Onsager's conjecture almost everywhere in time

TL;DR

This paper adapts recent iterative schemes to construct non-trivial solutions to the 3D Euler equations that are H"older continuous with regularity close to the Onsager critical exponent, valid for almost every time.

Contribution

It demonstrates the existence of H"older continuous solutions with regularity near the Onsager threshold that exist for almost all times, advancing understanding of energy dissipation in Euler flows.

Findings

Solutions are H"older continuous with regularity close to 1/3.

Solutions exist for almost every time with compact support.

The scheme adapts previous methods to achieve time-regularity results.

Abstract

In recent work by Isett (arXiv:1211.4065), and later by Buckmaster, De Lellis, Isett and Sz\'ekelyhidi Jr. (arXiv:1302.2815), iterative schemes where presented for constructing solutions belonging to the H\"older class $C^{1/5-\epsilon}$ of the 3D incompressible Euler equations which do not conserve energy. The cited work is partially motivated by a conjecture of Lars Onsager in 1949 relating to the existence of $C^{1/3-\epsilon}$ solutions to the Euler equations which dissipate energy. In this note we show how the later scheme can be adapted in order to prove the existence of non-trivial H\"older continuous solutions which for almost every time belong to the critical Onsager H\"older regularity $C^{1/3-\epsilon}$ and have compact temporal support.