Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise

Abstract: We construct H\"older continuous, global-in-time probabilistically strong solutions to 3D Euler equations perturbed by Stratonovich transport noise. Kinetic energy of the solutions can be prescribed a priori up to a stopping time, that can be chosen arbitrarily large with high probability. We also prove that there exist infinitely many H\"older continuous initial conditions leading to non-uniqueness of solutions to the Cauchy problem associated with the system. Our construction relies on a flow transformation reducing the SPDE under investigation to a random PDE, and convex integration techniques introduced in the deterministic setting by De Lellis and Sz\'ekelyhidi, here adapted to consider the stochastic case. In particular, our novel approach allows to construct probabilistically strong solutions on $[0,\infty)$ directly.