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LIT-1305: Nonuniqueness and existence of continuous, globally dissipative Euler flows

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StatusActive — read 2026-10-20 (NOTE-1027), skimmed, from arXiv:1710.11186v4: the introduction, Sections 2 and 3 (the dissipative Euler–Reynolds flow and the Main Lemma), Sections 4.1–4.5 (the shape of the construction, the coefficient equations and the energy increment), the opening of 4.10, Section 4.14, Section 5 and Section 6 in full were read; Sections 4.6–4.13, the estimates, were paged through and not checked. Worth reading for the first continuous, indeed C^α_{t,x} with α < 1/15, weak solutions of the 3D incompressible Euler equations that satisfy the local energy inequality with nonzero dissipation (Theorems 1 and 4); for nonuniqueness of such solutions from one initial datum, as a Cantor family with a common dissipation measure and positive Hausdorff dimension in C_tL²_x, and of energy-conserving ones (Theorems 4, 5 and 7); for C⁰ density of wild initial data (Theorem 8); for the notion of a dissipative Euler–Reynolds flow, whose relaxed local energy inequality carries an unresolved flux density and current; and, from the proof rather than any statement, for the fact that the dissipation measure is a function of time alone, equal at each time to the rate of loss of total energy and so spread uniformly over the torus (THEORY-821). Read it knowing that "globally dissipative" means the local energy inequality (not whole-box dissipation, and not strict); that the exponent 1/15 is far below 1/3 (De Lellis and Kwon later reached 1/7 in Hölder spaces, Giri, Kwon and Novack 1/3 in L³-based ones); that the dissipation happens only in a short final interval in which the construction drains the solution to zero, its timing an input; that the shared initial datum is produced by the construction, not prescribed; that the "probabilistic method" is a binary choice per stage, not a random construction; and that the Burgers example of §1.2.1 misstates two constants (−d/dt∫u₀² is 1/3, not 2/3, and the threshold (1/6)^{1/3} is sufficient but not sharp, which is (1/4)^{1/3}), without affecting its point.
Version2
Tagsmathematics, natural-sciences
Cited as “paper” byNOTE-1027 — Isett — Continuous, globally dissipative Euler flows
Cited as “source” byTHEORY-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

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Isett, P. (2022). Nonuniqueness and existence of continuous, globally dissipative Euler flows. Archive for Rational Mechanics and Analysis 244: 1223–1309. doi:10.1007/s00205-022-01780-6. arXiv:1710.11186 (30 October 2017).

Skimmed in NOTE-1027, from the arXiv copy. What it implies for the local coarse-grained energy flux is filed as THEORY-821.

Why this is here

Named unread by THEORY-776, whose Hölder solutions lose energy only for the box as a whole. QUESTION-219 needed to know whether one-way transfer at every place is possible for Hölder solutions, and what the dissipation looks like when it is. This paper gives the first such solutions, at a low exponent, and its construction makes the dissipation uniform in space.

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