A stochastic convex integration scheme for the intermittent Onsager theorem in two-dimensional domains

Abstract

For any $\gamma \in \left[ {0,\frac{1}{3}} \right)$, we construct $\gamma$-H\"{o}lder-continuous martingale solutions to the two-dimensional stochastic incompressible Euler equations. These martingale solutions exhibit dissipative behavior and intermittency, thereby establishing the flexible part of the intermittent Onsager theorem. The proof rests on a stochastic and intermittent variant of the Newton--Nash iteration combined with the Littlewood--Paley decomposition scheme. This composite iteration scheme incorporates new stochastic pressure, Reynolds stress and intermittent perturbations to formalize the stochastic and intermittent fluctuations.