A system of continuity equations with nonlocal interactions of Morse type
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914850060369920 |
|---|---|
| author | Di Francesco, Marco Iorio, Valeria |
| author_facet | Di Francesco, Marco Iorio, Valeria |
| contents | 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-Savaré 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A system of continuity equations with nonlocal interactions of Morse type Di Francesco, Marco Iorio, Valeria Analysis of PDEs Primary: 35D30, 35F55, 35Q70. Secondary: 35Q49, 49Q22 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-Savaré 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. |
| title | A system of continuity equations with nonlocal interactions of Morse type |
| topic | Analysis of PDEs Primary: 35D30, 35F55, 35Q70. Secondary: 35Q49, 49Q22 |
| url | https://arxiv.org/abs/2406.18771 |