Nonuniqueness of Hölder Continuous Solutions for Stochastic Euler and Hypodissipative Navier–Stokes Equations

Abstract. We construct infinitely many Hölder continuous, global-in-time, and stationary solutions to the stochastic Euler equations and the hypodissipative Navier–Stokes equations, taking values in the space [Formula: see text]. For the Euler case, the Hölder exponent [Formula: see text] satisfies [Formula: see text] with [Formula: see text], while for the hypodissipative Navier–Stokes equations, [Formula: see text] must additionally satisfy [Formula: see text]. The construction relies on a modified stochastic convex integration scheme, which is central to the analysis. This scheme incorporates Beltrami flows as building blocks and carefully tracks inductive estimates, both pathwise and in expectation. These refinements allow us to achieve improved Hölder regularity for solutions to the underlying stochastic equations, advancing the scope of convex integration techniques in the stochastic setting.