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;Cϑ) for 0<ϑ<5/7β, with 0<β< 1/24 and 0<β<min\1-2α/3,1/24\ 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.