Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier-Stokes equations perturbed by transport noise

Abstract: We show global existence and non-uniqueness of probabilistically strong, analytically weak solutions of the three-dimensional Navier-Stokes equations perturbed by Stratonovich transport noise. We can prescribe either: \emph{i}) any divergence-free, square integrable intial condition; or \emph{ii}) the kinetic energy of solutions up to a stopping time, which can be chosen arbitrarily large with high probability. Solutions enjoy some Sobolev regularity in space but are not Leray-Hopf.