Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field

Given a divergence-free vector field u ∈ L^∞ₜ W1,pₓ( Rᵈ) and a nonnegative initial datum ρ₀ ∈ Lʳ, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of L^∞ₜ Lʳₓ densities for 1/p + 1/r ≤ 1. This range was later improved in [BCDL21] to 1/p + d-1/dr ≤ 1. We prove that this range is sharp by providing a counterexample to uniqueness when 1/p + 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.