Alignment with nonlinear velocity couplings: collision-avoidance and micro-to-macro mean-field limits
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909317487132672 |
|---|---|
| author | Choi, Young-Pil Fabisiak, Michał Peszek, Jan |
| author_facet | Choi, Young-Pil Fabisiak, Michał Peszek, Jan |
| contents | We investigate the pressureless fractional Euler-alignment system with nonlinear velocity couplings, referred to as the $p$-Euler-alignment system. This model features a nonlinear velocity alignment force, interpreted as a density-weighted fractional $p$-Laplacian when the singularity parameter $α$ exceeds the spatial dimension $d$. Our primary goal is to establish the existence of solutions for strongly singular interactions ($α\ge d$) and compactly supported initial conditions. We construct solutions as mean-field limits of empirical measures from a kinetic variant of the $p$-Euler-alignment system. Specifically, we show that a sequence of empirical measures converges to a finite Radon measure, whose local density and velocity satisfy the $p$-Euler-alignment system. Our results are the first to prove the existence of solutions to this system in multi-dimensional settings without significant initial data restrictions, covering both nonlinear ($p>2$) and linear ($p=2$) cases. Additionally, we establish global existence, uniqueness, and collision avoidance for the corresponding particle ODE system under non-collisional initial conditions, extending previous results for $1 \le p \le α+ 2$. This analysis supports our mean-field limit argument and contributes to understanding alignment models with singular communication. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10501 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Alignment with nonlinear velocity couplings: collision-avoidance and micro-to-macro mean-field limits Choi, Young-Pil Fabisiak, Michał Peszek, Jan Analysis of PDEs 35Q70, 35D30, 35L81, 35Q83 We investigate the pressureless fractional Euler-alignment system with nonlinear velocity couplings, referred to as the $p$-Euler-alignment system. This model features a nonlinear velocity alignment force, interpreted as a density-weighted fractional $p$-Laplacian when the singularity parameter $α$ exceeds the spatial dimension $d$. Our primary goal is to establish the existence of solutions for strongly singular interactions ($α\ge d$) and compactly supported initial conditions. We construct solutions as mean-field limits of empirical measures from a kinetic variant of the $p$-Euler-alignment system. Specifically, we show that a sequence of empirical measures converges to a finite Radon measure, whose local density and velocity satisfy the $p$-Euler-alignment system. Our results are the first to prove the existence of solutions to this system in multi-dimensional settings without significant initial data restrictions, covering both nonlinear ($p>2$) and linear ($p=2$) cases. Additionally, we establish global existence, uniqueness, and collision avoidance for the corresponding particle ODE system under non-collisional initial conditions, extending previous results for $1 \le p \le α+ 2$. This analysis supports our mean-field limit argument and contributes to understanding alignment models with singular communication. |
| title | Alignment with nonlinear velocity couplings: collision-avoidance and micro-to-macro mean-field limits |
| topic | Analysis of PDEs 35Q70, 35D30, 35L81, 35Q83 |
| url | https://arxiv.org/abs/2409.10501 |