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arXiv · 2406.18771

A system of continuity equations with nonlocal interactions of Morse type

Marco Di Francesco·Valeria Iorio
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Abstract

We study a system of two continuity equations with nonlocal velocity fields using interaction potentials of both attractive and repulsive Morse type. Such a system is of interest in many contexts in multi-population modelling. We prove existence, uniqueness and stability in the 2-Wasserstein spaces of probability measures via Jordan-Kinderlehrer-Otto scheme and gradient flow solutions in the spirit of the Ambrosio-Gigli-Savar\'e theory. We then formulate a deterministic particle scheme for this model and prove that gradient flow solutions are obtained in the many particle limit by discrete densities constructed out of moving particles satisfying a suitable system of ODEs. The ODE system is formulated in a non standard way in order to bypass the Lipschitz singularity of the kernel, with difference quotients of the kernel replacing its derivative.

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Marco Di Francesco, Valeria Iorio. 2024-06-26. A system of continuity equations with nonlocal interactions of Morse type. https://arxiv.org/abs/2406.18771

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