Onsager’s Conjecture Almost Everywhere in Time

In recent works by Isett (Hölder continuous Euler flows in three dimensions with compact support in time, pp 1–173, 2012), and later by Buckmaster et al. (Ann Math 2015), iterative schemes were presented for constructing solutions belonging to the Hölder class C^1/5-ε of the 3D incompressible Euler equations which do not conserve the total kinetic energy. The cited work is partially motivated by a conjecture of Lars Onsager in 1949 relating to the existence of C^1/3-ε 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 Hölder continuous solutions which for almost every time belong to the critical Onsager Hölder regularity C^1/3-ε and have compact temporal support.