Anomalous Dissipation at Onsager-Critical Regularity

TL;DR

This paper constructs three-dimensional Euler solutions demonstrating anomalous energy dissipation at Onsager-critical regularity, extending previous models and establishing sharp energy criteria, including a fully 3D example driven by a rough external force.

Contribution

It extends 2.5D constructions to fully 3D Euler flows and establishes a sharp Onsager-critical energy criterion for anomalous dissipation.

Findings

Constructed solutions with anomalous dissipation via vanishing viscosity limit.

Extended 2.5D models to fully 3D Euler equations.

Established the sharpness of Onsager's energy criterion.

Abstract

We construct solutions to the three-dimensional Euler equations exhibiting anomalous dissipation in finite time through a vanishing viscosity limit. Inspired by \cite{BDL23} and \cite{cheskidov2023dissipation}, we extend the \(2\frac{1}{2}\)-dimensional constructions and establish an Onsager-critical energy criterion adapted to such flows, showing its sharpness. Moreover, we provide a fully three-dimensional dissipative Euler example, sharp in Onsager's sense, driven by a slightly rough external force, following the framework of \cite{CL21}.