NOTE-1027: Isett — Continuous, globally dissipative Euler flows

Contribution

Before this paper, convex integration gave either continuous Euler flows that dissipate total energy but satisfy the local energy inequality on no open ball (De Lellis and Székelyhidi; Buckmaster, De Lellis, Székelyhidi and Vicol, LIT-1300), or bounded, nowhere-continuous ones that satisfy it (De Lellis and Székelyhidi 2010). It constructs the first continuous, in fact C^α_{t,x} for every α < 1/15, weak solutions satisfying the local energy inequality with a nonzero dissipation measure. It shows these, and energy-conserving ones, are non-unique from one datum in a strong sense: a Cantor family with positive Hausdorff dimension in C_tL²_x and one dissipation measure. Wild data are C⁰ dense. The iteration is new: it reduces the error in the local energy balance along with the Reynolds stress.

Key insight

Carry the local energy balance through the iteration as a relaxed inequality with its own error terms, an unresolved flux density κ (half the trace of the stress) and current ϕ (from third-order interactions), and cancel them as the stress is cancelled. The density is cancelled by coupling the energy the waves add to the pressure; the current, a vector, by triads of transported plane waves whose phases sum to zero, so that the product of two “passive” and one “active” wave is a low-frequency field of any required direction. Dissipation enters only through one scalar per stage: the energy profile e(t) of the new waves. Where e decreases, the decrease −e′(t) is booked into the dissipation measure. Since e depends on time alone, so does the dissipation.

Assumptions

• Domain. T^d = (R/Z)^d with d ≥ 3, on an interval of time. The paper says d = 2 needs more work: waves must take values in three hyperplanes, and the needed cancellation (214) is lost.
• Notion of solution. Weak solutions with v ∈ L³_{t,x}; “globally dissipative” means the local energy inequality (2) in distributions, not necessarily strict. The dissipation measure is µ = −[∂_t(|v|²/2) + ∇_j((|v|²/2 + p)v^j)] ≥ 0.
• Regularity class. C^α_{t,x}, α < 1/15, uniformly; p ∈ L^q for all q < ∞. The title’s “continuous” undersells this.
• What is not assumed or obtained. No initial-value problem for given data is solved; the shared datum is produced. Nothing about Navier–Stokes.

Key results

• Theorem 1. Hölder-continuous weak solutions on (0, ∞) × T^d satisfying (2) “with strict inequality holding everywhere on an open interval of time”.
• Theorem 4. For α ∈ [0, 1/15), an infinite family v_β ∈ C^α_{t,x} satisfying (2), with one initial datum, uniformly bounded compact support in time, a common dissipation measure, homeomorphic to a Cantor set in C^α_{t,x} and of positive Hausdorff dimension in C_tL²_x.
• Theorem 5. The same with the local energy equality (3) on R × T³.
• Theorem 7 (Approximation). From any dissipative Euler–Reynolds flow with levels (Ξ₀, E₀, …) and errors supported in Ĩ(0): a family of C^α solutions, equal on Ĩ(0), with common µ∞ ≥ 0 equal to the initial µ on Ĩ(1), support in t ≤ sup Ĩ(1) + (Ξ₀E₀^{1/2})^{−1}, within CE₀^{1/2} of the initial velocity in C⁰. Hausdorff dimension at least 4 log 2/(3 log Z).
• Theorem 8. For α < 1/15 the data admitting uncountably many C^α solutions with (3) near t = 0 are C⁰ dense in continuous divergence-free fields. The time of existence is not uniform.
• Main Lemma 3.1. One stage maps levels (Ξ, e_v, e_ϕ, e_R, e_G) to (NĈΞ, Ĉẽ_ϕ, Ĉẽ_ϕ, Ĉe_G, e_G*), ẽ_ϕ = e_ϕ^{1/3}e_R^{2/3}, with e_G* a power of N^{−1} times powers of e_v and ẽ_ϕ (the displayed formula (21) was garbled in extraction and is not restated here); the new velocity differs by ‖V‖_{C⁰} ≤ C_Le_ϕ^{1/2}; µ* − µ = −∂_t e(t) ≥ 0 depends only on the levels, L and Ĩ[G], and µ* = µ on Ĩ[G]. It also gives a second flow, differing only in a short interval J̃, at C_tL²_x distance at least ((C_LM)^{−1} − N^{−1})^{1/2}e_ϕ
• The exponent (292), (306). (0, 15, 11, 5, 1, 2) is a left null vector of T − 1, so 15δL_G + 2δL_Ξ → 13L_Ĉ, and C^α convergence holds iff α < 1/15. Re-derived, with the numerical iteration giving thresholds 0.0655 and 0.0666 at L_Z/L_Ĉ = 50 and 500.

Claims

Not filled: this reading is Skimmed, and only a Read note may fill in a claims table. What the skim supports is in THEORY-821, filed Proposed.

Method

1. Start from a dissipative Euler–Reynolds flow (for Theorems 1, 4 and 5, the zero flow). Lemma 6.1 makes it well-prepared by adding a pressure −e_Z(t) and a stress −e_Z(t)δ[1∗], booking −(3/4)e_Z′ into µ.
2. At each stage, with axis l cycling through 1, 2, 3: regularize v by spatial mollification and the errors along the coarse flow; build waves V_I = P_I[e^{iλξ_I}v_I], λ = B_λNΞ, ξ_I transported by v_ε from times kτ, amplitudes v_I = e_I^{1/2}(t)η_I(t)γ_I f̃_I in the plane ⊥ ∇ξ_I.
3. Choose coefficients: for the current waves, constant passive amplitudes and active amplitudes linear in ϕ (68), (92); for the stress waves, positive square roots of a linear system (78), (93). The pressure correction (46) ties the energy density to the stress trace.
4. Absorb −∂_t e(t) into µ (44); collect new errors, using the cancellations (214)–(215), which hold because the wave amplitudes are orthogonal to the oscillation axis, to keep the too-large term V_ℓ P δ[2∗] out of ϕ.
5. Optionally flip the sign of one large wave inside J̃(k) (step k ∈ β).
6. Iterate with N(k) = Z²Q_vϕ^{1/2}Q_ϕϕ²Q_ϕG² inside the admissible sector (295); pass to the limit uniformly; read off (2) and µ∞ from the weak convergence of κ, R, ϕ to zero.

Concepts

• Globally dissipative: v ∈ L³_{t,x} and the local energy inequality (2). Not the whole-box sense used in THEORY-776.
• Dissipative Euler–Reynolds flow (Definition 2.1): (v, p, R, κ, ϕ, µ) with the Euler–Reynolds system (10) and the relaxed inequality (11); µ ≥ 0 the dissipation measure (12). Any ensemble average of globally dissipative flows is one (remark after the definition).
• Unresolved flux density κ and current ϕ: the errors in the local energy balance, the analogues of the Reynolds stress for energy.
• Compound frequency–energy levels (Definition 2.2): bounds on v, p and the error components, split by the subspace ker dx^I ⊗ ker dx^I they take values in.
• Well-prepared (Definition 4.1): the principal stress is dominated by −ē(t)δ[1∗], so its trace part controls the iteration.
• Cascade trios: (active, passive, passive) waves with n₁ + n₂ + n₃ = 0, whose product is low-frequency.

Connections

It descends from Isett and Vicol’s one-dimensional-wave scheme for active scalars (2015), made “multi-dimensional” here, and from the Euler–Reynolds formalism of De Lellis and Székelyhidi. It contrasts its approach with Beltrami flows (no gain on the local error) and Mikado flows (in Isett 2018 and Buckmaster, De Lellis, Székelyhidi and Vicol, which it says create and destroy energy where the waves are cut off). Later: De Lellis and Kwon raised the Hölder exponent to 1/7 (Analysis & PDE 2022, not read); Giri and Kwon extended the ideas to compressible Euler; Giri, Kwon and Novack reached 1/3 in L³-based spaces (LIT-1289, NOTE-1022). The Burgers example cites Dafermos; weak–strong uniqueness, Brenier, De Lellis and Székelyhidi.

Bearing on the record

• THEORY-821 is filed from this reading, Proposed. It states Theorems 1 and 4 and the record’s reading of the proof: the dissipation measure is m(t), a nonnegative integrable function of time alone, equal to −(d/dt)∫|v|²/2, zero while the local equality holds and positive on all of T³ during the final drain. Through the local coarse-grained balance, as in THEORY-772, the flux into scales below ℓ tends to m(t) in distributions.
• QUESTION-219. The question’s four answers on the negative half now read: two-signed (THEORY-763), one-signed for the box (THEORY-776), one-signed everywhere in an L³ class with dissipation in pipes (THEORY-772), and, here, one-signed everywhere in a Hölder class at α < 1/15 with dissipation uniform in space and confined to a final interval. On whether a pattern of transfer picks out a unit, this is the sharpest negative so far: the transfer has no spatial pattern at all, and it exists only while the construction is switching the flow off.
• THEORY-776. Its line “Isett (arXiv:1710.11186) … is not read” can now cite THEORY-821. Its claim that nothing excludes backscatter for its solutions gains the author’s remark that Mikado implementations “introduce both creation and dissipation of energy in the disjoint regions where the waves are supported and cut off”. That remark is not proved in this paper. The “1/7 (De Lellis and Kwon)” there is right, and 1/15 here is its predecessor.
• THEORY-772. Its “Not Hölder, not bounded” bullet can add that Hölder solutions with the local inequality exist below 1/15 (this paper) and 1/7 (De Lellis and Kwon), and that in this construction too the dissipation is an input, uniform rather than in pipes.
• THEORY-749. The local form of its balance is what this measure violates; above 1/3 it must vanish.
• QUESTION-218 and QUESTION-222 (closure). Definition 2.1’s remark puts the closure problem in energy form: an averaged flow’s local energy balance carries κ and ϕ besides the stress. 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 stages cycle through three coordinate axes and nest corrections coarse to fine, but as a device of the proof. The dissipative structures of THEORY-072 and THEORY-532 are not touched.

No instruction for machine-learning practice; nothing for the anthology.

Limitations

• Read in outline. Sections 4.6–4.13, the estimates on which the Main Lemma rests, were not checked.
• The exponent. 1/15, far from 1/3, and later improved by others.
• Errors in the Burgers example (§1.2.1). For (8), the computed total energy is ∫_R u_α² = 2 + t((4/3)α³ − 1/3) for 0 < t ≤ 2. So the entropy solution has −(d/dt)∫u₀² = 1/3, not the stated 2/3 (1/6 for ∫u²/2, two shocks of size 1 at 1/12 each). The family dissipates total energy iff α ≤ (1/4)^{1/3}; the stated α ≤ (1/6)^{1/3} is sufficient and not sharp. Checked numerically. The point of the example, that total-energy dissipation does not select the entropy solution, is unaffected.
• “Probabilistic method.” The paper calls the sign choices a coin toss and “the first use of a probabilistic method in the context of convex integration”. The construction is deterministic for each β ∈ 2^ℕ; no probability law on solutions is used or needed beyond Haar measure in the dimension count. The phrase describes the indexing, not a probabilistic result.
• The Kraichnan analogy (credited to Drivas) is an analogy between the triads used and LDIA, not a claim about turbulence.
• Spatial uniformity of µ is the record’s reading of (44), (117) and Lemma 6.1, not stated by the paper. The paper’s Theorem 1 says only that the inequality is strict everywhere on an open time interval, which is consistent with it.
• Copy. v4 is a 2022 revision (“Improvements in presentation”); v1 and the ARMA text were not compared.

Open questions

• Can a Hölder solution with the local energy inequality have a dissipation measure with spatial structure, or one that persists rather than draining the solution to zero? Giri, Kwon and Novack’s pipes do in L³; a Hölder analogue, or a proof that De Lellis and Kwon’s 1/7 construction has such a measure, would settle it for the Hölder class.
• Does any vanishing-viscosity limit of Navier–Stokes fall in this class with nonzero µ? Only a theorem about Navier–Stokes would say.
• The strong Onsager conjecture in L^∞_tC^{1/3}_x (Conjecture 1) is open.
• Applied to A5’s four conditions (QUESTION-219), a transfer uniform in space and present only during a drain fails conditions (1), (3) and (4) outright. Is there any convex-integration flux that could pass any of them, or is the method structurally unable to produce organized flux?