The rigorous derivation of Vlasov equations with local alignments from moderately interacting particle systems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929410481848320 |
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| author | Wang, Jinhuan Zhuang, Mengdi Huang, Hui |
| author_facet | Wang, Jinhuan Zhuang, Mengdi Huang, Hui |
| contents | In this paper, we present a rigorous derivation of the mean-field limit for a moderately interacting particle system in $\R^d$ $(d\geq 2)$. For stochastic initial data, we demonstrate that the solution to the interacting particle model, with an appropriately applied cut-off, converges in probabilistic sense to the solution of the characteristics of the regularized Vlasov models featuring local alignments and Newtonian potential. Notably, the cutoff parameter for the singular potential is selected to scale polynomially with the number of particles, representing an improvement over the logarithmic cut-off obtained in [38]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The rigorous derivation of Vlasov equations with local alignments from moderately interacting particle systems Wang, Jinhuan Zhuang, Mengdi Huang, Hui Analysis of PDEs Probability 35Q83, 82C22, 35Q70, 60K35 In this paper, we present a rigorous derivation of the mean-field limit for a moderately interacting particle system in $\R^d$ $(d\geq 2)$. For stochastic initial data, we demonstrate that the solution to the interacting particle model, with an appropriately applied cut-off, converges in probabilistic sense to the solution of the characteristics of the regularized Vlasov models featuring local alignments and Newtonian potential. Notably, the cutoff parameter for the singular potential is selected to scale polynomially with the number of particles, representing an improvement over the logarithmic cut-off obtained in [38]. |
| title | The rigorous derivation of Vlasov equations with local alignments from moderately interacting particle systems |
| topic | Analysis of PDEs Probability 35Q83, 82C22, 35Q70, 60K35 |
| url | https://arxiv.org/abs/2407.04387 |