Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field

Elia Bruè; Maria Colombo; Anuj Kumar

Paper

Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field

Abstract

Given a divergence-free vector field

and a nonnegative initial datum

, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of

densities for

. This range was later improved in [BCDL21] to

. We prove that this range is sharp by providing a counterexample to uniqueness when

. 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.