期刊
NONLINEARITY
卷 26, 期 10, 页码 2777-2808出版社
IOP PUBLISHING LTD
DOI: 10.1088/0951-7715/26/10/2777
关键词
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资金
- CIG Marie Curie grant 'DifNonLoc, Diffusive Partial Differential Equations with Nonlocal Interaction in Biology and Social Sciences' of the University of Bath (UK)
- Ramon y Cajal Sub-programme (MICINN-RYC) of the Spanish Ministry of Science and Innovation
- Ministerio de Ciencia y Innovacion [MTM2011-27739-C04-02]
- Universitat Autonoma de Barcelona
This paper presents a systematic existence and uniqueness theory of weak measure solutions for systems of non-local interaction PDEs with two species, which are the PDE counterpart of systems of deterministic interacting particles with two species. The main motivations behind those models arise in cell biology, pedestrian movements, and opinion formation. In case of symmetrizable systems (i.e. with cross-interaction potentials one multiple of the other), we provide a complete existence and uniqueness theory within (a suitable generalization of) the Wasserstein gradient flow theory in Ambrosio et al (2008 Gradient Flows in Metric Spaces and in the Space of Probability Measures (Lectures in Mathematics ETH Zurich) 2nd edn (Basel: Birkhauser)) and Carrillo et al (2011 Duke Math. J. 156 229-71), which allows the consideration of interaction potentials with a discontinuous gradient at the origin. In the general case of non-symmetrizable systems, we provide an existence result for measure solutions which uses a semi-implicit version of the Jordan-Kinderlehrer-Otto (JKO) scheme (Jordan et al 1998 SIAM J. Math. Anal. 29 1-17), which holds in a reasonable non-smooth setting for the interaction potentials. Uniqueness in the non-symmetrizable case is proven for C-2 potentials using a variant of the method of characteristics.
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