Stability of Anomalous Dissipation for the Forced 3D Navier--Stokes Equations under Geometric Perturbations
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913140670726144 |
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| author | Li, Changhong |
| author_facet | Li, Changhong |
| contents | The energy dissipation in the inviscid limit is a central problem in turbulence theory. Kolmogorov's K41 theory predicts a positive dissipation rate independent of viscosity -- a phenomenon known as anomalous dissipation. Brué and De Lellis gave the first rigorous construction, but it relies on extremely precise geometric conditions. Based on quasi-self-similar mixing, we prove structural stability under pure normal perturbations of the central curves. We establish C^2 stability of the maps and C^1 stability of the local fields, and obtain Hölder estimates and high-frequency energy concentration. A contradiction gives a positive dissipation lower bound independent of the perturbation, and embedding into the (2+1/2)-dimensional framework shows C^6 structural stability. The main novelty is that the Brué--De Lellis construction remains stable under such perturbations, so anomalous dissipation occurs in an open neighbourhood of function spaces, providing a rigorous foundation for K41 theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18126 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability of Anomalous Dissipation for the Forced 3D Navier--Stokes Equations under Geometric Perturbations Li, Changhong Analysis of PDEs 35Q30, 76D05, 35B35 The energy dissipation in the inviscid limit is a central problem in turbulence theory. Kolmogorov's K41 theory predicts a positive dissipation rate independent of viscosity -- a phenomenon known as anomalous dissipation. Brué and De Lellis gave the first rigorous construction, but it relies on extremely precise geometric conditions. Based on quasi-self-similar mixing, we prove structural stability under pure normal perturbations of the central curves. We establish C^2 stability of the maps and C^1 stability of the local fields, and obtain Hölder estimates and high-frequency energy concentration. A contradiction gives a positive dissipation lower bound independent of the perturbation, and embedding into the (2+1/2)-dimensional framework shows C^6 structural stability. The main novelty is that the Brué--De Lellis construction remains stable under such perturbations, so anomalous dissipation occurs in an open neighbourhood of function spaces, providing a rigorous foundation for K41 theory. |
| title | Stability of Anomalous Dissipation for the Forced 3D Navier--Stokes Equations under Geometric Perturbations |
| topic | Analysis of PDEs 35Q30, 76D05, 35B35 |
| url | https://arxiv.org/abs/2605.18126 |