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NOTE-974: Isett — A proof of Onsager's conjecture

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PaperLIT-1255 — A proof of Onsager's conjecture

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Contribution

It proves Conjecture 2, the negative direction of Onsager’s conjecture: for every α < 1/3 there are weak solutions of incompressible Euler on T³ in the Hölder class C^α that fail to conserve energy. The best previous results were 1/5 for C_tC^α_x (Isett 2013) and 1/3 in the weaker class L¹_tC^α_x (Buckmaster, De Lellis and Székelyhidi). With the positive half, conservation above 1/3 (Eyink; Constantin, E and Titi, LIT-1248), it makes 1/3 the threshold for the class L^∞_tC^α_x. The new device is a “gluing approximation”. Before each convex-integration step, the approximate solution is replaced by exact Euler solutions on short time windows, glued together, so that the remaining error lives on separated time intervals and the high-frequency corrections that cancel it never interact.

Key insight

The obstruction below 1/5 had been interference between the high-frequency waves added at each stage: they had to be cut off sharply in time, and the cutoffs cost regularity. If the error that the waves must cancel can first be confined to well-separated short time intervals, one wave per interval suffices and no two waves ever meet. The gluing step does this by solving Euler exactly between the intervals. Its difficulty is to make the glued error no larger than the original. That needs a “good anti-divergence” of the velocity difference, which is built by transporting along the flow and by using the structure of the linearized Euler equation. Mikado flows, straight pipe flows in disjoint cylinders, supply waves that are exact stationary solutions and stay coherent over the needed time scale.

Assumptions

  • Domain. The periodic box T³ = (R/Z)³, for all times t ∈ R. The paper says the proof extends to d ≥ 3, and does not cover d = 2 or the whole space with finite energy.
  • Notion of solution. Continuous v and p satisfying the Euler equations in the sense of distributions, equivalently the integral balances of momentum and mass (2)–(3) over every smooth subregion.
  • Regularity class. C^α_{t,x}: |f(t + Δt, x + Δx) − f(t, x)| ≤ C(|Δt| + |Δx|)^α uniformly. That is stronger than the L^∞tC^α_x of the conjecture and much stronger than Constantin, E and Titi’s L³_tB^α{3,∞}.
  • What is not assumed or obtained. No initial-value problem is posed: the solution is constructed, starting from the zero solution, and vanishes before some time. No energy inequality (admissibility) is claimed.

Key results

  • Theorem 1. For any α < 1/3 there is a nonzero weak solution v ∈ C^α_{t,x}(R × T³), p ∈ C^{2α}_{t,x}, with v ≡ 0 outside a finite time interval. It does not conserve energy.
  • Theorem 3 (Section 18). For any 0 < δ < 1/4 there is such a solution for every α satisfying (18.1). The left side of (18.1) tends to −1/2 + 3α/2 as δ → 0, which gives Theorem 1. Redone here by hand from the parameter-evolution matrix (eigenvalues 1 + δ, δ/2, 1).
  • Main Lemma 2.1. An Euler–Reynolds flow (v, p, R) with frequency–energy levels (Ξ, e_v, e_R) can be corrected to one with levels (CNΞ, log Ξ̂ e_R, (log Ξ̂)^{5/2} e_v^{1/2}e_R^{1/2}/N), with Ξ̂ = Ξ(e_v/e_R)^{1/2} and N ≥ max{Ξ^η, (e_v/e_R)^{1/2}}. The correction is bounded by C(log Ξ̂)^{1/2}e_R^{1/2} in C⁰ and by CNΞ(log Ξ̂)^{1/2}e_R^{1/2} in C¹. The time support grows by at most Ξ⁻¹e_v
  • The three sublemmas. Regularization (Lemma 3.1, by mollification and the quadratic commutator estimate (4.1)). Gluing (Lemma 3.2: the new stress is Σ_I R_I with the R_I supported in disjoint intervals of length θ ≈ δ(log Ξ̂)⁻²Ξ⁻¹e_v^{−1/2}, with the same size bounds up to logarithms and a bound on their advective derivatives). Convex integration (Lemma 3.3: one Mikado wave per R_I).
  • Proposition 8.1 (proof read). The tensor r_I = ρ_I + z_I built by the transport equations (7.8)–(7.10) is a symmetric anti-divergence of the velocity difference y_I. The proof uses that y_I and v are divergence free and uniqueness for transport.
  • Mikado flows (Section 11). For F = {e_i ± e_j}, |F| = 6, the fields ψ_f(X)f with ψ_f supported in disjoint thin cylinders around periodic lines in direction f are divergence free and have disjoint supports, so any combination Σγ_fψ_ff is a stationary Euler solution with zero pressure. The six tensors f⊗f form a basis of symmetric matrices with Σ ¼ f⊗f = Id (15.6)–(15.7), which lets the algebraic equation (15.5) absorb any small stress.
  • Nontriviality (Section 18.5). The base stage has ‖v₍₀₎‖ ≥ 0.9; the corrections sum to at most 0.01.

Claims

Not filled: this reading is Skimmed, and only a Read note may fill in a claims table. What the skim supports is in THEORY-763, filed Proposed.

Method

  1. Start from the zero flow and apply the Main Lemma of Isett’s 2013 paper once, with a prescribed energy profile e(t), e(0) = 1, to get a nonzero Euler–Reynolds flow with compact time support.
  2. Iterate the Main Lemma with frequencies Ξ_(k) and energy levels growing and shrinking double-exponentially, governed by the matrix of (18.20).
  3. Within each Main Lemma step:
    • regularize;
    • glue. Solve Euler exactly from the data v(8θI), call the differences y_I, and join them with a partition of unity η_I in time. The new stress, (6.7), comes from η_I′ times an anti-divergence of y_I − y_{I+1}, plus quadratic overlap terms;
    • convex-integrate. On each interval add V_J = e_I^{1/2}γ_J(∇Γ_I⁻¹)f ψ_f(λΓ_I) plus a divergence correction, with Γ_I the back-to-labels map of the mollified coarse flow and λ a large frequency. Choose γ_J by the algebraic equation so that Σ v_J⊗v_J + P + R_ε = 0, so the low-frequency part of the new stress cancels.
  4. Pass to the limit. The corrections are summable in C⁰ (so v is continuous and nonzero) and in C_tC^α_x for α in range (so v ∈ C^α). Time regularity follows from Isett’s earlier theory. The supports converge, so the time support is compact.

Concepts

  • Euler–Reynolds flow: (v, p, R) with ∂_t v + ∇·(v⊗v) + ∇p = ∇·R. R is what the approximate solution fails by, the analogue of a Reynolds stress.
  • Frequency–energy levels (Ξ, e_v, e_R): bounds on the derivatives of v and R by powers of a frequency Ξ times e_v^{1/2} and e_R (Definition 2.2).
  • Ideal Case Scenario: Isett’s 2013 target scaling for one step, e_R′ ≈ e_v^{1/2}e_R^{1/2}/N, which yields exponents up to 1/3.
  • Gluing approximation: replacing an approximate solution by exact local Euler solutions glued in time, so that the error is localized to short, separated intervals.
  • Mikado flow: a stationary Euler solution made of straight pipe flows in disjoint periodic cylinders (Daneri and Székelyhidi).
  • Back-to-labels map Γ_I: the inverse flow map of the coarse velocity from time t(I), used to transport the waves.

Connections

It completes the line of De Lellis and Székelyhidi (2013, 2014), who adapted Nash’s convex integration to Euler with Beltrami waves and reached 1/10. Isett’s 2013 work, with its “Ideal Case Scenario”, reached 1/5, and Buckmaster, De Lellis and Székelyhidi reached 1/3 in L¹_tC^α_x. Daneri and Székelyhidi (2016) supplied the Mikado flows. On the positive side it cites Eyink (1994), Constantin, E and Titi (1994, LIT-1248) and Cheskidov, Constantin, Friedlander and Shvydkoy (2008) for the endpoint space. The earliest non-uniqueness and non-conservation examples, in L² rather than Hölder classes, are Scheffer (1993) and Shnirelman (1997, 2000), and the paper suggests its gluing is related to Shnirelman’s construction. Later work: Buckmaster, De Lellis, Székelyhidi and Vicol (2019) for admissible, strictly dissipative solutions below 1/3, since read (LIT-1300, NOTE-1013).

Bearing on the record

  • THEORY-763 is filed from this reading, Proposed. It states Theorem 1 and the record’s consequence: by Constantin, E and Titi’s energy balance (11), re-derived for any weak solution in THEORY-749, the time-integrated flux across a mollification scale ε tends to the change in energy as ε → 0. For Isett’s solutions that change is nonzero. It is positive while the energy rises from zero and negative while it falls back.
  • QUESTION-219 and A3’s point. NOTE-982 found that the record held only the positive half: above 1/3 the flux into vanishing scales must die. This paper supplies the other half: below 1/3 it can survive. Two qualifications that the question should carry:
    • The surviving flux is not one-way. These solutions are not admissible. Energy first comes out of the vanishing scales and later goes back in, so they show transfer across vanishing scales, not a dissipative cascade. A one-way transfer, for the whole box, is Buckmaster, De Lellis, Székelyhidi and Vicol (2019), since read (THEORY-776).
    • It is not about turbulence. The solutions are constructed by convex integration, start from rest and are non-unique, and nothing ties them to vanishing-viscosity limits. So the question’s sentence “That a flux can survive for rougher fields is the negative half of Onsager’s conjecture (Isett 2018 and others), which the record does not hold” can now cite THEORY-763, with “in both directions” added. Nothing here bears on whether a pattern of transfer picks out a unit. If anything, the construction cuts against reading flux as organization. The flux in these solutions is produced by a superposition of waves with disjoint supports in time and space, one per short interval, and nothing persists across intervals except the coarse field the waves are transported by.
  • THEORY-749. Its “What this does not say” names Isett as the negative half, unread. It can now cite THEORY-763 for it, with the admissibility caveat.
  • THEORY-754. NOTE-982 found Eyink and Aluie’s non-zero mean inertial-range flux consistent only with exponents up to 1/3. Isett shows weak solutions with a surviving flux exist there, but says nothing about a constant-flux range or about locality of transfer.
  • QUESTION-222 and QUESTION-218 (closure). The Euler–Reynolds system is the closure problem in the form this field uses: v is a coarse field and R the unclosed stress. Convex integration runs the closure problem backwards: it chooses the fine-scale correction to make a prescribed stress vanish. It shows that a coarse field with a given Reynolds stress has many fine-scale completions, which is non-uniqueness in the opposite direction from closure. That is a contrast the record might use, and it is not filed.
  • The other key entries (THEORY-733, THEORY-257, THEORY-661, CLAIM-157, CLAIM-331, CLAIM-327, CLAIM-388, THEORY-755) and QUESTION-217, QUESTION-220, QUESTION-221: not addressed. The iteration’s frequencies Ξ_(k) form a coarse-to-fine sequence of nested corrections, v = v₍₀₎ + Σ V₍k₎. But it is a construction device, not a claim about optimal or natural hierarchies.

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

Limitations

  • Read in outline. About 40 of the 74 pages, the estimates for the gluing and convex-integration steps, were not read, so no step of the Main Lemma was checked. The record relies on the paper for it.
  • Not admissible, not physical. The abstract and Theorem 1 claim only non-conservation, and the paper is accurate about that. Readers, and A3’s gloss, may take “anomalous dissipation” from the conjecture’s motivation. These solutions gain energy as well as lose it.
  • The class. The conjecture as stated is about L^∞tC^α_x. The theorem gives the stronger C^α{t,x}, which is fine for the negative direction. It does not reach the endpoint α = 1/3, where Eyink and Cheskidov et al. suggest non-conservation may still be possible.
  • Scope stated by the author. d = 2, finite-energy solutions on R³, active scalars, Boussinesq and the steady equations are left open.
  • Copy. The arXiv v2 is a 2024 revision; whether it differs from the 2018 Annals text in substance was not checked.

Open questions

  • Are there weak solutions below 1/3 whose energy only decreases, and is the surviving flux then one-signed at every time? Buckmaster, De Lellis, Székelyhidi and Vicol (2019) answer the first part yes (THEORY-776, Proposed); the second, at every point in space, needs a local energy inequality they do not prove.
  • Does any vanishing-viscosity limit of Navier–Stokes fall in this class with non-zero flux? Only a theorem about Navier–Stokes would settle it.
  • What does the record’s flux question (A5’s four conditions) say of a flux that exists only during a construction’s lifetime and is produced by non-interacting waves? Applying the test to these solutions would be a sharp negative example.

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