  Given a divergence-free vector field ${\bf u} \in L^\infty_t
W^{1,p}_x(\mathbb R^d)$ 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 in [BCDL21] 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.
