Uniqueness and energy balance for isentropic Euler equation with stochastic forcing

TL;DR

This paper proves the uniqueness and energy balance for the stochastic isentropic Euler equations, confirming Onsager's conjecture in a stochastic setting with solutions of specific regularity.

Contribution

It establishes pathwise uniqueness and energy conservation for stochastic isentropic Euler equations with solutions in Besov spaces, extending classical results to stochastic contexts.

Findings

Proved pathwise uniqueness for solutions with Hölder regularity > 1/2.

Established energy balance (Onsager's conjecture) for solutions with Hölder regularity > 1/3.

Extended results to a general Besov space setting.

Abstract

In this article, we prove uniqueness and energy balance for isentropic Euler system driven by a cylindrical Wiener process. Pathwise uniqueness result is obtained for weak solutions having H\"older regularity $C^{\alpha},\alpha>1/2$ in space and satisfying one-sided Lipschitz bound on velocity. We prove Onsager's conjecture for isentropic Euler system with stochastic forcing, that is, energy balance equation for solutions enjoying H\"older regularity $C^{\alpha},\alpha>1/3$. Both the results have been obtained in a more general setting by considering regularity in Besov space.