Sharp nonuniqueness in the transport equation with Sobolev velocity field

Given a divergence-free vector field u in L1_t W1,p_x(R^d) and a non‑negative initial datum rho_0 in L^r, the DiPerna–Lions theory establishes uniqueness of the weak solution in the class of L1_t L^r_x densities when the condition 1/p + 1/r is less than or equal to 1. This range was later improved by Brue, Colombo, and De Lellis (2021) to the condition 1/p + (d minus 1) divided by (d times r) is less than or equal to 1. We prove that this improved range is sharp by providing a counterexample to uniqueness when 1/p + (d minus 1) divided by (d times r) is greater than 1. To achieve this, we introduce a new flow mechanism. It is not based on convex integration, which has produced non‑optimal results in this context, nor on purely self‑similar techniques, but it shares features of both. In particular, it has a locally discrete self‑similar structure and an intermittent localization in space and frequency.