Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves
Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves
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DOI：
10.1007/s00205-021-01628-5
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发表时间：
2021-03-08
影响因子：
2.5
通讯作者：
De Lellis, Camillo
中科院分区：
文献类型：
--
作者：
Brue, Elia;Colombo, Maria;De Lellis, Camillo
DiPerna和Lions（Invent Math 98（3）：511-547，1989）的开创性工作保证了Sobolev向量场的正则拉格朗日流的存在性和唯一性。后者是一个合适的选择轨迹的相关常微分方程满足额外的压缩性/半群性质。一个长期存在的问题是，对于a.e.，正则拉格朗日流的唯一性是否是常微分方程轨线唯一性的推论。初始数据使用Ambrosio的叠加原理，我们将后者与连续性方程正解的唯一性联系起来，然后使用Modena和Szekelyhidi在最近的开创性工作中引入的工具提供了一个否定的答案（Modena和Szekelyhidi在Ann PDE 4（2）：38，2018）。在另一方面，我们引入了一类新的非对称Lusin-Lipschitz不等式，并利用它们证明了连续性方程在可积范围内正解的唯一性，这一结果超出了DiPerna-Lions理论.
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 Szekelyhidi in the recent groundbreaking work (Modena and Szekelyhidi 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.