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

DiPerna和Lions（Invent Math 98（3）：511–547，1989）的开创性工作保证了Sobolev向量场的规则Lagrangian流的存在和唯一性。后者是满足相关ODE轨迹的合适选择，可以满足附加的可压缩性/半群性质。一个长期存在的悬而未决的问题是，规则拉格朗日流的唯一性是否是关于初始基准的ODE轨迹唯一性的必然结果。使用Ambrosio的叠加原理，我们将后者与连续性方程的正解的唯一性联系起来，然后使用Modena和Székelyhidi在最近的开创性工作中引入的工具提供否定答案（Ann PDE 4（2）中的Modena和Székelyhidi）： 38，2018）。在另一边，