Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves

Abstract The seminal work of DiPerna and Lions (Invent Math 98(3):511–547, 1989) guarantees the existence uniqueness regular Lagrangian flows for Sobolev vector fields. latter is a suitable selection trajectories related ODE satisfying additional compressibility/semigroup properties. A long-standing open question whether flow corollary trajectory a.e. initial datum. Using Ambrosio’s superposition principle, we relate to positive solutions continuity equation then provide negative answer using tools introduced by Modena Székelyhidi in recent groundbreaking (Modena Ann PDE 4(2):38, 2018). On opposite side, introduce new class asymmetric Lusin–Lipschitz inequalities use them prove an integrability range which goes beyond DiPerna–Lions theory.