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arXiv · 2605.18126

Stability of Anomalous Dissipation for the Forced 3D Navier--Stokes Equations under Geometric Perturbations

Changhong Li
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Abstract

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. Bru\'e 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 H\"older 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 Bru\'e--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.

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Changhong Li. 2026-05-18. Stability of Anomalous Dissipation for the Forced 3D Navier--Stokes Equations under Geometric Perturbations. https://arxiv.org/abs/2605.18126

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