The flexible part of intermittent Onsager theorem for the two-dimensional stochastic incompressible Euler equations

This work is concerned with the flexible part of Onsager theorem for the two-dimensional stochastic incompressible Euler equations. We develop a stochastic and intermittent variant of the Newton--Nash iteration scheme, which incorporates new stochastic pressure, Reynolds stress and intermittency perturbations to formalize the stochastic and intermittent fluctuations. For any $γ\in \left[ {0,\frac{1}{3}} \right)$, we employ this iteration scheme to construct $γ$-Hölder-continuous martingale solutions for the two-dimensional stochastic incompressible Euler equations. These martingale solutions exhibit dissipative behavior and intermittency, thereby settling the flexible part of intermittent Onsager theorem for the two-dimensional stochastic incompressible Euler equations.