THEORY-821: For every α < 1/15 there are weak solutions of the 3D incompressible Euler equations on the periodic box, Hölder-continuous of exponent α in space and time, that satisfy the local energy inequality; in the construction their dissipation measure is a nonnegative function of time alone, equal at each time to the rate at which the total energy falls, zero while the local energy equality holds and positive on all of space during a short final interval in which the solution's energy is drained to zero; uncountably many such solutions share one initial datum and one dissipation measure; for them the local coarse-grained energy flux into vanishing scales converges to that measure, so the transfer is one-way at every place and time in the limit and spread uniformly over the box, with its timing chosen by the construction, not forced by the equations

Source

Isett, “Nonuniqueness and existence of continuous, globally dissipative Euler flows”, Archive for Rational Mechanics and Analysis 244:1223–1309 (2022), LIT-1305, skimmed in NOTE-1027 from arXiv:1710.11186v4. The existence statements are its Theorems 1 and 4, both consequences of the Approximation Theorem 7. The form of the dissipation measure is read from the proof of the Main Lemma 3.1 (lines (43)–(44) and (117)) and from Lemma 6.1. The coarse-grained energy balance is Constantin, E and Titi’s (LIT-1248), re-derived in THEORY-749, taken in its local form as in THEORY-772 (LIT-1289).

What was actually shown

The terms. The paper calls a weak solution globally dissipative if v ∈ L³_{t,x} and it satisfies the local energy inequality

∂_t(|v|²/2) + ∇_j((|v|²/2 + p)v^j) ≤ 0 (2)

in the sense of distributions; the left side is then minus a nonnegative Radon measure µ, the dissipation measure. “Globally” does not mean “for the whole box”: it means the local inequality holds everywhere, and the inequality need not be strict. The record’s other entries use “global” for the whole-box energy (THEORY-776); the two usages are opposite.

The theorems.

• Theorem 1. “On (0, ∞) × T^d, for d ≥ 3, there exist Hölder-continuous weak solutions (v, p) that satisfy the local energy inequality (2) with strict inequality holding everywhere on an open interval of time.”
• Theorem 4. “For any α ∈ [0, 1/15), there exists an infinite family of weak solutions (v_β, p_β) to (1) of class v_β ∈ C^α_{t,x}([0, ∞) × T³) that satisfy the local energy inequality (2) and share the same initial data while having uniformly bounded, compact support. Such a family may be chosen to have a common dissipation measure µ … independent of β while being homeomorphic to a Cantor set as a subspace of C^α_{t,x}, and also having positive Hausdorff dimension as a subspace of C_tL²_x.”
• Theorem 5. The same with the local energy equality on R × T³: nonunique energy-conserving C^α solutions from one datum.
• Theorem 6 (8). For α < 1/15 the initial data admitting uncountably many C^α solutions with the local energy equality on a common interval are C⁰ dense among continuous divergence-free fields.

The exponent 1/15 does not depend on the dimension d ≥ 3.

How it is proved (read in outline; estimates not checked). A dissipative Euler–Reynolds flow (Definition 2.1) adds to the Euler–Reynolds system a relaxed local energy inequality with an unresolved flux density κ and current ϕ:

∂_t(|v|²/2) + ∇_j((|v|²/2 + p)v^j) = D_tκ + ∇_j(v_ℓR^{jℓ}) + ∇_jϕ^j − µ, µ ≥ 0.

The Main Lemma corrects (v, p) by plane waves V_I = P_I[e^{iλξ_I}v_I] whose phases are transported by the coarse flow, each oscillating essentially along one coordinate axis and taking values in the perpendicular plane, cycling the axis from stage to stage. Some waves cancel the low-frequency stress; others cancel the low-frequency current ϕ through “high-high-low” triads (two passive waves and one active wave with n₁ + n₂ + n₃ = 0), which the paper likens to Kraichnan’s Lagrangian direct interaction approximation. Beltrami and Mikado waves are set aside: the paper says Mikado implementations “introduce both creation and dissipation of energy in the disjoint regions where the waves are supported and cut off”. Nonuniqueness comes from flipping the sign of one large wave at each stage k ∈ β; this changes nothing in the estimates or in µ, and shifts the solution by at least a fixed fraction of the stage’s amplitude in C_tL²_x.

Checked here.

• The exponent. The row vector (0, 15, 11, 5, 1, 2) lies in the left null space of T − 1 for the recursion matrix T of (291), and its product with e₃ + e₆ is 13, giving (292). The reduced matrix T̊ of (294) has eigenvalues 1, 1/3, 0, 0 with fixed direction ζ = (1, 3/2, 1, 1), and ζ satisfies both admissibility inequalities (293) strictly (1 > 0, 6 > 0). Iterating the recursion numerically from L̊ = L_Z ζ, the frequency grows by about Z^{7.5} per stage while the error level e_G falls by Z, and the Hölder threshold of (306) tends to 1/15 as L_Ĉ/L_Z → 0 (0.0655 at L_Z/L_Ĉ = 50, 0.0666 at 500). So the C^α amplitude scales like frequency
• The dissipation measure. In the Main Lemma the energy increment is e^{1/2}(t) = 2(K₀e_ϕ)^{1/2} η_τ ∗ 1_{t ≤ sup Ĩ[G] + τ/100}(t), τ = (Ξe_v^{1/2})^{−1} (117): a function of t alone, constant up to just past the stage’s interval and zero shortly after. The new measure is µ* = µ − D_t e(t) (after (44)), and D_t e = ∂t e because e does not depend on x. Lemma 6.1 starts the iteration with µ̃ = µ − (3/4)∂t e_Z(t), tr δ[1∗] = 3/2. So from the zero flow (Theorems 1 and 4) every stage adds a nonnegative function of t alone, and the limit is µ∞(t, x) = m(t) = −(3/4)e_Z′(t) − Σ_k e(k)′(t). Integrating (2) over T³ (volume 1) gives (d/dt)∫|v|²/2 = −m(t): the dissipation density at every point equals the rate at which the total energy falls. Each stage’s drop is a bump of mass 4K₀e{ϕ,(k)} and width about τ_k/200, so m is integrable in t but not bounded: its bumps grow taller as they accumulate at the end of the support.
• Where in time. Theorem 7(3) gives µ∞ = 0 on Ĩ(1) × T³, which is (−∞, 2) for Theorem 4; property 4 puts the support of every v_β in t ≤ 2 + (Ξ₀E₀^{1/2})^{−1}. So each solution conserves energy locally until t = 2, loses all of it on the whole torus within a time of order (Ξ₀E₀^{1/2})^{−1}, and is identically zero afterwards. The solutions of a family coincide for t < 1 and branch near t ≈ 1, while the local energy equality still holds.

The consequence for the local flux (the record’s). v is bounded and uniformly continuous, and p ∈ L^q_{t,x} for every q < ∞ (Section 6.2). Mollifying in space at scale ℓ, as in THEORY-772,

∂_t(½|v^ℓ|²) + div((½|v^ℓ|² + p^ℓ)v^ℓ + τ_ℓv^ℓ) = τ_ℓ : ∇v^ℓ, τ_ℓ = (v⊗v)^ℓ − v^ℓ⊗v

Here τ_ℓ → 0 uniformly (|τ_ℓ| ≲ ℓ^{2α}), |v^ℓ|² and |v^ℓ|²v^ℓ converge uniformly, and p^ℓv^ℓ → pv in L¹_loc. So the left side tends in distributions to the left side of (2), and

−τ_ℓ : ∇v^ℓ → m(t) ≥ 0 in D′.

For every nonnegative test function ψ(t, x), the energy passed to scales below ℓ in the region ψ weights tends to ∫ψ(t, x)m(t) dt dx. It is positive whenever ψ’s time support meets the drain interval, wherever in space ψ sits, and zero otherwise. The pointwise bound |τ_ℓ : ∇v^ℓ| ≲ ℓ^{3α−1} grows as ℓ → 0 for α < 1/3, so only the distributional limit is controlled.

What could have come out otherwise: above 1/3 the measure must vanish (Duchon and Robert; THEORY-749 for the total). Isett’s 2018 solutions (THEORY-763) and Buckmaster, De Lellis, Székelyhidi and Vicol’s (THEORY-776) satisfy no proved local sign. A scheme whose waves create energy locally where they are cut off, which the paper says of Mikado schemes, would not give (2).

What this does not say

• Not near 1/3. α < 1/15 in C^α_{t,x}. De Lellis and Kwon improved the Hölder exponent to 1/7, drawing on this paper’s ideas (Analysis & PDE 15:2003–2059, 2022, not read); Giri, Kwon and Novack reach 1/3 only in L³-based Besov spaces (THEORY-772). The strong Onsager conjecture in L^∞_tC^{1/3}_x, the paper’s Conjecture 1, stays open.
• Not dissipation the dynamics selects. The functions e_(k)(t) are fixed by the frequency–energy levels and the intervals; the constructor chooses when the drain happens. The sign is not free, since µ ≥ 0 is what the scheme is built to keep, but the place, the time and the rate are inputs, as in THEORY-772 and THEORY-776.
• Not a spatially structured dissipation. In this construction the measure has no spatial structure at all: it is uniform on T³ at each time. That is the record’s reading of the proof, not a statement of the paper. It is the opposite of the concentrated, intermittent dissipation that turbulence phenomenology expects, and of Giri, Kwon and Novack’s pipes.
• Not sustained dissipation. The energy is lost only while the solution is being shut off: before the drain interval the local energy equality holds, and after it the solution is zero. There is no stationary or persistent cascade.
• Not one-signed at a positive scale. As in THEORY-772, the sign holds only for the limit ℓ → 0 tested against fixed functions.
• Not non-uniqueness for given smooth data, nor turbulence. The common initial datum of Theorem 4 is produced by the construction. By weak–strong uniqueness, which the paper cites, solutions satisfying the energy inequality cannot branch from data that launch a classical solution; Theorem 8’s wild data are only C⁰-close to smooth data. The paper notes that L³ limits of suitable Navier–Stokes solutions satisfy (2); it does not show these are such limits. The Kraichnan analogy is an analogy.
• Not a probabilistic result. The “coin toss” is a deterministic binary choice at each stage indexed by β ∈ 2^ℕ; no law on solutions is used except Haar measure on 2^ℕ in the Hausdorff-dimension count.
• Not a checked proof. The estimates of Sections 4.6–4.13 were not checked.

Connections

• QUESTION-219: answers in part. It gives the one-way local transfer that THEORY-776 lacked in a Hölder class, as THEORY-772 does in an L³ one, and so makes the class no longer the obstruction. For the question proper it bears negatively, more sharply than the pipes did: the flux has no spatial pattern to pick out a unit, its timing is the constructor’s, and it exists only while the solution is being drained. None of A5’s four conditions can be met by a transfer with no spatial structure and no persistence.
• THEORY-772. Both prescribe the dissipation. Giri, Kwon and Novack set a smooth density in pipes, positive for all time, with β < 1/3 in L³ Besov spaces; here it is uniform in space, confined to a final interval, with α < 1/15 in Hölder spaces. Neither contains the other.
• THEORY-776. Its qualification “not one-way at every point” holds for its solutions. This paper is the earlier Hölder construction that removes it at a lower exponent, and its author says that the Mikado cutoffs of that scheme create energy locally.
• THEORY-763. Isett’s two-signed flux; here, one sign everywhere in the limit, at a fifth of the exponent.
• THEORY-749. The positive half; the measure here is what its local form forbids above 1/3.
• QUESTION-218 and QUESTION-222 (closure). Definition 2.1’s remark: any ensemble average of globally dissipative flows is a dissipative Euler–Reynolds flow, with κ half the trace of the Reynolds stress and ϕ from the third-order cumulants. The coarse field’s local energy balance thus carries two unclosed terms besides the stress. Convex integration again runs closure backwards. Not filed.