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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2401.01840 |
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| _version_ | 1866916080475176960 |
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| author | Kim, Inwon Mellet, Antoine Wu, Jeremy Sheung-Him |
| author_facet | Kim, Inwon Mellet, Antoine Wu, Jeremy Sheung-Him |
| contents | This paper reviews (and expands) some recent results on the modeling of aggregation-diffusion phenomena at various scales, focusing on the emergence of collective dynamics as a result of the competition between attractive and repulsive phenomena - especially (but not exclusively) in the context of attractive chemotaxis phenomena.
At microscopic scales, particles (or other agents) are represented by spheres of radius $δ>0$ and we discuss both soft-sphere models (with a pressure term penalizing the overlap of the particles) and hard-sphere models (in which overlap is prohibited). The first case leads to so-called ``blob models" which have received some attention recently as a tool to approximate non-linear diffusion by particle systems. The hard-sphere model is similar to a classical model for congested crowd motion. We review well-posedness results for these models and discuss their relationship to classical continuum description of aggregation-diffusion phenomena in the limit $δ\to0$: the classical nonlinear drift diffusion equation and its incompressible counterpart.
In the second part of the paper, we discuss recent results on the emergence and evolution of sharp interfaces when a large population of particles is considered at appropriate space and time scales: At some intermediate time scale, phase separation occurs and a sharp interface appears which evolves according to a Stefan free boundary problem (and the density function eventually relaxes to a characteristic function - metastable steady state for the original problem). At a larger time scale the attractive forces lead to surface tension phenomena and the evolution of the sharp interface can be described by a Hele-Shaw free boundary problem with surface tension. At that same time scale, we will also discuss the emergence of contact angle conditions for problems set in bounded domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01840 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Aggregation-diffusion phenomena: from microscopic models to free boundary problems Kim, Inwon Mellet, Antoine Wu, Jeremy Sheung-Him Analysis of PDEs This paper reviews (and expands) some recent results on the modeling of aggregation-diffusion phenomena at various scales, focusing on the emergence of collective dynamics as a result of the competition between attractive and repulsive phenomena - especially (but not exclusively) in the context of attractive chemotaxis phenomena. At microscopic scales, particles (or other agents) are represented by spheres of radius $δ>0$ and we discuss both soft-sphere models (with a pressure term penalizing the overlap of the particles) and hard-sphere models (in which overlap is prohibited). The first case leads to so-called ``blob models" which have received some attention recently as a tool to approximate non-linear diffusion by particle systems. The hard-sphere model is similar to a classical model for congested crowd motion. We review well-posedness results for these models and discuss their relationship to classical continuum description of aggregation-diffusion phenomena in the limit $δ\to0$: the classical nonlinear drift diffusion equation and its incompressible counterpart. In the second part of the paper, we discuss recent results on the emergence and evolution of sharp interfaces when a large population of particles is considered at appropriate space and time scales: At some intermediate time scale, phase separation occurs and a sharp interface appears which evolves according to a Stefan free boundary problem (and the density function eventually relaxes to a characteristic function - metastable steady state for the original problem). At a larger time scale the attractive forces lead to surface tension phenomena and the evolution of the sharp interface can be described by a Hele-Shaw free boundary problem with surface tension. At that same time scale, we will also discuss the emergence of contact angle conditions for problems set in bounded domains. |
| title | Aggregation-diffusion phenomena: from microscopic models to free boundary problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.01840 |