Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves
By Elia Brué, Maria Colombo, Camillo De Lellis. Archive for Rational Mechanics and Analysis, 2020
67 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Control and Systems Engineering
View Paper: https://www.semanticscholar.org/paper/b1ed7db5ae1016b8340d4fa9871845eaf5d4c73e
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In Applied Mathematics
In Control and Systems Engineering
Citations in the corpus
- Distribution dependent SDEs driven by additive fractional Brownian motion (2021, background): “The above and similar inequalities (see [7, 8] for recent asymmetric extensions) allow to control the map μ → b ∗ μ in Wasserstein spaces.”