The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime
We consider an aggregation-diffusion equation modelling particle interaction with non-linear diffusion and non-local attractive interaction using a homogeneous kernel (singular and non-singular) leading to variants of the Keller-Segel model of chemotaxis. We analyse the fair-competition regime in wh...
| Главные авторы: | , , |
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| Другие авторы: | |
| Формат: | Conference item |
| Язык: | English |
| Опубликовано: |
Springer, Cham
2017
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| _version_ | 1826293560675139584 |
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| author | Calvez, V Carrillo, JA Hoffmann, F |
| author2 | Bonforte, M |
| author_facet | Bonforte, M Calvez, V Carrillo, JA Hoffmann, F |
| author_sort | Calvez, V |
| collection | OXFORD |
| description | We consider an aggregation-diffusion equation modelling particle interaction with non-linear diffusion and non-local attractive interaction using a homogeneous kernel (singular and non-singular) leading to variants of the Keller-Segel model of chemotaxis. We analyse the fair-competition regime in which both homogeneities scale the same with respect to dilations. Our analysis here deals with the one-dimensional case, building on the work in Calvez et al. (Equilibria of homogeneous functionals in the fair-competition regime), and provides an almost complete classification. In the singular kernel case and for critical interaction strength, we prove uniqueness of stationary states via a variant of the Hardy-Littlewood-Sobolev inequality. Using the same methods, we show uniqueness of self-similar profiles in the sub-critical case by proving a new type of functional inequality. Surprisingly, the same results hold true for any interaction strength in the non-singular kernel case. Further, we investigate the asymptotic behaviour of solutions, proving convergence to equilibrium in Wasserstein distance in the critical singular kernel case, and convergence to self-similarity for sub-critical interaction strength, both under a uniform stability condition. Moreover, solutions converge to a unique self-similar profile in the non-singular kernel case. Finally, we provide a numerical overview for the asymptotic behaviour of solutions in the full parameter space demonstrating the above results. We also discuss a number of phenomena appearing in the numerical explorations for the diffusion-dominated and attraction-dominated regimes. |
| first_indexed | 2024-03-07T03:32:00Z |
| format | Conference item |
| id | oxford-uuid:bb0ac1b7-f766-4320-a907-c20bbc11ec78 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T03:32:00Z |
| publishDate | 2017 |
| publisher | Springer, Cham |
| record_format | dspace |
| spelling | oxford-uuid:bb0ac1b7-f766-4320-a907-c20bbc11ec782022-03-27T05:14:09ZThe geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regimeConference itemhttp://purl.org/coar/resource_type/c_5794uuid:bb0ac1b7-f766-4320-a907-c20bbc11ec78EnglishSymplectic ElementsSpringer, Cham2017Calvez, VCarrillo, JAHoffmann, FBonforte, MGrillo, GWe consider an aggregation-diffusion equation modelling particle interaction with non-linear diffusion and non-local attractive interaction using a homogeneous kernel (singular and non-singular) leading to variants of the Keller-Segel model of chemotaxis. We analyse the fair-competition regime in which both homogeneities scale the same with respect to dilations. Our analysis here deals with the one-dimensional case, building on the work in Calvez et al. (Equilibria of homogeneous functionals in the fair-competition regime), and provides an almost complete classification. In the singular kernel case and for critical interaction strength, we prove uniqueness of stationary states via a variant of the Hardy-Littlewood-Sobolev inequality. Using the same methods, we show uniqueness of self-similar profiles in the sub-critical case by proving a new type of functional inequality. Surprisingly, the same results hold true for any interaction strength in the non-singular kernel case. Further, we investigate the asymptotic behaviour of solutions, proving convergence to equilibrium in Wasserstein distance in the critical singular kernel case, and convergence to self-similarity for sub-critical interaction strength, both under a uniform stability condition. Moreover, solutions converge to a unique self-similar profile in the non-singular kernel case. Finally, we provide a numerical overview for the asymptotic behaviour of solutions in the full parameter space demonstrating the above results. We also discuss a number of phenomena appearing in the numerical explorations for the diffusion-dominated and attraction-dominated regimes. |
| spellingShingle | Calvez, V Carrillo, JA Hoffmann, F The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime |
| title | The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime |
| title_full | The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime |
| title_fullStr | The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime |
| title_full_unstemmed | The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime |
| title_short | The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime |
| title_sort | geometry of diffusing and self attracting particles in a one dimensional fair competition regime |
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