Contribution
Isett (LIT-1255, NOTE-974) reached the Onsager exponent 1/3, but his solutions are only shown to be non-conservative, and, the authors say, their energy “fails to be monotonic on any interval of time”. This paper modifies Isett’s scheme so that the energy can be prescribed exactly: any smooth, strictly positive e(t) is the kinetic energy of some C^β weak solution, for every β < 1/3. A strictly decreasing e gives solutions whose energy strictly decreases, which the title calls admissible and the text calls dissipative. An h-principle adds that these solutions are weak-* dense among smooth strict subsolutions. The proof is also, the authors note, “considerably shorter” than Isett’s.
Key insight
The energy of a convex-integration solution is decided step by step by how much energy each perturbation adds. In Isett’s scheme the perturbations sit only where the glued Reynolds stress lives, on separated time intervals, so there is nothing to add in the gaps. Here the perturbations are placed in disjoint space-time regions with wavy (“squiggling”) boundaries whose projections on the time axis cover every time, so energy ρ_q(t) = ⅓(e(t) − δ_{q+2}/2 − ∫|v̄_q|²) can be pumped in everywhere, and the energy gap is closed to a prescribed target at each step. For that bookkeeping to work the glued stress must stay trace-free, which is why the gluing stress is rebuilt with De Lellis and Székelyhidi’s inverse divergence and a new commutator estimate.
Assumptions
- Domain. The periodic box T³ = R³/Z³ and a finite time interval [0, T]. No initial-value problem is posed: v(0) is an output.
- Notion of solution. Distributional solutions of ∂_t v + v·∇v + ∇p = 0, div v = 0, with v Hölder-continuous.
- Regularity class. C^β(T³ × [0, T]), Hölder jointly in space and time; the inductive estimates use spatial Hölder norms only (Appendix A), and time regularity is recovered at the end.
- The energy profile. e : [0, T] → R smooth and strictly positive. Positivity is needed: the inductive gap (2.10) requires e(t) − ∫|v_q|² ≥ δ_{q+1}λ_q^(−α) > 0 at every step, so the solution can never be at rest. By scaling, sup|e′| ≤ 1 (2.1) and (2.13) are assumed without loss of generality.
- “Admissible”. The word occurs only in the title. What is obtained is a prescribed, hence possibly strictly decreasing, total energy. The local energy inequality is not claimed.
Key results
- Theorem 1.1. For any strictly positive smooth e and any 0 < β < 1/3 there is a weak solution v ∈ C^β(T³ × [0, T]) with ∫_{T³}|v(x, t)|² dx = e(t) for all t.
- Theorem 1.3 (h-principle). If (v̄, p̄, R̄) is a smooth strict subsolution, ∂_t v̄ + div(v̄⊗v̄) + ∇p̄ = −div R̄ with R̄ positive definite, then for β < 1/3 there are C^β weak solutions v_k with v_k ⇀* v̄, v_k⊗v_k ⇀* v̄⊗v̄ + R̄ in L^∞ uniformly in time, and ∫|v_k|² = ∫(|v̄|² + tr R̄) for all t.
- Proposition 2.1 (the iteration). With λ_q = 2π⌈a^(b^q)⌉, δ_q = λ_q^(−2β), and 1 < b < (1 − β)/(2β) (2.11), the inductive bounds ‖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), δ_{q+1}λ_q^(−α) ≤ e(t) − ∫|v_q|² ≤ δ_{q+1} propagate from q to q + 1, with ‖v_{q+1} − v_q‖₀ + λ_{q+1}^(−1)‖v_{q+1} − v_q‖₁ ≤ Mδ_{q+1}^(1/2), M geometric (Definition 5.6). Holds when: α below a threshold depending on β and b, and a large.
- The parameter inequality (2.38)–(2.39). It reduces to −β − βb + 1 − b + 2b²β < 0, which factors as (b − 1)(β(2b + 1) − 1) < 0 (checked here), i.e. (2.11). A b > 1 exists iff β < 1/3.
- Scales. Mollification at ℓ = δ_{q+1}^(1/2)/(δ_q^(1/2)λ_q^(1+3α/2)) (2.19); gluing time τ_q = ℓ^(2α)/(δ_q^(1/2)λ_q) (2.26), with the CFL-like condition τ_q‖v_ℓ‖_{1+α} ≲ ℓ^α ≪ 1 (2.27).
- Gluing (Section 4, Propositions 4.2–4.4). v̄_q = Σχ_i v_i with v_i the exact Euler solutions from v_ℓ(t_i); on I_i the stress is ∂tχ_i R(v_i − v{i+1}) − χ_i(1 − χ_i)(v_i − v_{i+1})⊗̊(v_i − v_{i+1}), trace-free, bounded by δ_{q+1}ℓ^(−N+α) in C^{N+α}, with energy within δ_{q+1}ℓ^α of v_ℓ’s.
- New stress (Proposition 6.1). ‖R̊_{q+1}‖₀ ≲ δ_{q+1}^(1/2)δ_q^(1/2)λ_q/λ_{q+1}^(1−4α), from the Nash, transport and oscillation errors (5.21).
- Energy (Proposition 6.2). |e(t) − ∫|v_{q+1}|² − δ_{q+2}/2| ≲ δ_q^(1/2)δ_{q+1}^(1/2)λ_q
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | For every β < 1/3 and smooth e > 0 on [0, T] there is a C^β weak Euler solution on T³ with ∫|v|² = e(t) | strong | Theorem 1.1, via Proposition 2.1 and Sections 3–6 (estimates followed, not each re-derived here) |
| C2 | Hence there are C^β solutions, β < 1/3, with strictly decreasing energy | strong | C1 with e′ < 0 |
| C3 | Smooth strict subsolutions are weak-* limits of such solutions (h-principle) | moderate | Theorem 1.3, resting on Theorem 7.1 quoted from Daneri and Székelyhidi without proof |
| C4 | The dissipative solutions are “in a sense typical” among subsolutions | weak | abstract’s gloss of C3; density is shown, not genericity, and the h-principle solutions have energy ∫(|v̄|² + tr R̄), which need not decrease |
| C5 | Isett’s solutions’ energy fails to be monotonic on any time interval | weak | asserted in the introduction without proof or reference; Isett’s own Theorem 1 claims only non-conservation (NOTE-974) |
| C6 | Dissipative solutions satisfy weak–strong uniqueness | moderate | cited (Lions; Brenier, De Lellis and Székelyhidi), not proved here |
| C7 | The proof is self-contained apart from classical appendix facts and shorter than Isett’s | moderate | true of Theorem 1.1 as read; Proposition D.1 for N = 0 is Constantin’s lemma adapted to the torus by a sketch |
Method
- Normalize e by the Euler scaling v ↦ Γv(x, Γt) so that (2.13) holds; start from (v₀, R₀, p₀) = (0, 0, 0).
- At step q:
- mollify v_q at scale ℓ (Proposition 2.2, using the Constantin–E–Titi commutator estimate, Proposition A.2);
- glue: solve Euler exactly from v_ℓ(t_i), t_i = iτ_q; join with a partition of unity in time; the stress lives on I_i, is trace-free, and is estimated through vector potentials z_i = (−Δ)^(−1)curl v_i (Proposition 3.4), gaining a factor ℓ;
- perturb: choose cut-offs η_i on disjoint squiggling space-time slabs covering every time, distribute the energy deficit ρ_q(t) as ρ_{q,i} ∝ η_i²ρ_q, form R̃_{q,i} = ∇Φ_i(ρ_{q,i}Id − η_i²R̊_q)∇Φ_iᵀ/ρ_{q,i} near the identity, and add w_o = Σ_i ρ_{q,i}^(1/2)(∇Φ_i)^(−1)W(R̃_{q,i}, λ_{q+1}Φ_i) with W the Mikado flows and Φ_i the back-to-labels map, plus a corrector w_c making w_{q+1} a curl.
- The new stress is the inverse divergence of the Nash, transport and oscillation errors; the cross terms between different i vanish because the supports are disjoint.
- v_q converges uniformly; R̊_q → 0, so v solves Euler; (2.10) with δ_q → 0 gives ∫|v|² = e; Section 2.2 gives joint Hölder regularity.
Concepts
- Euler–Reynolds system: ∂_t v + div(v⊗v) + ∇p = div R̊, R̊ symmetric and trace-free, p of mean zero.
- Smooth strict subsolution: (v̄, p̄, R̄) solving the same with −div R̄ and R̄ positive definite. Its “energy” is ∫(|v̄|² + tr R̄).
- Gluing (Isett’s): replacing an approximate solution by exact local solutions joined in time, so the error is confined to separated intervals.
- Mikado flows (Daneri and Székelyhidi): stationary Euler solutions W(R, ξ) with mean zero and ∫W⊗W dξ = R, for R in a compact set of positive-definite matrices (Lemma 5.1).
- Squiggling stripes: space-time regions bounded by t = t_i + (τ_q/6)(sin 2πx₁ ± ½), so that each time slice meets some region while the regions stay disjoint.
- Dissipative / admissible (as used here): total kinetic energy strictly decreasing in time.
Connections
It is a direct modification of Isett (LIT-1255), with Mikado flows from Daneri and Székelyhidi (2016) and the convex-integration framework of De Lellis and Székelyhidi (2009–2014) after Nash and Kuiper; earlier exponents were 1/10, 1/5 (Isett; Buckmaster, De Lellis, Isett and Székelyhidi) and 1/3 in L¹_tC^β_x (Buckmaster, De Lellis and Székelyhidi). The positive half is Constantin, E and Titi (LIT-1248), after Eyink, with Cheskidov et al. for the endpoint space. The non-L^∞ precursors are Scheffer and Shnirelman. Later work: Isett, “Nonuniqueness and existence of continuous, globally dissipative Euler flows” (arXiv:1710.11186); Daneri, Runa and Székelyhidi, “Non-uniqueness for the Euler equations up to Onsager’s critical exponent” (arXiv:2004.00391), for admissible non-uniqueness from given data; and Giri, Kwon and Novack, “The L³-based strong Onsager theorem” (arXiv:2305.18509), which obtains the local energy inequality in an L³-based class, since skimmed (LIT-1289, NOTE-1022). The first two are not read.
Bearing on the record
- THEORY-776 is filed from this reading, Proposed. It states Theorem 1.1 with e strictly decreasing and the record’s consequence: by THEORY-749’s energy balance, for every s < t the integrated flux ∫_s^t Π_ε across a mollification scale tends to ½(e(t) − e(s)) < 0. So the resolved field loses energy to vanishing scales over every interval.
- QUESTION-219 and A3’s dissipative cascade. This is the reading
NOTE-974 said the question needed. The answer to “does a rough Euler
solution carry a one-way, dissipative flux into vanishing scales?” is yes,
with three qualifications the question should carry:
- Global, not local. “Admissible” here is the total-energy inequality. Nothing is proved about the sign of the local defect, so backscatter in parts of the box is not excluded.
- Chosen, not forced. The same theorem gives increasing energy. At β < 1/3 the Euler equations alone do not set the direction of transfer; the energy profile is a free input. That fits A3’s separation of “direction of transfer” from “that scales exist” and “locality”.
- Not turbulence. No link to vanishing viscosity, to a constant-flux range, or to uniqueness. The question’s text, “The admissible version (Buckmaster, De Lellis, Székelyhidi and Vicol 2019) is not held”, can now cite LIT-1300 and THEORY-776. On the question proper, whether flux picks out a unit: if anything negative. The dissipation rate is −½e′, an arbitrary input, and the transfer is carried by non-interacting waves on disjoint space-time slabs; nothing persists that A5’s conditions (2)–(4) could recognise as self-maintained organization.
- THEORY-763. Its “What this does not say” names this paper as the later theorem the record did not hold; it can now cite THEORY-776. Its promote_when names this paper as a possible independent proof, but only of the Hölder non-conservation statement, not of Isett’s compact time support, which this construction cannot give (e > 0).
- THEORY-749. Its summary says “that dissipative solutions exist below 1/3 is a later theorem (Isett 2018 and others), which the record has not read”. Isett’s solutions are not dissipative; this is the dissipative one, now read. The sentence should cite THEORY-776 for the dissipative case and THEORY-763 for non-conservation.
- QUESTION-218 and QUESTION-222 (closure). As with Isett: the Euler–Reynolds system is the closure problem with the unclosed stress R̊, and convex integration runs it backwards. Theorem 1.3 sharpens the point: every smooth coarse field with a positive-definite “Reynolds stress” R̄ is the weak-* limit of exact solutions whose fine scales reproduce R̄ on average. A coarse description plus its stress does not determine the fine field. Not filed.
- The other key entries (THEORY-733, THEORY-257, THEORY-661, CLAIM-157, CLAIM-331, CLAIM-327, CLAIM-388, THEORY-755) and QUESTION-217, QUESTION-220, QUESTION-221: not addressed. The frequencies λ_q form a coarse-to-fine sequence of corrections, but as a construction device.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- The meaning of “admissible”. The title promises admissible solutions and the text never defines the word. What is proved is a prescribed, and so possibly strictly decreasing, total energy. A reader who takes “admissible” in the stronger, local sense (the local energy inequality) would be overreading the paper.
- The abstract’s “typical”. Theorem 1.3 proves weak-* density of solutions near subsolutions, not genericity in any Baire or measure sense, and its solutions have energy ∫(|v̄|² + tr R̄), which need not decrease, so they are not all “dissipative solutions” as the abstract says.
- The claim about Isett’s solutions. “Fails to be monotonic on any interval of time” is stated without argument. It may follow from details of Isett’s construction; it is not in Isett’s theorem.
- Small slips. Lemma 5.5 states a |k|^(−5) bound and its proof gives |k|^(−4); Definition 5.6 sums |k|^(−4), which converges in three dimensions, so nothing breaks. (2.37) and Proposition 6.2 omit the absolute value. Several subscripts are misprinted (ρ_{q+1,i}, w_q).
- Scope. Finite time interval, torus, e > 0 (never at rest), β < 1/3 not the endpoint, no initial-value problem, no non-uniqueness for given data, nothing on Navier–Stokes.
Open questions
- Are there C^β solutions, β < 1/3, satisfying the local energy inequality, so that transfer to vanishing scales is one-way at every point? In the L³-Besov class, yes (Giri, Kwon and Novack, THEORY-772, Proposed). For C^β it is still open; the best known is β < 1/7.
- Is any vanishing-viscosity limit of Navier–Stokes of this kind? Only a theorem about Navier–Stokes would settle it.
- Given an initial datum, how many admissible solutions start from it? Daneri, Runa and Székelyhidi (arXiv:2004.00391), not held.
- For QUESTION-219: what in a flow, if not regularity, fixes the direction of transfer? Here the answer is “the constructor’s choice of e”, which suggests direction is a selection criterion (admissibility) imposed on the equations, not a consequence of them.