Source
Isett (2018), Annals of Mathematics 188(3):871–963, LIT-1255, skimmed in NOTE-974 from arXiv:1608.08301v2. The existence statement is its Theorem 1. The energy balance used for the flux is Constantin, E and Titi’s (11) (LIT-1248), as re-derived for any weak solution in THEORY-749, Steps 4–5.
What was actually shown
The theorem (Isett, Theorem 1). For any α < 1/3 there is a nonzero weak solution (v, p) of ∂t v + ∇·(v⊗v) + ∇p = 0, ∇·v = 0 on R × (R/Z)³, with v ∈ C^α{t,x} and p ∈ C^{2α}{t,x}, such that v ≡ 0 outside a finite time interval. Here C^α{t,x} means |f(t + Δt, x + Δx) − f(t, x)| ≤ C(|Δt| + |Δx|)^α uniformly. Since v is continuous, nonzero somewhere and zero at early and late times, E(t) = ½∫|v(t)|² is not constant.
How it is proved (outline, not checked line by line).
- An Euler–Reynolds flow (v, p, R) solves ∂_t v + ∇·(v⊗v) + ∇p = ∇·R with a symmetric stress R. Its frequency–energy levels (Ξ, e_v, e_R) bound ‖∇^k v‖ ≤ Ξ^k e_v^{1/2} and ‖∇^k R‖ ≤ Ξ^k e_R.
- The Main Lemma (Lemma 2.1) turns such a flow into one with levels (CNΞ, log Ξ̂·e_R, (log Ξ̂)^{5/2} e_v^{1/2}e_R^{1/2}/N), where Ξ̂ = Ξ(e_v/e_R)^{1/2}, for any N ≥ max{Ξ^η, (e_v/e_R)^{1/2}}. The correction has ‖V‖ ≲ (log Ξ̂)^{1/2}e_R^{1/2}, and the time support grows by at most Ξ⁻¹e_v^{−1/2}. Up to logarithms this is the “Ideal Case Scenario” of Isett’s 2013 work: the new stress is e_v^{1/2}e_R
- The Main Lemma is three sublemmas: a regularization (Lemma 3.1); a gluing approximation (Lemma 3.2), which replaces v by exact Euler solutions started at times 8θI and glued with a partition of unity in time, so that the new stress is a sum of pieces R_I supported in separated intervals of length about θ; and a convex integration step (Lemma 3.3), which cancels each R_I with one Mikado-flow wave, stationary pipe flows in six lattice directions, transported by the coarse flow. Waves from different I never overlap in time, and Mikado pipes of different directions never overlap in space, so the cross terms that limited earlier schemes to 1/5 vanish.
- The iteration (Section 18, read in full). The base case comes from Isett’s earlier Main Lemma, which forces the energy profile close to a prescribed e(t) with e(0) = 1. That gives ‖v₍₀₎‖ ≥ 0.9, while all later corrections sum to at most 0.01, so the limit is nonzero. The parameters evolve by a lower-triangular matrix with eigenvalues 1 + δ, δ/2 and 1, so they grow double-exponentially. The C^α norms of the corrections decay double-exponentially when [1/2 − δα, 1/2, α]·ψ₊ < 0, ψ₊ being the (1 + δ)-eigenvector. That condition tends to −1/2 + 3α/2 < 0, i.e. α < 1/3, as δ → 0. The sum of the time-support increments converges, so the support stays compact.
Checked by hand here. The 1/3 can be read off the Main Lemma’s scaling, ignoring logarithms. Let stage k have frequency λ_k and velocity energy e_v = δ_k = λ_k^{−2α}, let its stress be the energy of the next correction, e_R = δ_{k+1}, and take N = λ_{k+1}/λ_k with λ_{k+1} = λ_k^b. The new stress is then δ_{k+2} = δ_k^{1/2}δ_{k+1}^{1/2}λ_k/λ_{k+1}. Comparing exponents of λ_k gives −2αb² = −α − αb + 1 − b, so α(2b² − b − 1) = b − 1 and α = 1/(2b + 1). As b ↓ 1, α ↑ 1/3. Frequencies that grow only slightly faster than geometrically therefore reach any α < 1/3, and no b > 1 reaches 1/3. The eigenvector computation of Section 18 was redone by hand and gives the same limit.
The consequence for the flux (the record’s). Mollify in space at scale ε, as in THEORY-749. For any weak solution in C([0, T]; L²) ∩ L³, which a continuous v on the torus is, Steps 4–5 there give, for s < t,
½∫|v^ε(t)|² − ½∫|v^ε(s)|² = ∫_s^t Π_ε(τ) dτ, Π_ε = ∫τ_ε : ∇v^ε, τ_ε = (v⊗v)^ε − v^ε⊗v
Π_ε is the rate at which energy enters the resolved field v^ε from the scales below ε. Since v(t) ∈ L², the left side tends to E(t) − E(s) as ε → 0, for each s and t. Take s before the support, where E(s) = 0, and t with E(t) > 0. Then ∫_s^t Π_ε → E(t) > 0. Take instead s with E(s) > 0 and t after the support. Then ∫_s^t Π_ε → −E(s) < 0. So the time-integrated flux across ε survives the limit ε → 0, and its sign differs between the two intervals. First the resolved field gains energy from vanishing scales, then it loses energy to them. Nothing more local is claimed. Whether Π_ε(τ) converges pointwise in time, and where in space the transfer happens, would need Duchon and Robert’s local balance, which the record does not hold.
What could have come out otherwise: by THEORY-749, any weak solution regular enough (Besov 3, ∞ above 1/3) has ∫_s^t Π_ε → 0. Isett’s solutions escape this only because C^α with α < 1/3 lies below the threshold. That is consistent, and it shows that the threshold is sharp in the exponent.
What this does not say
- Not that the surviving flux is a dissipative cascade. Isett’s solutions are not admissible: their energy rises from zero before it falls. Weak solutions with energy that only decreases, at every α < 1/3, are a later theorem (Buckmaster, De Lellis, Székelyhidi and Vicol, 2019), filed as THEORY-776 (Proposed). With it, the record can say that energy can cross vanishing scales one way for the whole box; this entry alone says only that it can cross in either direction.
- Not that all rough solutions transfer energy. The rough shear flows of THEORY-749 conserve energy and have Π_ε = 0 at every ε. The theorem is an existence result. Roughness permits a surviving flux; it does not force one.
- Not anything about turbulence or Navier–Stokes. The solutions are built by convex integration, starting from rest, not as limits of viscous flows. They also show non-uniqueness: v and 0 share the same initial data. Whether inviscid limits of Navier–Stokes belong to this class is open.
- Not the endpoint, the plane or the whole space. α = 1/3 is not covered. The paper says the proof extends to dimensions d ≥ 3 but not to d = 2, for lack of a substitute for Mikado flows, and does not give finite-energy solutions on R³.
- Not a checked proof. The 90-page estimates were not read. The entry rests on the paper’s having proved what it states. The statement is the one the field takes as settled, but the record’s check is of the architecture and the exponent bookkeeping only.
Connections
- QUESTION-219: answers in part. It supplies the half of A3’s point that THEORY-749 could not: below Onsager’s exponent a coarse-grained flux can survive the limit of vanishing scale. The question’s sentence that this “is the negative half …, which the record does not hold” can now cite this entry, with the qualification that the surviving flux here takes both signs. It does not bear on whether a pattern of transfer picks out a unit. A5’s warning that persistent flux alone is not an individual is untouched. Here the flux is not even persistent; it exists only while the solution does.
- THEORY-749. Its “What this does not say” names this result as the negative half. Together the two make 1/3 sharp for the Hölder class: above it, for every solution, the integrated flux vanishes; below it, for some solutions, it does not.
- THEORY-754. Eyink and Aluie’s conditional account assumes a non-zero mean flux through an inertial range, which NOTE-982 found consistent only with exponents at most 1/3. This entry shows that weak solutions with a non-vanishing flux exist in that range. It does not show that they have a constant-flux inertial range, or the scaling that account assumes.