Calculus of Variations and Geometric Measure Theory
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M. Di Francesco - S. Fagioli

Measure solutions for nonlocal interaction PDEs with two species

created by difrancesco on 16 Jan 2013


Submitted Paper

Inserted: 16 jan 2013

Year: 2013
Links: pdf file


This paper presents a systematic existence and uniqueness theory of weak measure solutions for systems of nonlocal 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 \emph{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-Gigli-Savaré and other papers, which allows to consider interaction potentials with 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 scheme, which holds in a reasonable \emph{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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