Carrillo, J. A. and Delgadino, M. G. and Frank, R. L. and Lewin, M. (2022) Fast Diffusion leads to partial mass concentration in Keller–Segel type stationary solutions. Mathematical Models and Methods in Applied Sciences, 32 (4). pp. 831-850. ISSN 0218-2025. doi:10.1142/S021820252250018X. https://resolver.caltech.edu/CaltechAUTHORS:20211004-222702303
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Abstract
We show that partial mass concentration can happen for stationary solutions of aggregation–diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions N≥6. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions N≥3, for homogeneous interaction potentials with higher power.
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| Additional Information: | © 2022 The Author(s). This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited. Received 23 March 2021; Accepted 18 December 2021; Published: 10 March 2022. The authors would like to thank Jean Dolbeault, David Gomez-Castro, Juan Luis Vazquez and an anonymous referee for pointing out references and fruitful comments. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Advanced Grant Nonlocal-CPD 883363 of J.A.C. and Consolidator Grant MDFT 725528 of M.L.). M.G.D. was partially supported by CNPq-Brazil (#308800/2019-2) and Instituto Serrapilheira. R.L.F. was partially supported by the U.S. National Science Foundation through grants DMS-1363432 and DMS-1954995 and through Germany’s Excellence Strategy EXC-2111-390814868. | ||||||||||||||||
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| Subject Keywords: | Keller–Segel; aggregation–diffusion; mass concentration | ||||||||||||||||
| Issue or Number: | 4 | ||||||||||||||||
| Classification Code: | AMS Subject Classification 2020: 35A23, 26D15, 35K55, 46E35, 49J40 | ||||||||||||||||
| DOI: | 10.1142/S021820252250018X | ||||||||||||||||
| Record Number: | CaltechAUTHORS:20211004-222702303 | ||||||||||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20211004-222702303 | ||||||||||||||||
| Official Citation: | Fast Diffusion leads to partial mass concentration in Keller–Segel type stationary solutions. Mathematical Models and Methods in Applied Sciences 2022 32:04, 831-850; DOI: 10.1142/S021820252250018X | ||||||||||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||||||
| ID Code: | 111199 | ||||||||||||||||
| Collection: | CaltechAUTHORS | ||||||||||||||||
| Deposited By: | George Porter | ||||||||||||||||
| Deposited On: | 04 Oct 2021 22:53 | ||||||||||||||||
| Last Modified: | 03 Jun 2022 17:36 |
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