Non-uniqueness of Leray-Hopf solutions for stochastic forced Navier-Stokes equations

We consider stochastic forced Navier–Stokes equations on ${\mathbb{R}}^3$ starting from zero initial condition. The noise is linear multiplicative and the equations are perturbed by an additional body force. Based on the ideas of Albritton, Brué and Colombo \cite{ABC22}, we prove non-uniqueness of local-in-time Leray–Hopf solutions as well as joint non-uniqueness in law for solutions on ${\mathbb{R}}^{+}$. In the deterministic setting, we show that the set of forces, for which Leray–Hopf solutions are non-unique, is dense in $L_t^1{L}_x^2$. In addition, by a simple controllability argument we show that for every divergence-free initial condition in $L_x^2$ there is a force so that non-uniqueness of Leray–Hopf solutions holds.