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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.14309 |
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| _version_ | 1866912002252734464 |
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| author | Mellet, Antoine Rozowski, Michael |
| author_facet | Mellet, Antoine Rozowski, Michael |
| contents | The Patlak-Keller-Segel system of equations (PKS) is a classical example of aggregation-diffusion equation in which the repulsive effect of diffusion is in competition with the attractive chemotaxis term. Recent work on the Parabolic-Elliptic PKS model have shown that when the repulsion is modeled by a nonlinear diffusion term $ρ\nabla ρ^{m-1}$ with $m>2$, this competition leads to phase separation phenomena. Furthermore, in some asymptotic regime corresponding to a large population observed over a long enough time, the interface separating regions of high and low density evolves according to the Hele-Shaw free boundary problem with surface tension. In the present paper, we consider the counterpart of that model, namely the Elliptic-Parabolic PKS model and we prove that the same phase separation phenomena occurs, but the motion of the interface is now described (asymptotically) by a volume-preserving mean-curvature flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14309 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Volume-preserving mean-curvature flow as a singular limit of a diffusion-aggregation equation Mellet, Antoine Rozowski, Michael Analysis of PDEs The Patlak-Keller-Segel system of equations (PKS) is a classical example of aggregation-diffusion equation in which the repulsive effect of diffusion is in competition with the attractive chemotaxis term. Recent work on the Parabolic-Elliptic PKS model have shown that when the repulsion is modeled by a nonlinear diffusion term $ρ\nabla ρ^{m-1}$ with $m>2$, this competition leads to phase separation phenomena. Furthermore, in some asymptotic regime corresponding to a large population observed over a long enough time, the interface separating regions of high and low density evolves according to the Hele-Shaw free boundary problem with surface tension. In the present paper, we consider the counterpart of that model, namely the Elliptic-Parabolic PKS model and we prove that the same phase separation phenomena occurs, but the motion of the interface is now described (asymptotically) by a volume-preserving mean-curvature flow. |
| title | Volume-preserving mean-curvature flow as a singular limit of a diffusion-aggregation equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.14309 |