Source
Buckmaster, De Lellis, Székelyhidi and Vicol (2019), Communications on Pure and Applied Mathematics 72(2):229–274, LIT-1300, read in full in NOTE-1013 from arXiv:1701.08678v1. The existence statement is its Theorem 1.1. The energy balance used for the flux is Constantin, E and Titi’s (LIT-1248), as re-derived for any weak solution in THEORY-749, Steps 4–5. The construction is a modification of Isett’s (LIT-1255, THEORY-763).
What was actually shown
The theorem (Theorem 1.1). “Assume e : [0, T] → R is a strictly positive smooth function. Then for any 0 < β < 1/3 there exists a weak solution v ∈ C^β(T³ × [0, T]) to (1.1) such that ∫_{T³} |v(x, t)|² dx = e(t).” Here (1.1) is ∂_t v + v·∇v + ∇p = 0, div v = 0 on the torus T³ = R³/Z³, and C^β(T³ × [0, T]) is Hölder continuity in space and time jointly. Nothing restricts the sign of e′. Taking e′ < 0 gives a solution whose kinetic energy strictly decreases, which is what the paper calls dissipative and the title calls admissible.
How it is proved (read; estimates followed, not each re-derived).
- Inductive Proposition 2.1. A sequence of Euler–Reynolds flows (v_q, p_q, R̊_q), ∂t v_q + div(v_q⊗v_q) + ∇p_q = div R̊_q with R̊_q trace-free, at frequency λ_q = 2π⌈a^(b^q)⌉ and amplitude δ_q = λ_q^(−2β), satisfying ‖R̊_q‖₀ ≤ δ{q+1}λ_q^(−3α), ‖v_q‖₁ ≤ Mδ_q^(1/2)λ_q, ‖v_q‖₀ ≤ 1 − δ_q^(1/2), and the energy gap δ_{q+1}λ_q^(−α) ≤ e(t) − ∫|v_q|² ≤ δ_{q+1} (2.7)–(2.10). The step to q + 1 holds whenever 1 < b < (1 − β)/(2β) (2.11), with ‖v_{q+1} − v_q‖₀ + λ_{q+1}^(−1)‖v_{q+1} − v_q‖₁ ≤ Mδ_{q+1}^(1/2) (2.12), M a geometric constant. The iteration starts from (0, 0, 0).
- Each step is Isett’s three stages: mollify at scale ℓ; glue exact Euler solutions started at times t_i = iτ_q with a partition of unity in time, so that the new stress lives on disjoint intervals I_i of length about τ_q; then add Mikado-flow perturbations transported by the glued flow. Two changes from Isett. The glued stress is built with the standard inverse divergence R of De Lellis and Székelyhidi, so it stays trace-free, which is “crucial to control the energy”; this needs a commutator estimate for the advective derivative of negative-order operators (Proposition D.1). And the perturbations are placed in disjoint “squiggling” space-time regions whose time projections cover all of [0, T] (Lemma 5.3), so energy ρ_q(t) = ⅓(e(t) − δ_{q+2}/2 − ∫|v̄_q|²) can be added even where the stress vanishes. That second change is what lets the energy follow any prescribed e.
- Energy. Proposition 6.2 gives |e(t) − ∫|v_{q+1}|² − δ_{q+2}/2| ≲ δ_q^(1/2)δ_{q+1}^(1/2)λ_q^(1+2α)/λ_{q+1}. Since δ_q → 0, (2.10) forces ∫|v|² = e(t) in the limit, at every t.
- Regularity. (2.12) gives C⁰_tC^{β′}_x for every β′ < β by interpolation, and Section 2.2 recovers time regularity by a short mollification argument, so v ∈ C^{β″} jointly for every β″ < β < 1/3.
Checked here. The parameter condition (2.39), −β − βb + 1 − b + 2b²β < 0, factors as (b − 1)(β(2b + 1) − 1) < 0, i.e. b < (1 − β)/(2β), which is (2.11). A b > 1 exists if and only if β < 1/3. This is the same bookkeeping as Isett’s α = 1/(2b + 1) in THEORY-763, read the other way. The factorization was confirmed numerically on random (b, β). The derivation of (2.39) from (2.38), by taking logarithms base λ_q, was redone by hand, as was the claim that (2.10) with δ_{q+1} → 0 gives the exact profile.
The consequence for the flux (the record’s). Mollify in space at scale ε, as in THEORY-749. v is continuous on [0, T] × T³, so it lies in C([0, T]; L²) ∩ L³, and 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
with Π_ε the rate at which energy enters the resolved field v^ε from the scales below ε. As ε → 0 the left side tends to ½(e(t) − e(s)). With e strictly decreasing this is strictly negative for every s < t. So for every time interval the integrated flux converges to a negative number: the resolved field loses energy to vanishing scales over every interval, never gains it back over any interval, and the loss rate in the limit is −½e′, which the construction prescribes. Compare THEORY-763, where the same integral takes both signs.
What could have come out otherwise: by THEORY-749 a solution regular enough (Besov 3, ∞ above 1/3) has ∫_s^t Π_ε → 0 and e constant. These solutions escape that only because β < 1/3.
What this does not say
- Not one-way at every point. “Admissible” here is the global energy inequality: the total energy does not increase. The word appears only in the title and is never defined in the text; the introduction speaks of “strictly monotonic decreasing energy”. The local energy inequality, ∂_t(½|v|²) + div((½|v|² + p)v) ≤ 0 in the sense of distributions, equivalently a non-negative Duchon–Robert dissipation measure, is not claimed or proved. So nothing excludes backscatter: energy may return from vanishing scales in some regions while more leaves in others. Later work reaches the local inequality below 1/3 in a different class: Giri, Kwon and Novack (THEORY-772, Proposed), in L³-based Besov spaces, not Hölder. For Hölder solutions the local inequality came first at α < 1/15 (Isett, THEORY-821, Proposed), then at β < 1/7 (De Lellis and Kwon, not read), so this qualification stands for the solutions of this entry. Isett calls such solutions “globally dissipative”, meaning the local inequality holds everywhere, the opposite of this entry’s use of “global” for the whole box.
- Not that the direction is forced. Theorem 1.1 accepts any smooth positive e. An increasing profile gives solutions that steadily draw energy out of vanishing scales, by the same proof. At this regularity the Euler equations do not pick the direction of transfer; the construction does. A one-way cascade is admissible here, not compelled.
- Not a pointwise-in-time flux. The limit is of time integrals over intervals. Whether Π_ε(τ) itself converges for each τ, or where in space and in scale the energy goes, is not shown. There is no constant-flux inertial range and nothing on locality of transfer between scales (THEORY-754’s conditional account is not touched).
- Not turbulence or Navier–Stokes. The solutions are built by convex integration from the zero flow. Nothing shows they are vanishing-viscosity limits of Navier–Stokes, and Onsager’s quoted “turbulent dissipation … without the final assistance by viscosity” is motivation, not a result.
- Not uniqueness, nor non-uniqueness for given data. Theorem 1.1 prescribes the energy, not the initial datum, and does not show two admissible solutions with the same initial datum. That came later (Daneri, Runa and Székelyhidi, arXiv:2004.00391, not read). The paper cites weak–strong uniqueness for dissipative solutions (Brenier, De Lellis and Székelyhidi). The record’s inference from it: the initial datum of a strictly dissipative solution here cannot launch a classical solution, which would conserve energy, so these solutions start from rough data.
- Not the endpoint or other settings. β = 1/3 is not reached; the domain is T³ on a finite interval, e must be strictly positive (the energy gap (2.10) needs room at every step), so the solutions are never at rest, unlike Isett’s.
- Not a checked proof. The estimates of Sections 3–6 were followed, not re-derived. The h-principle (Theorem 1.3) further rests on Theorem 7.1, quoted from Daneri and Székelyhidi without proof; this entry does not use it.
Connections
- QUESTION-219: answers in part. It supplies what THEORY-763 could not: below 1/3 a coarse-grained energy flux can survive the limit of vanishing scale and run one way, into the small scales, on every time interval. This is the dissipative transfer A3’s point needed, in the integrated, whole-box sense. Two qualifications the question should carry: the one-way property is global, not local; and it is selected by the prescribed energy, not produced by the dynamics. On the question proper, whether a pattern of transfer picks out a unit, it bears negatively if at all: the dissipation rate −½e′ is a free input of the construction, and the flux is carried by non-interacting waves on disjoint space-time regions, so nothing in it is a persisting organization of the kind A5’s four conditions ask for.
- THEORY-763. Isett’s solutions take both signs; these take one. With this entry the record holds both the bidirectional and the dissipative negative half, and the promote_when of THEORY-763 names this paper as an independent proof.
- THEORY-749. The positive half. Together with this entry: above 1/3 no flux survives for any weak solution; below it, a flux of either sign, or of one prescribed sign throughout, can.
- THEORY-754. Eyink and Aluie’s conditional account assumes a non-zero mean flux through an inertial range. This entry shows weak solutions with a non-vanishing, one-signed mean flux exist at exponents below 1/3; it shows nothing about a range of scales across which that flux is constant.