Clays Millennium Prize Submission (all 7)
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2025
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| _version_ | 1866901515641290752 |
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| author | Riley, Casey |
| author_facet | Riley, Casey |
| contents | <p>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.</p> <p> </p> <p>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.</p> <p> </p> <p>Building on this identity, the paper derives:</p> <p> </p> <p>A Caccioppoli inequality with ledger mass, sharpening the standard local energy tools and revealing how $\mu$ obstructs—or permits—local regularity.</p> <p> </p> <p>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.</p> <p> </p> <p>An interpretation of Duchon–Robert coarse-grained dissipation as the weak limit of commutator defects in the mollified momentum flux.</p> <p> </p> <p> </p> <p>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.</p> <p> </p> <p>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.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17872229 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Clays Millennium Prize Submission (all 7) Riley, Casey Clays Millennium <p>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.</p> <p> </p> <p>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.</p> <p> </p> <p>Building on this identity, the paper derives:</p> <p> </p> <p>A Caccioppoli inequality with ledger mass, sharpening the standard local energy tools and revealing how $\mu$ obstructs—or permits—local regularity.</p> <p> </p> <p>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.</p> <p> </p> <p>An interpretation of Duchon–Robert coarse-grained dissipation as the weak limit of commutator defects in the mollified momentum flux.</p> <p> </p> <p> </p> <p>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.</p> <p> </p> <p>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.</p> |
| title | Clays Millennium Prize Submission (all 7) |
| topic | Clays Millennium |
| url | https://doi.org/10.5281/zenodo.17872229 |