Non-uniqueness of H\"older continuous solutions for stochastic Euler and Hypodissipative Navier-Stokes equations

Here we construct infinitely many H\"older continuous global-in-time and stationary solutions to the stochastic Euler and hypodissipative Navier-Stokes equations in the space C (R;C^) for 0<<57, with 0<< 124 and 0<<\1-2{3, 124\} respectively. A modified stochastic convex integration scheme, using Beltrami flows as building blocks and propagating inductive estimates both pathwise and in expectation, plays a pivotal role to improve the regularity of H\"older continuous solutions for the underlying equations. As a main novelty with respect to the related literature, our result produces solutions with noteworthy H\"older exponents.