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 ∈ L^1_t W^1,q , the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class L^∞ _t L^p when 1/p + 1/q≤ 1 . For such vector fields, we show that in the regime 1/p + 1/q > 1 , weak solutions are not unique in the class L^1_t L^p . One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.