Measure solutions for non-local interaction PDEs with two species

Abstract

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 cross-interaction potentials one multiple of the other), we provide a complete existence and uniqueness theory within (a suitable generalization of) the Wasserstein gradient ow theory in [3, 20], 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 JKO scheme [43], 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.

Mean-field Analysis of Multipopulation Dynamics with Label Switching

“…In some of the applications, the interacting agents are assumed to belong to a number of different species, or populations [3,4,20,22,26]. This is a useful modeling assumption, e.g., in the theory of mean-field games, or in control theory, where it can been used to Date: July 12, 2019.…”

“…considered for instance in [22]. In the presence of exchange rates, the correct mean-field description of the above particle system is given by (3.27) in the product space R d ×P 1 ({F, L}).…”

Two-species flocking particles immersed in a fluid

“…A mathematical study of existence, stability, finite-time blow up, and the large-time behavior for two competitive populations of biological species which are attracted by random diffusion and chemotaxis is another recent active research area [12,20,23,31]. We also refer to [21,25] for nonlocal interaction PDEs with two-species.…”

Discrete and continuum links to a nonlinear coupled transport problem of interacting populations

“…Note that the system in (18) can be seen as a generalisation of the many-particle system arising in [8] to our model of coupled interactions of two species. Remark also that in [4,5], the authors studied similar systems but in the absence of the stochastic noise.…”