Source
Giri, Kwon and Novack, “The L³-based strong Onsager theorem”, Annals of Mathematics 204(1):265–421 (2026), LIT-1289, skimmed in NOTE-1022 from arXiv:2305.18509v3. The existence statement is its Theorem 1.1, and the identity D[u] = E is in the proof of that theorem. The coarse-grained energy balance is Constantin, E and Titi’s (LIT-1248), re-derived in THEORY-749, taken here in its local form. The whole-box predecessor is Buckmaster, De Lellis, Székelyhidi and Vicol (LIT-1300, THEORY-776).
What was actually shown
The theorem (Theorem 1.1). “For any fixed β ∈ (0, 1/3) there exist weak solutions to the Euler equations (1.1) which belong to C⁰([0, T]; (B^β_{3,∞} ∩ L^{1/(1−3β)})(T³)), satisfy the local energy inequality (1.6), and dissipate the total kinetic energy.” Here (1.6) is
∂_t(½|u|²) + div((½|u|² + p)u) = −D[u] ≤ 0
in the sense of distributions, with D[u] the Duchon–Robert measure (1.5), and B^β_{3,∞} the Besov space of (1.7): ‖u(· + z) − u‖_{L³} ≤ C|z|^β. The paper calls this the L³-based strong Onsager conjecture, part (b); part (a), the local energy equality above 1/3, is Duchon and Robert’s.
What the proof gives beyond the statement. The inductive Proposition 2.11 takes a “continuous positive function E(t, x) ≥ 0” and carries it unchanged from step to step in the relaxed local energy identity (2.8),
∂_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,
whose error terms R_q, π_q, φ_q tend to zero (footnote 6: “The dissipation measure E is the Duchon-Robert measure of the limiting solution”). The first step fixes E. Step 1 of the proof takes
u₀ = e(t)ρ₀(x₁, x₂)e₃, e(t) = c·exp(−τ₀⁻¹(t + 1)), p₀ = φ₀ = 0,
with ρ₀ a smooth function supported in a periodic family of straight pipes along e₃ of thickness λ_{n̄−1}⁻¹, written as ρ₀ = div a for a vector field a(x₁, x₂), with R₀ the trace-free matrix whose only nonzero entries are e′(t)a₁, e′(t)a₂ in the third row and column, and π₀ a spatial constant. Step 2 concludes, for the limit,
∂_t(½|u|²) + div((½|u|² + p)u) = e(t)e′(t)|ρ₀|² ≤ 0,
“In particular, the strict local energy inequality holds in the interior of supp(ρ₀)”. So D[u] = E = τ₀⁻¹e(t)²ρ₀(x₁, x₂)²: smooth, nonnegative, independent of x₃, decaying in time at rate 2τ₀⁻¹, positive where ρ₀ ≠ 0 and zero outside the pipes. The total energy satisfies (d/dt)½∫|u|² = −τ₀⁻¹e(t)²∫ρ₀² < 0 at every time.
How it is proved (read in outline; estimates not checked). An L³-based convex integration with intermittent pipe flows at the “Goldilocks” intermittency r_q = λ_{q+n̄/2}/λ_{q+n̄}. Each new increment corrects only the lowest-frequency stress and current errors (R_q^q, φ_q^q), so successive increments overlap in frequency (“partial wavelet sums”); they are kept apart in space by “intermittent Mikado bundles” placed to dodge earlier pipes (Section 4), and every error is dominated pointwise by an “intermittent pressure” π_q built in advance (Sections 2.4 and 6). The proofs of Section 3 are in a companion paper (Giri, Kwon and Novack, Annals of PDE 10:19, 2024, with errata on the third author’s site).
Checked here.
- The first step. For u₀ = e(t)ρ₀e₃ with ρ₀ = div a, a = a(x₁, x₂), and R₀ as above: ∂_t u₀ + div(u₀⊗u₀) = div R₀, R₀ is trace-free, and ∂_t(½|u₀|²) + div(½|u₀|²u₀) = e e′ρ₀². With û₀ = 0, φ₀ = 0 and π₀ constant, the right side of (2.8) is −E, so E = −e e′ρ₀². Done symbolically (sympy) for generic a and e.
- The heuristics of §1.2. The Hölder Nash error gives (b − 1)(β(2b + 1) − 1) ≤ 0, β ≤ 1/(2b + 1); the Nash current error in a homogeneous scheme gives β ≤ 1/(3b + 1) → 1/4; with the extra factor r_{q+1}^{−2/3} of the L³ scheme the condition is (b − 1)(9bβ + 3β − b − 3)/3 ≤ 0, i.e. β ≤ (b + 3)/(3(3b + 1)) → 1/3 as b → 1. Checked symbolically. The extracted text of the last threshold is garbled; this is what its preceding line gives.
- E is carried unchanged. The algebra of (5.4)–(5.8) and of Lemma 6.10’s (6.41), where the new errors and the time-only functions m(t) are absorbed into φ_{q+1} and R_{q+1} and −E passes through untouched, was followed.
The consequence for the local flux (the record’s). u ∈ C([0, T]; L³) and, from the paper, p ∈ C([0, T]; L^{3/2}). Mollify in space at scale ℓ. Then u^ℓ is C¹ in time with values in smooth fields and
∂_t u^ℓ + div(u^ℓ⊗u^ℓ) + ∇p^ℓ = −div τ_ℓ, τ_ℓ = (u⊗u)^ℓ − u^ℓ⊗u
Dotting with u^ℓ, using div u^ℓ = 0 and the symmetry of τ_ℓ:
∂_t(½|u^ℓ|²) + div((½|u^ℓ|² + p^ℓ)u^ℓ + τ_ℓu^ℓ) = τ_ℓ : ∇u
The right side is the local rate at which the resolved field u^ℓ gains energy from scales below ℓ; its integral over T³ is THEORY-749’s Π_ℓ. As ℓ → 0, uniformly in t (the range of t ↦ u(t) is compact in L³): |u^ℓ|² → |u|² in L^{3/2}; |u^ℓ|²u^ℓ → |u|²u and p^ℓu^ℓ → pu in L¹; and τ_ℓ → 0 in L^{3/2}, so τ_ℓu^ℓ → 0 in L¹. Hence the left side tends in distributions to ∂_t(½|u|²) + div((½|u|² + p)u) = −E, and so
−τ_ℓ : ∇u^ℓ → E ≥ 0 in D′((0, T) × T³).
For every nonnegative test function ψ(t, x), the energy passed from the resolved field to scales below ℓ in the region ψ weights tends to ∫ψE ≥ 0. It is strictly positive whenever ψ’s support meets the set where ρ₀ ≠ 0, and zero when ψ is supported outside the pipes. The transfer is one-way at every place and time in this sense, not only for the whole box.
What could have come out otherwise: by Duchon and Robert, D[u] = 0 for solutions in C⁰_tB^α_{3,∞} with α > 1/3; and THEORY-776’s solutions, built in Hölder spaces with prescribed total energy, satisfy no proved local sign at all.
What this does not say
- Not one-signed at any positive scale. The convergence is of −τ_ℓ : ∇u^ℓ tested against fixed functions as ℓ → 0. At fixed ℓ the flux density may take both signs; backscatter at finite resolution is neither shown nor excluded. “One-way at every place” means that the limiting measure is nonnegative.
- Not that the equations choose the place or the rate. E is fixed by the first approximate solution and passes through the iteration unchanged. Proposition 2.11 accepts any continuous E ≥ 0 for which a starting triple exists, but the paper builds only this one, so the record cannot say which dissipation densities are attainable. In the same way that THEORY-776 chooses the total energy profile, this construction chooses the local dissipation density. The sign is not free here: E ≥ 0 is the local inequality the theorem was built to satisfy, and Proposition 2.11 requires it.
- Not the dissipation turbulence phenomenology expects. The dissipation is a smooth density on a periodic array of straight pipes, set at the coarsest active scale, not a measure concentrated on a thin or fractal set. All the later, finer increments together contribute no net local dissipation in the limit. The paper motivates L³ and intermittency by structure-function exponents of turbulent velocity fields, and its solutions are intermittent in that sense; their dissipation is not.
- Not that the solution is rough everywhere. Nothing is shown about where u fails to lie in B^{1/3+}_{3,∞}. Where E > 0 it must be rough, since a local form of Duchon and Robert’s estimate (not re-derived) gives D[u] = 0 wherever u is locally smoother than 1/3. Outside the pipes the record cannot say.
- Not Hölder, not bounded. u ∈ C⁰_t(B^β_{3,∞} ∩ L^p) for p = 1/(1 − 3β), which is finite for each β < 1/3. Hölder solutions with the local inequality are known, per the paper, only for β < 1/7 (De Lellis and Kwon), after Isett’s α < 1/15 (THEORY-821, Proposed); Isett’s strong C⁰ Onsager conjecture is left open. Both constructions prescribe the dissipation. Here it sits in pipes for all time; in Isett’s it is uniform in space and confined to a final drain to zero.
- Not Navier–Stokes. The paper notes that L³ limits of suitable weak Navier–Stokes solutions satisfy the local inequality, so such flows “may” be inviscid limits. It does not show that these ones are.
- Not uniqueness or an initial-value problem. No datum is prescribed, and the energy is never at rest (e > 0).
- Not a checked proof. The estimates of Sections 3–6 and the companion paper’s results quoted in Section 3 were not checked.
Connections
- QUESTION-219: answers in part. It removes the first qualification THEORY-776 put on the dissipative cascade. Below 1/3, a coarse-grained energy flux can survive the limit of vanishing scale and run one way into the small scales at every place and time, in the distributional sense, and not only for the whole box. The second qualification stands and is made sharper. The direction is fixed by the local inequality, and where and at what rate the transfer happens are fixed by the first step, so the dynamics selects nothing. On whether a pattern of transfer picks out a unit, the bearing is negative again. The set where the flux is nonzero is a periodic array of pipes the constructor placed. Inside it the rate is prescribed and decays exponentially. Nothing in it restores itself or depends on internal organization (A5’s conditions (3) and (4)).
- THEORY-776. Its “What this does not say” names this paper for the local inequality. The two differ in class: Hölder C^β (BDSV) against L³ Besov (here). They also differ in what is prescribed: the total energy e(t) against the local dissipation density E(t, x). Neither result contains the other.
- THEORY-749. The class C⁰_tB^β_{3,∞} lies in the L³_tB^α_{3,∞} of Constantin, E and Titi, so the 1/3 threshold is sharp in their own scale of spaces, for the local balance as well as the total.
- THEORY-763. Isett’s flux takes both signs over time; BDSV’s has one sign for the box; this one has one sign at every place, in the limit.
- THEORY-754. Eyink and Aluie’s conditional locality concerns a non-zero flux through an inertial range of scales. This entry gives a non-zero, one-signed flux density but says nothing about how it is spread over scales.