Energy-balance for the incompressible Euler equations with stochastic forcing

TL;DR

This paper proves energy conservation for weak solutions of the stochastic incompressible Euler equations with Hölder regularity above 1/3, extending Onsager's conjecture to stochastic settings and including inhomogeneous cases driven by Wiener processes.

Contribution

It establishes energy-balance for stochastic Euler equations with Hölder regularity above 1/3, extending Onsager's conjecture to stochastic and inhomogeneous cases.

Findings

Energy-balance proven for stochastic Euler solutions with Hölder regularity > 1/3

Extension of Onsager's conjecture to stochastic incompressible Euler equations

Energy conservation shown for inhomogeneous stochastic Euler system driven by Wiener process

Abstract

We establish energy-balance for weak solutions of the stochastically forced incompressible Euler equations, enjoying H\"older regularity $C^{\alpha}$, $\alpha>1/3$. It is well known as the Onsager's conjecture for the deterministic incompressible Euler equations, which describes the energy conservation of weak solutions having H\"older regularity $C^{\alpha}$, $\alpha>1/3$. Additionally, we obtain energy-balance for the inhomogeneous incompressible Euler system driven by a cylindrical Wiener process.