Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift

We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli and Priola. We consider periodic solutions in $\rho \in L_t^{\mathrm{\infty }}{L}_x^p$ for divergence-free drifts $u\in L_t^{\mathrm{\infty }}{W}_x^{\theta ,\tilde{p}}$ for a large class of parameters. We prove local-in-time pathwise nonuniqueness and compare them to uniqueness results by Beck, Flandoli, Gubinelli and Maurelli, addressing a conjecture made by these authors, in the case of bounded-in-time drifts for a large range of spatial parameters. To this end, we use convex integration techniques to construct velocity fields $u$ for which several solutions $\rho$ exist in the classes mentioned above. The main novelty lies in the ability to construct deterministic drift coefficients, which makes it necessary to consider a convex integration scheme $\mathit{\text{with}}\mathit{\text{a}}\mathit{\text{constraint}}$, which poses a series of technical difficulties.