Sharp nonuniqueness in the transport equation with Sobolev velocity field

Given a divergence-free vector field {\mathbf{u}} \in L^\infty_t W^{1,p}_x(\mathbb{R^d}) and a nonnegative initial datum \rho_{0} \in L^{r} , the celebrated DiPerna–Lions theory established the uniqueness of the weak solution in the class of L^{\infty}_{t} L^{r}_{x} densities for \frac{1}{p}+ \frac{1}{r} \leq 1 . This range was later improved by Bruè, Colombo and De Lellis (2021) to \frac{1}{p} + \frac{d-1}{dr} \leq 1 . We prove that this range is sharp by providing a counterexample to uniqueness when \frac{1}{p} + \frac{d-1}{dr} > 1 . To this end, we introduce a novel flow mechanism. It is not based on convex integration, which has provided a non-optimal result in this context, nor on purely self-similar techniques, but shares features of both, such as a local (discrete) self-similar nature and an intermittent space-frequency localization.