论文标题
运输方程的积极解决方案和特征曲线的经典非独特性
Positive solutions of transport equations and classical nonuniqueness of characteristic curves
论文作者
论文摘要
Diperna和Lions的开创性工作[发明。 Math。，98，1989]保证了Sobolev Vector Fields常规Lagrangian流的存在和独特性。后者是满足其他可压缩性/半群属性的相关ODE的合适轨迹选择。一个长期的开放问题是，常规拉格朗日流的独特性是否是A.E.颂歌轨迹轨迹的独特性的推论。初始基准。使用Ambrosio的叠加原则，我们将后者与连续性方程式正面解决方案的唯一性联系起来，然后我们使用Modena和SzékelyHidi在最近的开创性工作[Ann [Ann]中提出的工具提供了负面答案。 PDE，4，2018]。在另一侧，我们引入了一类新的不对称lusin-lipschitz不平等现象，并使用它们来证明连续性方程的正溶液在整合性范围内的唯一性，这超出了Diperna-Lions理论。
The seminal work of DiPerna and Lions [Invent. Math., 98, 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 [Ann. PDE, 4, 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.
