NOTE-982: Constantin, E and Titi — Onsager's conjecture on energy conservation

Contribution

It proves the sharp positive half of Onsager’s 1949 conjecture, in the Besov scale. A weak solution of incompressible Euler whose velocity increments obey ‖u(·+y) − u‖_{L³} ≤ C|y|^α with α > 1/3, uniformly enough in time, conserves kinetic energy exactly. Before it, by the paper’s own account, Eyink had proved conservation for Hölder exponent α > 1/2, and for α > 1/3 only under the stronger norm Σ_k |k|^α |û_k|. The proof is short enough to be checked in full, and it became the template for later work.

Key insight

Energy can leave a weak solution only through the flux into vanishingly small scales, and that flux is a commutator. Mollify at scale ε. The energy of u^ε changes by ∫(u⊗u)^ε : ∇u^ε. The part u^ε⊗u^ε contributes nothing (it is transport by a divergence-free field), so only the subgrid stress (u⊗u)^ε − u^ε⊗u^ε matters, and it is quadratic in increments over distances below ε. Increments of size ε^α, twice, against a gradient of size ε^(α−1), give a flux of size ε^(3α−1). Above α = 1/3 the flux into the unresolved scales dies as the resolution is refined, so nothing can be lost there. At α = 1/3 the bound stops decaying, which is exactly the Kolmogorov scaling (4). Regularity decides whether the coarse-grained flux can survive the limit.

Assumptions

• Domain. The periodic box D = [0, 1]³, “to avoid questions regarding boundaries”. Three dimensions are stated; nothing in the proof uses d = 3 except the notation.
• Weak solution. u satisfies the distributional form (1) of ∂_t u + ∇·(u⊗u) + ∇p = 0 against every smooth, compactly supported test field ψ, with initial data u(·, 0). Divergence-free u is implicit. The pressure’s regularity is not stated. On the torus it is fixed, up to a function of time that has no gradient, by −Δp = ∂_i∂_j(u_iu_j), so p ∈ L^{3/2} follows from u ∈ L³; the proof needs only that ∇p^ε is a gradient.
• Regularity. u ∈ L³([0, T]; B^α_{3,∞}(D)) ∩ C([0, T]; L²(D)) with α > 1/3. For 0 < α < 1, B^α_{3,∞} is the space with ‖u‖{L³} + sup_y |y|^(−α)‖u(·+y) − u‖{L³} finite; that is what (6) uses. The time integrability is L³, so the bound is integrated in time, not uniform.
• Mollifier. φ ∈ C₀^∞(R³), “a standard mollifier supported in B(0, 1)”: ∫φ = 1, and, for (12) as written, φ ≥ 0.

Key results

• Theorem. Under the assumptions above, with α > 1/3, ∫_D |u(x, t)|² dx = ∫_D |u(x, 0)|² dx for t ∈ [0, T). The manuscript writes the integrals in (2) over R³; on the periodic box they are over D.
• Besov facts (6)–(8). ‖u(·+y) − u‖{L³} ≤ C|y|^α‖u‖{B^α_{3,∞}}; ‖u − u^ε‖{L³} ≤ Cε^α‖u‖; ‖∇u^ε‖{L³} ≤ Cε^(α−1)‖u‖. Re-derived in THEORY-749: (6) is the definition, and (7) and (8) follow from it by Minkowski’s inequality, using ∫φ_ε = 1 and ∫∇φ_ε = 0.
• Main identity (10), pointwise: with r_ε(u, u)(x) = ∫φ_ε(y) δ_yu(x) ⊗ δ_yu(x) dy and δ_yu(x) = u(x−y) − u(x), (u⊗u)^ε = u^ε⊗u^ε + r_ε(u, u) − (u − u^ε)⊗(u − u^ε). Re-derived by expanding r_ε; checked numerically to 10⁻¹⁴.
• Energy balance (11) of the mollified field: ∫|u^ε(t)|² − ∫|u^ε(0)|² = 2∫₀ᵗ dτ ∫ Tr((u⊗u)^ε ∇u
• Commutator bound (12). ‖r_ε(u, u)‖{L^{3/2}} ≤ Cε^(2α)‖u‖²{B
• Flux bound (unnumbered, last display): |∫|u^ε(t)|² − ∫|u^ε(0)|²| ≤ Cε^(3α−1)∫₀ᵗ ‖u(τ)‖³_{B^α_{3,∞}} dτ → 0. Holds when: α > 1/3. At α = 1/3 the bound is constant in ε and decides nothing.
• Physical reading (stated, not proved): “if ⟨|u(x + r) − u(x)|³⟩^(1/3) ≤ Cr^α for α > 1/3 then ε̄ = 0.”

Claims

id	claim	strength	support
C1	Weak Euler solutions in L³(0,T; B^α_{3,∞}) ∩ C(0,T; L²) with α > 1/3 conserve energy	strong	the proof; re-derived step by step in THEORY-749
C2	This implies the Hölder version of the positive half	strong	C^α ⊂ B^α_{3,∞} on the bounded box; needs L³ integrability of the Hölder norm in time
C3	This is the “sharp result”	moderate as written	sharp in the exponent, as the later negative half shows; not proved sharp here, and the endpoint α = 1/3 is not covered (see Limitations)
C4	Onsager conjectured that (2) “ceases to be true for α ≤ 1/3”	weak as phrased	Onsager’s claim, and the later theorems, are that dissipative solutions exist, not that every rough solution dissipates (see Limitations)
C5	If ⟨	δu(r)	³⟩^(1/3) ≤ Cr^α with α > 1/3, then ε̄ = 0
C6	Eyink proved the α > 1/2 Hölder case and α > 1/3 under norm (5)	not checked	the paper’s report of [2], which the record does not hold

Method

1. Mollify at scale ε. The mollified field u^ε is smooth in space and satisfies ∂_t u^ε + ∇·(u⊗u)^ε + ∇p^ε = 0. Multiply by u^ε and integrate: the pressure term vanishes since ∇·u^ε = 0, and integration by parts gives (11).
2. Split the mollified flux by identity (10). The term u^ε⊗u^ε contributes ∫u^ε_i u^ε_j ∂_i u^ε_j = ∫u^ε·∇(½|u^ε|²) = 0.
3. Bound what remains by Hölder’s inequality, L^{3/2} against L³, using (12) for r_ε, (7) twice for (u − u^ε)⊗(u − u^ε), and (8) for ∇u
4. Let ε → 0. The right side vanishes for α > 1/3, and ∫|u^ε(t)|² → ∫|u(t)|² at each t because u(t) ∈ L².

The paper does step 1 “as if the solution is differentiable in time” and says the time mollification is “straightforward”. It is: testing (1) with ψ = θ(t)φ_ε(x − ·)e_j shows ∂_t u^ε ∈ L^{3/2}(0, T; C^∞), so t ↦ ∫|u^ε|² is absolutely continuous and its derivative is 2∫u^ε·∂_t u

Concepts

• Weak solution: a field satisfying the Euler equations only after integration against smooth test fields, (1). Such a field need not have derivatives, so energy conservation, automatic for smooth solutions, is not.
• B^α_{3,∞}: fields whose L³-averaged increments scale at most as |y|^α. An L³ analogue of Hölder continuity, weaker than C^α, and the space-averaged form of the scaling Kolmogorov’s theory assigns to turbulent velocity at α = 1/3.
• r_ε(u, u): the mollifier-weighted second moment of increments over distances below ε. With (u − u^ε)⊗(u − u^ε) it makes up the subgrid stress, (u⊗u)^ε − u^ε⊗u^ε = r_ε − (u − u^ε)⊗(u − u
• Path-wise (the paper’s word): a statement about each solution, as against Kolmogorov’s statements about ensemble averages.

Connections

It proves what Onsager (1949) asserted, in a sharper space than his Hölder one, and improves on Eyink’s preprints of 1992 (published 1994), which the paper credits with α > 1/2 and with α > 1/3 under norm (5). Neither reference is held by the record. The mollified flux in step 2 is exactly the subgrid energy flux of the filtering approach to turbulence: ∫(u⊗u)^ε : ∇u^ε = ∫τ_ε : ∇u^ε with τ_ε = (u⊗u)^ε − u^ε⊗u^ε, the Π_ℓ of Eyink and Aluie (LIT-1239, NOTE-969) up to sign and the choice of kernel. The estimate is the one THEORY-754 re-derives, read in the other direction (see Bearing). Later work the record does not hold, named here so that what this paper did not do is placed: the local energy balance with a dissipation measure (Duchon and Robert 2000); the endpoint, conservation in a space just below B^{1/3}_{3,∞} (Cheskidov, Constantin, Friedlander and Shvydkoy 2008); and the negative half, dissipative weak solutions in C^α for every α < 1/3 (Isett 2018, after De Lellis and Székelyhidi’s convex integration; Buckmaster, De Lellis, Székelyhidi and Vicol 2019 for admissible ones).

Bearing on the record

• THEORY-749 is filed from this reading, Active: the theorem and its proof, re-derived step by step, with the bound O(ε^(3α−1)) on the flux into the unresolved scales.
• A3’s point, which QUESTION-219 carries in spirit. A3 cited the “Constantin–E–Titi work” for “Onsager’s 1/3, where the regularity of a field decides whether the unresolved scales vanish in the limit”, and the LIT took it as the ground for “a coarse-grained flux can survive as the scale goes to zero”. The paper proves half of that. Above 1/3, the flux of energy into the unresolved scales must vanish as the resolution is refined. It does not prove that below 1/3 it survives. Roughness below 1/3 permits a surviving flux; it does not produce one. A shear flow u = (f(x₂), 0, 0), with f as rough as one likes, is a stationary weak solution whose flux is zero at every ε: the only non-zero component of (u⊗u)^ε is (f²)^ε in position (1, 1), and it meets ∂₁u^ε₁ = 0, so Σ_{ij}(u_iu_j)^ε ∂_iu^ε_j = 0 pointwise. That “a coarse-grained flux can survive” is true is the negative half, from Isett (2018) on, which the record has not read. The LIT’s “Why this is here” and QUESTION-219 should not cite this paper for survival.
• THEORY-754. Its hypothesis, ‖δu(r)‖₃ ≤ C₀u_rms(r/L)^σ, is the B^σ_{3,∞} condition, and its prefactor (u_rms³/L)(ℓ/L)^(3σ−1) is this paper’s ε^(3α−1). Read at a fixed field as ℓ → 0, that prefactor is the Constantin–E–Titi bound on the whole flux: it vanishes for σ > 1/3 and is constant at σ = 1/3. So a non-zero mean flux through an inertial range, which Eyink and Aluie assume, is consistent only with σ₃ ≤ 1/3 in the limit, and Kolmogorov’s 1/3 is the largest exponent at which it can be. THEORY-754’s Connections could say so. The two results use the same three estimates; Eyink and Aluie split the flux by which scales it comes from, and this paper bounds it whole.
• QUESTION-219 itself. Nothing here bears on whether a pattern of transfer picks out a unit. The paper concerns one field and the energy crossing an analyst-chosen scale ε. Its relevance is narrower: it shows that a flux across scales, as a quantity, is fixed by a balance law and a regularity class, and that in a regular enough class there is no flux to the smallest scales to be organized at all. A5’s requirement, “a balance law for a specified transferred quantity”, is met here in its purest form.
• THEORY-661, CLAIM-157, THEORY-733, THEORY-257, CLAIM-331, CLAIM-327, CLAIM-388, THEORY-755. Not addressed. There is no decomposition into blocks, no aggregation, no time scales and no closure here. The mollified equation for u^ε is not closed: it depends on (u⊗u)^ε, which u^ε does not determine. That is a small instance of QUESTION-222’s and QUESTION-218’s closure problem, and the paper’s point is that the unclosed term’s effect on energy vanishes in the limit for regular fields, not that it can be closed.
• QUESTION-217, QUESTION-220, QUESTION-221. Nothing. Mollifiers at different ε are nested only in the trivial sense that u^ε → u; the paper uses one scale at a time.

No instruction for machine-learning practice; nothing for the anthology.

Limitations

• Two printed formulas are wrong in this manuscript. In (2) the integrals are over R³; the solution lives on the periodic box D. In the final estimate the middle line reads ‖r_ε‖^{2/3}{L^{3/2}}‖∇u^ε‖^{1/3}{L³} + ‖u − u^ε‖^{2/3}{L³}‖∇u^ε‖^{1/3}{L³}. That is not what Hölder’s inequality gives, and it is not homogeneous: the left side is cubic in u and these terms are of degree 5/3. Inserting (7), (8) and (12) into it would give ε^((5α−1)/3), not the ε^(3α−1) of the next line. The correct middle line is ‖r_ε‖{L^{3/2}}‖∇u^ε‖{L³} + ‖u − u^ε‖²_{L³}‖∇u^ε‖_{L³}, which gives ε^(3α−1) and the theorem. Whether the typeset article corrects either could not be checked.
• The proof is terse, not gapped. Three steps are left to the reader: the time mollification, the pressure’s regularity, and the convergence ∫|u^ε(t)|² → ∫|u(t)|². Each is routine and is written out in THEORY-749.
• “Ceases to be true for α ≤ 1/3” misstates the converse. What is conjectured, and now proved by others, is that energy-dissipating weak solutions exist below 1/3. Conservation does not fail for every rough solution: the rough shear flows above conserve energy. The paper’s “sharp” is sharp in this sense only.
• The endpoint. At α = 1/3 the bound is constant in ε and the theorem says nothing. Conservation in a space slightly smaller than B^{1/3}_{3,∞} came later (Cheskidov et al. 2008, not held).
• The physical reading overreaches the proof. “If ⟨|δu|³⟩^(1/3) ≤ Cr^α for α > 1/3 then ε̄ = 0” moves from a statement about Euler weak solutions to one about the viscous dissipation of Navier–Stokes in the limit, and from a path-wise bound to an ensemble average ⟨·⟩. The theorem needs the bound for the solution itself, integrated in time. The passage to Navier–Stokes needs a convergence argument the paper does not give, and the record has not checked one.
• Global, not local. The theorem conserves the total energy. A local energy balance, with a defect measure saying where dissipation happens, is Duchon and Robert’s (2000), not here.
• Nothing about existence or the negative half. It does not show that solutions with this regularity exist for given data, and it does not construct dissipative solutions.

Open questions

• Does any weak solution arising as an inviscid limit of Navier–Stokes belong to a class with α ≤ 1/3 while dissipating? The negative half gives dissipative weak solutions, constructed by convex integration, not as vanishing-viscosity limits. Whether turbulence’s limit is one of them is a question about Navier–Stokes, which only a theorem about the equations would settle.
• For the record: does a surviving flux into the unresolved scales correspond to anything the record would call organization, or is it, in A5’s term, mere transport? This paper only says when there is none.