Onsager's conjecture for the 3D incompressible Euler equations

Consider the 3D incompressible Euler equations on $[0,T]\times \mathbb{T}^3$. A weak solution is understood in the usual distributional sense, and $C^{\beta}$ denotes the Hölder space of exponent $\beta$. Onsager's conjecture. (a) Any weak solution $u$ belonging to

$$C^{\beta}([0,T]\times \mathbb{T}^3)$$

for $\beta>1/3$ conserves kinetic energy, meaning that $\|u(t)\|_{L^2}$ is conserved in time. (b) For any $\beta<1/3$, there exist weak solutions

$$u\in C^{\beta}([0,T]\times \mathbb{T}^3)$$

which dissipate kinetic energy. The first assertion was fully proven by Constantin, E, and Titi in 1994 using a commutator argument; the second assertion is the anomalous-dissipation part of Onsager's conjecture and has since been established by convex integration. In this paper, the corresponding energy-conservation threshold is confirmed for the stochastic 3D Euler equations.