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LIT-1300: Onsager's conjecture for admissible weak solutions

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StatusActive — read 2026-10-20 (NOTE-1013) in full, from arXiv:1701.08678v1: worth reading for the admissible half of Onsager's negative direction: for every β < 1/3 and every smooth, strictly positive energy profile e(t) on [0, T] there is a weak solution of the 3D incompressible Euler equations on the periodic box, Hölder-continuous with exponent β in space and time, whose energy ∫|v|² equals e(t) at every time; taking e strictly decreasing gives solutions whose kinetic energy strictly decreases, so the energy flux into vanishing scales is one-signed on every time interval (THEORY-776); and for a shorter, more self-contained proof of the 1/3 threshold than Isett's (LIT-1255), with a trace-free glued stress and perturbations added everywhere in time so as to steer the energy. Read it knowing that "admissible" here means only that the total energy does not increase (the word appears in the title alone), not the local energy inequality, so the transfer is one-way for the whole box and may run backwards in places; that the same theorem gives solutions with increasing energy just as readily, so the direction is chosen by the prescribed profile and is not forced by the equations; that nothing is shown about vanishing-viscosity limits, about uniqueness or non-uniqueness for given initial data, or about the endpoint 1/3; that the h-principle (Theorem 1.3) rests on a result of Daneri and Székelyhidi stated without proof; and that the abstract's "typical" is looser than the weak-* density that Theorem 1.3 proves.
Version2
Tagsmathematics, natural-sciences
Cited as “paper” byNOTE-1013 — Buckmaster, De Lellis, Székelyhidi and Vicol — Onsager's conjecture for admissible weak solutions
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-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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Buckmaster, T., De Lellis, C., Székelyhidi, L., Jr. & Vicol, V. (2019). Onsager’s conjecture for admissible weak solutions. Communications on Pure and Applied Mathematics 72(2): 229–274. doi:10.1002/cpa.21781. arXiv:1701.08678 (30 January 2017).

Read in full in NOTE-1013, from the arXiv copy. The theorem, and what it implies for the coarse-grained energy flux, is filed as THEORY-776.

Why this is here

Named by the reading of Isett (NOTE-974, THEORY-763) as the one-way version of the negative half of Onsager’s conjecture. Isett’s solutions start from rest and return to it, so the energy flux they carry across vanishing scales runs in both directions. QUESTION-219 needed to know whether a rough Euler solution can carry a flux that runs only into the small scales, the dissipative cascade A3 had in mind. This paper gives one, for the energy of the whole box: any smooth decreasing energy profile is attained exactly.

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