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

TL;DR

This paper proves the structural stability of anomalous energy dissipation in the 3D Navier--Stokes equations under geometric perturbations, supporting the robustness of turbulence theory predictions.

Contribution

It demonstrates that Brué and De Lellis's construction of anomalous dissipation remains stable under certain geometric perturbations, establishing a rigorous foundation for K41 turbulence theory.

Findings

Positive dissipation lower bound independent of perturbations

C^2 stability of maps and C^1 stability of local fields

Anomalous dissipation occurs in an open neighborhood of function spaces

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.