Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field

Given a divergence-free vector field u L^ₜ W^1, pₓ (Rᵈ) and a nonnegative initial datum ₀ Lʳ, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of L^ₜ Lʳₓ densities for 1p + 1r 1. This range was later improved in BCDL21 to 1p + d-1dr 1. We prove that this range is sharp by providing a counterexample to uniqueness when 1p + d-1dr > 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.