Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field

Given a divergence-free vector field $\mathbf{u}\in L_t^{\mathrm{\infty }}{W}_x^{1,p}({\mathbb{R}}^d)$ 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 $L_t^{\mathrm{\infty }}{L}_x^r$ densities for $\frac{1}{p}+\frac{1}{r}\le 1$. This range was later improved in [BCDL21] to $\frac{1}{p}+\frac{d-1}{dr}\le 1$. 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.