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LIT-1289: The L³-based strong Onsager theorem

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StatusActive — read 2026-10-20 (NOTE-1022), skimmed, from arXiv:2305.18509v3: the introduction and its heuristics, the whole of Section 2 with the inductive Proposition 2.11 and the proof of Theorem 1.1 from it, the derivation of the new current errors (§5.1) and the closing of the relaxed local energy identity (§6.3, Lemma 6.10) were read; the estimates of Sections 3–6, the dodging lemmas of Section 4, the parameter choices of Section 7 and the appendix were paged through and not checked, and Section 3's proofs are in a companion paper (Annals of PDE 2024) that the record does not hold. Worth reading for the local, L³-based version of the negative half of Onsager's conjecture: for every β < 1/3 there is a weak solution of the 3D incompressible Euler equations on the periodic box in C⁰_t(B^β_{3,∞} ∩ L^{1/(1−3β)}) that satisfies the local energy inequality and whose total energy strictly decreases; and, from the proof rather than the theorem's statement, for the fact that its Duchon–Robert dissipation is exactly the smooth, nonnegative density E(t, x) = −e(t)e′(t)ρ₀(x₁, x₂)² fixed at the iteration's first step, positive in a periodic array of straight pipes and zero outside them (THEORY-772). So the coarse-grained energy flux into vanishing scales converges, as a distribution in space and time, to a one-signed density: one-way at every place and time in the limit, which Buckmaster, De Lellis, Székelyhidi and Vicol (LIT-1300) showed only for the whole box. Read it knowing that the solutions are not Hölder or even bounded (the strong Hölder version is open; the best Hölder result with the local inequality is β < 1/7, De Lellis and Kwon); that where and how fast energy is dissipated is an input of the construction, carried unchanged through every step, not something the dynamics selects; that the one-signed statement is about the limit ℓ → 0 tested against nonnegative functions, and nothing is said about the sign of the flux at any finite scale; that nothing ties these solutions to vanishing-viscosity limits, though the paper notes that the local inequality is what such limits would satisfy; and that the arXiv v3 copy read is the authors' final version, not compared with the Annals text.
Version2
Tagsmathematics, natural-sciences
Cited as “paper” byNOTE-1022 — Giri, Kwon and Novack — The L³-based strong Onsager theorem
Cited as “source” byTHEORY-772 — For every β < 1/3 there is a weak solution of the 3D incompressible Euler equations on the periodic box, in C⁰_t(B^β_{3,∞} ∩ L^{1/(1−3β)}), whose Duchon–Robert dissipation is the smooth nonnegative density E(t, x) = −e(t)e′(t)ρ₀(x₁, x₂)² fixed at the construction's first step; for such a solution the local coarse-grained energy flux into scales below ℓ converges, as ℓ → 0 and as a distribution in space and time, to E, so the transfer into vanishing scales is one-way at every place and time in the limit, nonzero inside a periodic array of pipes and zero outside them; where and how fast energy is dissipated is chosen by the construction, not forced by the equations, and nothing is shown about the sign at any positive scale, THEORY-821 — For every α < 1/15 there are weak solutions of the 3D incompressible Euler equations on the periodic box, Hölder-continuous of exponent α in space and time, that satisfy the local energy inequality; in the construction their dissipation measure is a nonnegative function of time alone, equal at each time to the rate at which the total energy falls, zero while the local energy equality holds and positive on all of space during a short final interval in which the solution's energy is drained to zero; uncountably many such solutions share one initial datum and one dissipation measure; for them the local coarse-grained energy flux into vanishing scales converges to that measure, so the transfer is one-way at every place and time in the limit and spread uniformly over the box, with its timing chosen by the construction, not forced by the equations

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Giri, V., Kwon, H. & Novack, M. (2026). The L³-based strong Onsager theorem. Annals of Mathematics 204(1): 265–421. doi:10.4007/annals.2026.204.1.4. arXiv:2305.18509 (29 May 2023).

Skimmed in NOTE-1022, from the arXiv copy. What it implies for the local coarse-grained energy flux is filed as THEORY-772.

Why this is here

Named by the reading of Buckmaster, De Lellis, Székelyhidi and Vicol (NOTE-1013, THEORY-776), whose rough Euler solutions lose energy only for the box as a whole. QUESTION-219 needed to know whether the one-way transfer into vanishing scales can hold at every place as well as overall. This paper gives solutions whose dissipation measure is nonnegative everywhere and, in its proof, a known smooth function.

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