exponent α in space and time, that vanishes outside a finite time interval; its energy is not conserved, so the energy flux across a mollification scale ε does not vanish as ε → 0, and it runs in both directions, out of the vanishing scales while the energy rises and into them while it falls, THEORY-776 — For every β < 1/3 and every smooth, strictly positive and strictly decreasing energy profile e(t) on [0, T], there is a weak solution of the 3D incompressible Euler equations on the periodic box, Hölder-continuous of exponent β in space and time, with ∫|v(t)|² = e(t) at every time; for such a solution the energy flux across a mollification scale ε, integrated over any time interval, tends as ε → 0 to ½ times the drop in e, which is strictly negative, so energy leaves the resolved field for vanishing scales on every interval; this is one-way for the energy of the whole box, not at every point, and it is chosen by the prescribed profile, not forced by the equations

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Isett, P. (2018). A proof of Onsager’s conjecture. Annals of Mathematics 188(3): 871–963. doi:10.4007/annals.2018.188.3.4. arXiv:1608.08301 (30 August 2016).

Skimmed in NOTE-974, from the arXiv copy. The theorem, and what it implies for the coarse-grained energy flux, is filed as THEORY-763.

Why this is here

The negative half of Onsager’s conjecture, which the reading of Constantin, E and Titi (NOTE-982, THEORY-749) found missing. A3 had cited “Onsager’s 1/3” for the point that the regularity of a field decides whether the unresolved scales vanish in the limit, and QUESTION-219 carries it. Constantin, E and Titi show the flux into vanishing scales must die above 1/3. This paper shows that below 1/3 it need not: there are rough solutions for which it survives.

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