Source
Eyink and Aluie (2009), Physics of Fluids 21:115107, LIT-1239, read in full in NOTE-969 from arXiv:0909.2386v1. The bounds are its (23) and (24). The paper sketches them heuristically and refers to Eyink (2005), LIT-1252 (registered unread), for the proofs. The derivation below is the record’s, written from the hypothesis the paper states as its (20).
What was actually shown
Setting. u is a velocity field on a periodic or unbounded domain in d dimensions, ⟨·⟩ is the space average, and ‖f‖_p = ⟨|f|^p⟩^(1/p). G is a filter kernel: smooth, with ∫G = 1, and decaying fast enough that ∫|G(s)| |s|^(2σ) ds and ∫|∇G(s)| |s|^σ ds are finite. Write G_ℓ(r) = ℓ^(−d)G(r/ℓ), u_ℓ = G_ℓ ∗ u, u′_δ = u − u_δ, τ_ℓ(v, w) = (vw)_ℓ − v_ℓ w_ℓ, and
Π_ℓ(u, v, w) = −∇u_ℓ : τ_ℓ(v, w),
the subgrid energy flux across ℓ, a source in the small-scale and a sink in the large-scale energy budget. Hypothesis (H): ‖δu(r)‖₃ ≤ C₀ u_rms (|r|/L)^σ for every r, where δu(r)(x) = u(x + r) − u(x).
Step 1. Everything is an increment. Since ∫G = 1 and ∫∇G = 0:
- ∇u_Δ(x) = −∫(∇G_Δ)(r) δu(r)(x) dr, and (∇G_Δ)(r) = Δ
- u′_δ(x) = −∫G_δ(r) δu(r)(x) dr;
- τ_ℓ(v, w) = ∫G_ℓ(r) δv(r)δw(r) dr − (∫G_ℓ δv dr)(∫G_ℓ δw dr). Expanding both terms, the products f·g_ℓ and f_ℓ·g cancel and what is left is (vw)_ℓ − v_ℓw_ℓ. The paper uses this form, after Vreman et al., for the positivity of the subgrid energy.
Step 2. Norm bounds from (H). By Minkowski’s inequality, then (H), then the change of variable r = Δs:
- ‖∇u_Δ‖₃ ≤ ∫|∇G_Δ(r)| ‖δu(r)‖₃ dr ≤ C₁ u_rms Δ^(σ−1) L^(−σ). Filtering again at ℓ multiplies the norm by at most ‖G‖₁ (Young), so the same bound holds for ∇(u_Δ)_ℓ. This is the paper’s (21) in gradient form.
- ‖u′_δ‖₃ ≤ ∫|G_δ(r)| ‖δu(r)‖₃ dr ≤ C₂ u_rms (δ/L)^σ. This is (22).
- ‖τ_ℓ(u, u)‖{3/2} ≤ ∫|G_ℓ(r)| ‖δu(r)‖₃² dr + (∫|G_ℓ(r)| ‖δu(r)‖₃ dr)² ≤ C₃ u_rms² (ℓ/L)^(2σ), using ‖|f|²‖{3/2} = ‖f‖₃². This is (15).
- |τ_ℓ(f, f)| ≤ (|G| ∗ |f|²) + |f_ℓ|², so ‖τ_ℓ(u′_δ, u′δ)‖{3/2} ≤ (‖G‖₁ + ‖G‖₁²)‖u′_δ‖₃² ≤ C₄ u_rms² (δ/L)
Step 3. Hölder. |⟨a : b⟩| ≤ ‖a‖₃ ‖b‖_{3/2}. So
- large-scale strain: |⟨Π_ℓ(u_Δ, u, u)⟩| ≤ C (u_rms³/L) (ℓ/L)^(3σ−1) (ℓ/Δ)
- small-scale stress: |⟨Π_ℓ(u, u′_δ, u′_δ)⟩| ≤ C (u_rms³/L) (ℓ/L)^(3σ−1) (δ/ℓ)
The exponents check: Δ^(σ−1)ℓ^(2σ) = ℓ^(3σ−1)(ℓ/Δ)^(1−σ), and ℓ^(σ−1)δ^(2σ) = ℓ^(3σ−1)(δ/ℓ)^(2σ). At σ = 1/3 the prefactor is u_rms³/L, of order the mean dissipation ε, and both exponents are exactly 2/3. These are the paper’s (23) and (24). The paper writes them with ε, which is right only at σ₃ = 1/3; the general form is the record’s.
Where 0 < σ < 1 enters. The large-scale factor (ℓ/Δ)^(1−σ) vanishes as Δ/ℓ → ∞ only if σ < 1. The small-scale factor (δ/ℓ)^(2σ) vanishes as δ/ℓ → 0 only if σ > 0. A field rough enough for σ ≤ 0 is not ultraviolet-local, and one smooth enough for σ ≥ 1 is not infrared-local, by this argument.
Checked numerically (NOTE-969): on a synthetic one-dimensional Gaussian field with σ₂ = 1/3, the L² share of the large-scale filtered field in an increment fell as (r/Δ)^0.62, and that of the high-passed field as (δ/r)^0.36. The predictions were 2/3 and 1/3. That is a check on the increment estimates behind (21)–(22) for one field, not on Navier–Stokes.
What could have come out otherwise. A flux built from the absolute velocity, rather than from increments, would not have been bounded this way. The large-scale velocity itself is not small, and sweeping by it is non-local. The bound holds because the Galilean-invariant flux depends only on increments within about ℓ, and the hypothesis makes increments local.
What this does not say
- That turbulence satisfies (H). The paper takes the scaling of structure functions, with 0 < σ_p < 1 and σ₃ ≈ 1/3, from experiment and simulation. It is not proved for Navier–Stokes solutions. The result is a conditional inequality.
- That non-local contributions are a small fraction of the flux. Turning the bounds into fractions needs |⟨Π_ℓ⟩| bounded below, of order ε, across the range. The paper assumes this (“it must be assumed that the mean flux is non-zero”). It is realistic in a forced constant-flux inertial range, and it is not shown here.
- The 4/3 exponent. Kraichnan’s faster decay, ε(ℓ/Δ)^(1+ζ₁), rests on a decorrelation estimate and a multiplicative-cascade or fusion-rule ansatz. The paper calls this “plausible but non-rigorous”. Its 512³ simulation gives signed-flux slopes of 1.1 (large scales) and 1.9 (small scales), which bracket 4/3 rather than confirm it.
- Pointwise statements. The paper’s pointwise version uses local Hölder exponents, and the authors say themselves that this “cannot make precise statements about the fractional contributions”. Only the space-averaged bounds are re-derived here.
- Locality in space. “Local” means local in scale: the flux across ℓ is set mainly by modes within a bounded ratio of ℓ. Transport of energy in space by sweeping is dominated by large scales, and the paper says so.
- Weak coupling. Adjacent scales are strongly coupled: in the integrated band budget, d/dt E_n = Π_(n−1) − Π_n, and in a constant-flux range the whole throughput ε crosses every scale ℓ_n, and by locality mostly between neighbouring bands. Locality says the coupling falls off with scale ratio, as a power of it, that is geometrically in band number. It does not say any band is weakly coupled to its neighbours.
- Anything about units or organization. The result concerns transfer between scales of one field. The decomposition into scales is the analyst’s choice of filter, ratio and outer scale. Nothing in it picks out a subsystem, a boundary in space, or something that persists while its constituents are replaced. Reading flux as a form of unity is the exchange’s move (A3, A5 of 2026-10-19), not the paper’s.
Connections
- THEORY-661 and CLAIM-157. Near-decomposability is a block-diagonal interaction pattern with a gap: strong within blocks, weak, of order ε, between them. Scale locality is a banded pattern: strongest between neighbouring scales, decaying as a power of the scale ratio, with no gap anywhere along the chain. Cut the chain of bands anywhere and the flux across the cut is the whole throughput. So no contiguous group of scales is a gap-isolated block, and CLAIM-157’s test would find none. The two are different organizing principles. This contrast is the record’s; the paper does not mention near-decomposability.
- QUESTION-219. The paper provides what A5 required before any flux test: a balance law for a specified quantity, with an exact band-by-band budget (its (11) and the telescoping flux equations). It does not touch the question asked, whether a pattern of transfer picks out a unit.
- LIT-1252. Eyink (2005) holds the original proofs, including the pointwise and all-subsets versions this account does not re-derive.
- THEORY-749 (Constantin, E and Titi). The same Besov hypothesis makes the flux across ℓ at most of order ℓ^(3σ−1), which vanishes as ℓ → 0 when σ > 1/3. So the non-zero mean flux that this account’s reading as fractions assumes is consistent only with σ₃ ≤ 1/3, and Kolmogorov’s 1/3 is the largest exponent at which a cascade can carry energy to arbitrarily small scales.