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LIT-1248: Onsager's conjecture on the energy conservation for solutions of Euler's equation

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StatusActive — read 2026-10-20 (NOTE-982) in full, from the authors' manuscript (the `url:`); worth reading for a complete one-page proof that a weak solution of incompressible Euler with L³-averaged increments scaling as |y|^α, α > 1/3, conserves energy (THEORY-749), and for its mechanism: the energy flux into scales below a mollification scale ε is a commutator of size ε^(3α−1). Read it knowing that it is the positive half of Onsager's conjecture only and shows nothing about rougher fields: that dissipative solutions exist below 1/3 is Isett (2018) and others, and some rough solutions conserve energy anyway; that its "physical interpretation" (ε̄ = 0) passes to Navier–Stokes and to ensemble averages without proof; that the endpoint 1/3 is not covered; and that the manuscript misprints the exponents in the middle line of its final estimate and integrates (2) over R³ for the box D, neither affecting the result.
Version3
Tagsmathematics, natural-sciences
Cited as “paper” byNOTE-982 — Constantin, E and Titi — Onsager's conjecture on energy conservation
Cited as “source” byTHEORY-749 — A weak solution of the incompressible Euler equations on the periodic box that lies in L³(0,T; B^α_{3,∞}) ∩ C([0,T]; L²) with α > 1/3 conserves kinetic energy, because the energy flux into scales below a mollification scale ε is bounded by Cε^(3α−1)∫‖u‖³ and so vanishes as ε → 0; this does not show that rougher solutions dissipate, THEORY-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-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, 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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Constantin, P., E, W. & Titi, E. S. (1994). Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Communications in Mathematical Physics 165(1): 207–209. doi:10.1007/BF02099744.

Read in full on 2026-10-20 from the authors’ manuscript, in NOTE-982. The proof is held, re-derived, as THEORY-749.

Why this is here

A3 cited it for a general pattern: whether the effect of unresolved structure vanishes in a limit depends on the regularity of the field. The exchange named it as “the classic Constantin–E–Titi work” without a title. Bears on QUESTION-219. The reading narrows what it can be cited for. It proves that above exponent 1/3 the flux into unresolved scales must vanish as the scale goes to zero. It does not prove that a coarse-grained flux can survive for rough fields: that rests on the later negative half (Isett 2018 and others), which the record does not hold. Its estimate is the one behind THEORY-754, read at a fixed field as the scale goes to zero.

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