Energy-balance for the incompressible Euler equations with stochastic forcing

Abstract

We establish energy-balance for weak solutions of the stochastically forced incompressible Euler equations, enjoying Hölder 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 Hölder regularity $C^{\alpha }$, $\alpha >1/3$. Additionally, we obtain energy-balance for the inhomogeneous incompressible Euler system driven by a cylindrical Wiener process.