PUB - Publikationen an der Universität Bielefeld

We study the question of non-uniqueness of Leray-Hopf solutions to stochastic forced Navier-Stokes equations on R3R3 starting from zero initial condition. Specifically, we consider a linear multiplicative noise and the equations are perturbed by an additional body force f. This type of noise is particularly appealing due to the regularization by noise phenomena established in [RZZ14], which provides global uniqueness for arbitrary f is an element of L-2 (0,T;Hs-1) for s > 3/2 with high probability. Based on the ideas of Albritton, Bru & eacute; and Colombo [ABC22], we prove that there exists a force f is an element of L-1 (0,T;L-p) p<3, so that non-uniqueness of local-in-time probabilistically strong Leray-Hopf solutions as well as joint non-uniqueness in law of Leray-Hopf solutions on R+ hold true. In the deterministic setting, we show that the set of forces, for which Leray-Hopf solutions are non-unique, is dense in L-1(0,T;L-2). In addition, by a simple controllability argument we show that for every divergence-free initial condition in L-2 there is a force so that non-uniqueness of Leray-Hopf solutions holds.