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NOTE-1022: Giri, Kwon and Novack — The L³-based strong Onsager theorem

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PaperLIT-1289 — The L³-based strong Onsager theorem

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Contribution

Isett (LIT-1255) and Buckmaster, De Lellis, Székelyhidi and Vicol (LIT-1300) settled the negative half of Onsager’s conjecture for the total energy, in Hölder spaces. Whether rough solutions below 1/3 can satisfy the local energy inequality, the condition that limits of suitable Navier–Stokes solutions satisfy, was open except for β < 1/7 (De Lellis and Kwon, Hölder). This paper settles it in the L³-based scale: for every β < 1/3 there are weak solutions in C⁰_t(B^β_{3,∞} ∩ L^{1/(1−3β)}) with D[u] ≥ 0 and strictly decreasing total energy. The method is new: an intermittent convex integration measured in L³, with velocity increments that overlap in frequency.

Key insight

The local energy inequality brings in a “current” error, and in a homogeneous (Hölder) scheme its Nash term caps the exponent at 1/4. Making the building blocks intermittent, concentrated on thin pipes, weakens that error by a factor r^{2/3} in L¹. With the right amount of intermittency (r_q = λ_{q+n̄/2}/λ_{q+n̄}, the “Goldilocks” ratio) the cap rises to 1/3. Keeping that ratio at every step forces each increment to reach well above the frequency of the error it corrects. Increments of different steps then overlap in frequency, which homogeneous schemes forbid. The paper restores the separation in space: new pipes are bundled and placed to dodge all recent ones. The local energy balance itself is held exact at every step, with a fixed dissipation term −E carried along, so the limit’s Duchon–Robert measure is whatever E was at the start.

Assumptions

  • Domain. The periodic box T³ = [−π, π]³, a finite interval [0, T]. No initial-value problem: the datum is an output.
  • Notion of solution. Distributional solutions of ∂_t u + (u·∇)u + ∇p = 0, div u = 0, with u ∈ C⁰_t L³ and, from the proof, p ∈ C⁰_t L
  • Regularity class. C⁰([0, T]; B^β_{3,∞} ∩ L^{1/(1−3β)}): L³-Besov in space, continuous in time, not Hölder and not shown bounded. The abstract’s C⁰_t(W^{1/3−,3} ∩ L^{∞−}) is the same statement as β → 1/3.
  • Local energy inequality (1.6). ∂_t(½|u|²) + div((½|u|² + p)u) = −D[u] ≤ 0 in distributions, D[u] the Duchon–Robert measure (1.5).
  • Exponent. §2.1 fixes β ∈ [1/7, 1/3), Proposition 2.11 says β ∈ (0, 1/3); either way the proof of Theorem 1.1 runs the iteration at some β̄ ∈ (β, 1/3) close to 1/3, so smaller β come for free.

Key results

  • Theorem 1.1. For any β ∈ (0, 1/3) there are weak solutions in C⁰([0, T]; B^β_{3,∞} ∩ L^{1/(1−3β)}) that satisfy (1.6) and dissipate the total kinetic energy.
  • Proposition 2.11 (the iteration). Given (u_q, p_q, R_q, φ_q, −π_q) satisfying the Euler–Reynolds system ∂_t u_q + div(u_q⊗u_q) + ∇p_q = div(R_q − π_q Id) and the relaxed local energy identity ∂_t(½|u_q|²) + div((½|u_q|² + p_q)u_q) = (∂_t + û_q·∇)κ_q + div((R_q − π_q Id)û_q) + div φ_q − E (2.8), with κ_q = ½tr(R_q − π_q Id) and the many inductive bounds of §§2.3–2.6, there is a next step with the same E. Holds when: n̄ and b satisfy (2.1)–(2.2) and a ≥ a*.
  • The limiting identity (Step 2 of the proof). With the base case below, ∂_t(½|u|²) + div((½|u|² + p)u) = e(t)e′(t)|ρ₀|² ≤ 0, so D[u] = −e e′ρ₀² = τ₀⁻¹e(t)²ρ₀(x₁, x₂)². “In particular, the strict local energy inequality holds in the interior of supp(ρ₀), which leads to the total kinetic energy dissipation.”
  • Base case (Step 1). u₀ = e(t)ρ₀(x₁, x₂)e₃ with e(t) = c·exp(−τ₀⁻¹(t + 1)) with c a small constant built from the parameters (Γ_{n̄}^{−100}δ_{2n̄} times an intermittency factor); ρ₀ a pipe flow along e₃ of thickness λ_{n̄−1}⁻¹, periodic, written as a d-fold divergence; R₀ trace-free with ∂t u₀ = div R₀; p₀ = φ₀ = 0; π₀ = Σ_k Γ_kδ{k+n̄}, a constant. Checked here.
  • Regularity bounds. ‖ŵ_q‖{C⁰B^{β′}{3,∞}} ≤ λ_q^{−β+(b−1)/2+β′} (2.47), summable for β′ < β̄ − (b − 1)/2; ‖ŵ_q‖_{L^p} summable for p < p*(β̄, b) (2.48), which (2.46) makes larger than 1/(1 − 3β).
  • Heuristic thresholds (§1.2). Hölder Nash error: β ≤ 1/(2b + 1). Homogeneous Nash current error: β ≤ 1/(3b + 1) → 1/4. Intermittent L³ Nash current error: β ≤ (b + 3)/(3(3b + 1)) → 1/3. Recomputed here.

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-772, filed Proposed.

Method

  1. Fix β̄ ∈ (β, 1/3), n̄ ∈ 6N large and b close to 1 (2.1)–(2.2), (2.46). Frequencies λ_q ≈ a^{b^q}, amplitudes δ_q = λ_q^{−2β̄}, intermittency r_q = λ_{q+n̄/2}Γ_q/λ_{q+n̄}.
  2. Start from the pipe flow u₀ = e(t)ρ₀e₃, which sets E.
  3. At step q, correct only the lowest-frequency errors R_q^q, φ_q^q with ŵ_{q+n̄} = Σ a(∇û_q, φ_q^q) B_φ∘Φ + Σ a(∇û_q, φ_q^q, R_q^q) B_R∘Φ: the current part takes amplitude (−2φ_q^q)^{1/3}, the stress part a matrix square root built from r_q^{1/3}(−2φ_q^q)^{1/3} and the negative-definite R_q^q − π_q^q Id (1.18). The B are intermittent Mikado bundles, thin intermittent pipes at frequencies between λ_{q+n̄/2} and λ_{q+n̄} multiplied by a thicker “bundling pipe” at λ_{q+1}, transported by local flow maps Φ of û_q, cut off in space-time by velocity, pressure and checkerboard cutoffs.
  4. Place each bundle to dodge every increment of the last n̄ steps (Lemmas 4.1–4.2), so cross terms with u_q − û_q vanish.
  5. Sort the new stress and current errors into frequency bins λ_{q+1}, …, λ_{q+n̄} with a synthetic, compactly supported Littlewood–Paley decomposition, invert the divergence locally, and absorb the time-only means m(t) into the stress as (2/3)m Id.
  6. Build the new intermittent pressure π_{q+1} from all the pressure increments, including “anticipated” rescaled copies, so that π dominates every error and its derivatives pointwise (Section 6).
  7. In the limit R_q, π_q → 0 in C⁰L^{3/2} and φ_q → 0 in C⁰L¹, and (2.8) becomes (1.4) with D[u] = E.

Concepts

  • Duchon–Robert measure D[u]: the distribution defined by (1.5), equal to minus the left side of the local energy balance; a non-negative D[u] is the local energy inequality.
  • L³-based strong Onsager conjecture: (a) local energy equality above 1/3 in C⁰_tB^β_{3,∞}; (b) below 1/3, solutions in that class with the local inequality and dissipated total energy.
  • Relaxed local energy inequality / identity: (1.11), (2.8), the energy analogue of the Euler–Reynolds system, with a current error φ_q.
  • Goldilocks intermittency: r_q = (λ_q/λ_{q+1})^{1/2} in Novack and Vicol’s scheme, here r_q = λ_{q+n̄/2}/λ_{q+n̄}.
  • Partial wavelet sum: u_q = û_q + ŵ_{q+1} + … + ŵ_{q+n̄−1}, increments overlapping in frequency, unlike a partial Fourier sum.
  • Intermittent Mikado bundle: a thin intermittent pipe flow multiplied by a thicker, mildly intermittent “bundling pipe” whose placement can be shifted.
  • Dodging: choosing supports of new increments disjoint from those of recent ones.
  • Intermittent pressure π_q: a positive scalar such that π_q Id − R_q is positive definite and that bounds all errors pointwise.

Connections

The scheme extends Novack and Vicol’s intermittent Onsager theorem (C⁰_t H^{1/2−}, non-conservative) and the intermittent Navier–Stokes and Euler schemes of Buckmaster, Vicol and others, with Modena and Székelyhidi’s intermittent Mikado flows. The earlier local-inequality results are De Lellis and Székelyhidi’s (admissible non-uniqueness), Isett’s strong C⁰ conjecture and globally dissipative continuous flows, and De Lellis and Kwon’s Hölder β < 1/7. The positive half, local energy equality above 1/3, is Duchon and Robert’s, after Constantin, E and Titi (LIT-1248) and Eyink. The Hölder negative half is Isett (LIT-1255) and Buckmaster, De Lellis, Székelyhidi and Vicol (LIT-1300). The technical core, Section 3, is the authors’ companion paper, “A wavelet-inspired L³-based convex integration framework for the Euler equations” (Annals of PDE 10:19, 2024), not held.

Bearing on the record

  • THEORY-772 is filed from this reading, Proposed. It states Theorem 1.1 with what the proof adds, D[u] = E, and the record’s consequence. The local coarse-grained flux −τ_ℓ : ∇u^ℓ tends to E ≥ 0 in distributions. So energy leaves the resolved field for vanishing scales at every place and time in the limit: in the pipes where ρ₀ ≠ 0, and nowhere else.
  • QUESTION-219. This is the reading NOTE-1013 named for the open question “Are there C^β solutions, β < 1/3, satisfying the local energy inequality?”. Its answer, with the precise terms the question asked for:
    • Class. L³-Besov, C⁰_t(B^β_{3,∞} ∩ L^{1/(1−3β)}), not C^β. For Hölder solutions, the local inequality is known only for β < 1/7.
    • Form. The inequality (1.6), D[u] ≥ 0 as a distribution, everywhere in T³ × (0, T). It is not a pointwise statement, but here D[u] is in fact a smooth function, −e e′ρ₀².
    • Nonzero and where. Positive where ρ₀ ≠ 0, inside a periodic array of straight pipes along e₃ of width about λ_{n̄−1}⁻¹. Zero outside them. Constant along the pipes and decaying exponentially in time. The paper’s “strict … in the interior of supp(ρ₀)” is loose: ρ₀ has zeros inside its support (it is a divergence, so of mean zero). D[u] still charges every open subset of that interior.
    • Hence the flux. One-signed locally in space and time, in the limit ℓ → 0 and in distributions. At no finite ℓ is a sign shown. This removes the first qualification THEORY-776 carried (“global, not local”), in a different class. It strengthens the second (“chosen, not forced”): here the constructor chooses the local density, not just the total. On whether a pattern of transfer picks out a unit, the bearing is negative. The dissipating set is placed by hand, and the rate is prescribed. All later fine-scale structure adds no net local dissipation. Nothing self-maintains in A5’s sense.
  • THEORY-776. Its “What this does not say” cites this paper as “reported to reach the local inequality” and unread. It can now cite THEORY-772, with the class difference: L³-Besov, not Hölder.
  • THEORY-749. This paper’s class lies inside Constantin, E and Titi’s L³_tB^α_{3,∞}. So their 1/3 is sharp in their own space, for the local balance as well as the total. Above 1/3, the local equality is Duchon and Robert’s.
  • THEORY-763. Isett: flux of both signs in time. BDSV: one sign, for the box. This paper: one sign at every place. The three now form a graded sequence in the record.
  • THEORY-754. Not touched. Nothing here concerns how the flux is spread over scales or whether it is constant through a range.
  • QUESTION-218 and QUESTION-222 (closure). As with the Hölder constructions, the relaxed system is a closure problem run backwards. Here the energy balance is relaxed too, with a current error φ_q beside the stress. A coarse field together with its stress and its current still does not determine the fine field. 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 “partial wavelet sum” is a construction device in which coarse-to-fine corrections overlap in scale and are separated in space. It is not a claim about any system.

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

Limitations

  • The theorem says less than the proof. Theorem 1.1 states D[u] ≥ 0 and dissipation. That D[u] equals the prescribed E is in the proof (footnote 6, Step 2). A reader of the statement alone would not learn where the dissipation is.
  • “Strict local energy inequality … in the interior of supp(ρ₀)”. Strict as a measure on open sets, not pointwise, since ρ₀ vanishes inside its support.
  • A slip in the introduction. De Lellis and Kwon’s solutions are said to satisfy “the strict local energy inequality (1.9)”. (1.9) is the Navier–Stokes inequality; (1.6) is meant.
  • Not self-contained. Section 3 quotes its results from the companion paper, whose errata the authors cite, and Section 4 leaves the general pipe direction “to the reader”.
  • Scope. Torus, finite time, not Hölder or bounded, no initial datum, no uniqueness or non-uniqueness for given data, nothing about Navier–Stokes, β = 1/3 not reached. Only one dissipation density is constructed.

Open questions

  • Hölder solutions, C^β with β < 1/3, satisfying the local inequality: Isett’s strong C⁰ Onsager conjecture. Open, per the paper.
  • Which dissipation densities E can be prescribed? Proposition 2.11 accepts any continuous E ≥ 0 with a starting triple. A result building such triples for general E, for instance concentrated on thin or fractal sets, would say whether the place of dissipation is as free as the total profile in THEORY-776.
  • Is the flux one-signed at small positive scales, or only in the limit? Only a pointwise or scale-resolved estimate on −τ_ℓ : ∇u^ℓ would say.
  • Are any such solutions limits of Navier–Stokes as ν → 0? Only a theorem about Navier–Stokes would settle it.

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