Conference item
The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime
- Abstract:
- 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.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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- Files:
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(Preview, Accepted manuscript, 3.6MB, Terms of use)
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- Publisher copy:
- 10.1007/978-3-319-61494-6_1
- Publication website:
- https://link.springer.com/book/10.1007/978-3-319-61494-6
- Publisher:
- Springer, Cham
- Host title:
- Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions
- Volume:
- 2186
- Pages:
- 1-71
- Series:
- Lecture Notes in Mathematics
- Publication date:
- 2017-10-04
- Acceptance date:
- 2016-09-01
- Event location:
- Cetraro, Italy
- DOI:
- EISSN:
-
1617-9692
- ISSN:
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0075-8434
- EISBN:
- 978-3-319-61494-6
- ISBN:
- 978-3-319-61493-9
- Language:
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English
- Keywords:
- Pubs id:
-
1098251
- Local pid:
-
pubs:1098251
- Deposit date:
-
2020-04-07
Terms of use
- Copyright holder:
- Springer International Publishing AG
- Copyright date:
- 2017
- Rights statement:
- © Springer International Publishing AG 2017
- Notes:
- This is the accepted manuscript version of the paper. The final version is available online from Springer at https://doi.org/10.1007/978-3-319-61494-6_1
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