Quantitative approximation of a Keller--Segel PDE by a branching moderately interacting particle system and suppression of blow-up
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| Format: | Preprint |
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2025
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| _version_ | 1866917166765309952 |
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| author | Cavallazzi, Thomas Richard, Alexandre Tomasevic, Milica |
| author_facet | Cavallazzi, Thomas Richard, Alexandre Tomasevic, Milica |
| contents | The Keller--Segel PDE is a model for chemotaxis known to exhibit possible finite-time blow-up. Following a seminal work by Tello and Winkler, a logistic damping term is added in this PDE and local well-posedness of mild solutions is proven. When the space dimension is $2$ or when the damping is strong enough, the solution is global in time. In the second part of this work, a microscopic description of this model is introduced in terms of a system of stochastic moderately interacting particles. This system features two main characteristics: the interaction between particles happens through a singular (Coulomb-type) kernel which is attractive; and the particles are subject to demographic events, birth and death due to local competition with other particles. The latter induces a branching structure of the particle system. Then the main result of this work is the convergence of the empirical measure of the particle system towards the Keller--Segel PDE with logistic damping, with a rate of order $N^{-\frac{1}{2(d+1)}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_20504 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative approximation of a Keller--Segel PDE by a branching moderately interacting particle system and suppression of blow-up Cavallazzi, Thomas Richard, Alexandre Tomasevic, Milica Probability Analysis of PDEs The Keller--Segel PDE is a model for chemotaxis known to exhibit possible finite-time blow-up. Following a seminal work by Tello and Winkler, a logistic damping term is added in this PDE and local well-posedness of mild solutions is proven. When the space dimension is $2$ or when the damping is strong enough, the solution is global in time. In the second part of this work, a microscopic description of this model is introduced in terms of a system of stochastic moderately interacting particles. This system features two main characteristics: the interaction between particles happens through a singular (Coulomb-type) kernel which is attractive; and the particles are subject to demographic events, birth and death due to local competition with other particles. The latter induces a branching structure of the particle system. Then the main result of this work is the convergence of the empirical measure of the particle system towards the Keller--Segel PDE with logistic damping, with a rate of order $N^{-\frac{1}{2(d+1)}}$. |
| title | Quantitative approximation of a Keller--Segel PDE by a branching moderately interacting particle system and suppression of blow-up |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2512.20504 |