Uniqueness and energy balance for isentropic Euler equation with stochastic forcing

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 Hölder 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 Hölder regularity $C^{\alpha },\alpha >1/3$. Both the results have been obtained in a more general setting by considering regularity in Besov space.