The Emerald Tablet: Foundations of Onu Calculus
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2026
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| _version_ | 1866901949216980992 |
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| author | Riley, Casey |
| author_facet | Riley, Casey |
| contents | <p>We introduce a scale-covariant reformulation of the three-dimensional incompressible Navier–Stokes equations based on a logarithmic radial coordinate (s = ln r), which maps multiplicative scaling to translation on an infinite line. Under this transformation, dilation-invariant dynamics are governed by a one-dimensional transport–diffusion equation for a scale-energy density E(s,t), generated by a scale-covariant operator (OSCO) with bounded transport velocity and positive scale viscosity.</p> <p>Within this framework, potential finite-time singularities in Euclidean space correspond to continuous energy flux toward the horizon (s → -∞), rather than local blow-up. We establish a conserved energy ledger identity on the scale line and prove global smoothness for small initial data in a scale-weighted Sobolev space H¹_Onu. For arbitrary smooth data, we obtain conditional global regularity assuming uniform control of angular concentration on ℝ², isolating vortex stretching as the sole obstruction within the scale-covariant dynamics.</p> <p>The approach unifies elements of Leray self-similarity, Mellin convolution, and multiresolution analysis, and provides a geometric reinterpretation of the Navier–Stokes cascade that separates coordinate artifacts from physical singularities. Limitations and directions toward removing the angular regularity condition via Littlewood–Paley analysis are discussed.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20291854 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Emerald Tablet: Foundations of Onu Calculus Riley, Casey (4-(m-Chlorophenylcarbamoyloxy)-2-butynyl)trimethylammonium Chloride math.ap math-ph physics.flu-dyn onu calculus navier-stokes Fluid dynamics Computational fluid dynamics <p>We introduce a scale-covariant reformulation of the three-dimensional incompressible Navier–Stokes equations based on a logarithmic radial coordinate (s = ln r), which maps multiplicative scaling to translation on an infinite line. Under this transformation, dilation-invariant dynamics are governed by a one-dimensional transport–diffusion equation for a scale-energy density E(s,t), generated by a scale-covariant operator (OSCO) with bounded transport velocity and positive scale viscosity.</p> <p>Within this framework, potential finite-time singularities in Euclidean space correspond to continuous energy flux toward the horizon (s → -∞), rather than local blow-up. We establish a conserved energy ledger identity on the scale line and prove global smoothness for small initial data in a scale-weighted Sobolev space H¹_Onu. For arbitrary smooth data, we obtain conditional global regularity assuming uniform control of angular concentration on ℝ², isolating vortex stretching as the sole obstruction within the scale-covariant dynamics.</p> <p>The approach unifies elements of Leray self-similarity, Mellin convolution, and multiresolution analysis, and provides a geometric reinterpretation of the Navier–Stokes cascade that separates coordinate artifacts from physical singularities. Limitations and directions toward removing the angular regularity condition via Littlewood–Paley analysis are discussed.</p> |
| title | The Emerald Tablet: Foundations of Onu Calculus |
| topic | (4-(m-Chlorophenylcarbamoyloxy)-2-butynyl)trimethylammonium Chloride math.ap math-ph physics.flu-dyn onu calculus navier-stokes Fluid dynamics Computational fluid dynamics |
| url | https://doi.org/10.5281/zenodo.20291854 |