期刊
KINETIC AND RELATED MODELS
卷 14, 期 1, 页码 89-113出版社
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/krm.2020050
关键词
Measure differential equations; superposition principle; measure-valued solution; probability vector fields; measure dynamics
资金
- Cariplo Foundation
- Regione Lombardia via project Variational Evolution Problems and Optimal Transport
- MIUR PRIN 2015 project Calculus of Variations
- FAR funds of the Department of Mathematics of the University of Pavia
- INdAM-GNAMPA Project 2019 Optimal transport for dynamics with interaction (Trasporto ottimo per dinamiche con interazione)
- National Science Foundation under the CPS SynergyGrant [CNS-1837481]
The paper analyzes the properties of Measure Differential Equations (MDE) and highlights their connection with nonlocal continuity equations. It proves a representation result similar to the Superposition Principle by Ambrosio-Gigli-Savare, and provides alternative schemes converging to a solution of MDE, focusing on uniqueness/non-uniqueness phenomena.
Measure Differential Equations (MDE) describe the evolution of probability measures driven by probability velocity fields, i.e. probability measures on the tangent bundle. They are, on one side, a measure-theoretic generalization of ordinary differential equations; on the other side, they allow to describe concentration and diffusion phenomena typical of kinetic equations. In this paper, we analyze some properties of this class of differential equations, especially highlighting their link with nonlocal continuity equations. We prove a representation result in the spirit of the Superposition Principle by Ambrosio-Gigli-Savare, and we provide alternative schemes converging to a solution of the MDE, with a particular view to uniqueness/non-uniqueness phenomena.
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