Nonuniqueness of Weak Solutions for the Transport Equation at Critical Space Regularity

We consider the linear transport equations driven by an incompressible flow in dimensions d≥ 3. For divergence-free vector fields u∈Lt1W1,q, the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class Lt∞Lp when 1p+1q≤1. For such vector fields, we show that in the regime 1p+1q>1, weak solutions are not unique in the class Lt1Lp. One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.