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Bibliographic Details
Main Author:	Riley, Casey
Format:	Recurso digital
Language:	English
Published:	Zenodo 2025
Subjects:	Clays Millennium
Online Access:	https://doi.org/10.5281/zenodo.17872229
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This work presents a refined and fully ledgered formulation of the local energy identity for Leray–Hopf weak solutions of the incompressible Navier–Stokes equations. By introducing a ledger mass—a nonnegative Radon defect measure $\mu$—the classical Duchon–Robert dissipation anomaly is placed into a transparent conservation-law structure that exposes both its geometric origin and its analytical role.   The central result establishes that every Leray–Hopf solution satisfies a ledgered local energy balance, making it automatically a suitable weak solution. The measure $\mu$ quantifies the failure of nonlinear flux closure at infinitesimal scales; for smooth solutions it vanishes, while for rough flows it encodes the precise locus and intensity of anomalous dissipation.   Building on this identity, the paper derives:   A Caccioppoli inequality with ledger mass, sharpening the standard local energy tools and revealing how $\mu$ obstructs—or permits—local regularity.   A clean ε-regularity criterion of Caffarelli–Kohn–Nirenberg type in which smallness of both the scaled energy $\mathcal E(r)$ and the ledger mass $\Lambda(r)$ forces local Hölder continuity.   An interpretation of Duchon–Robert coarse-grained dissipation as the weak limit of commutator defects in the mollified momentum flux.     The result is a compact, rigorous, and conceptually unifying treatment of the local energy mechanism behind Navier–Stokes regularity theory. The “ledger” viewpoint clarifies how energy transport, flux imbalance, and scale-to-scale transfer interact, offering a sharper analytical handle on the structure of potential singularities.   This submission is part of an ongoing program to understand nonlinear PDEs through geometric, variational, and phase-based identities—placing the Navier–Stokes energy budget within a wider context of recursive, topological, and harmonic structures.

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