A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912777375842304 |
|---|---|
| author | Elbar, Charles Fernández-Jiménez, Alejandro Santambrogio, Filippo |
| author_facet | Elbar, Charles Fernández-Jiménez, Alejandro Santambrogio, Filippo |
| contents | We prove Li-Yau and Aronson-Bénilan type estimates for the parabolic-elliptic Keller-Segel system with critical exponent $m=2-\frac 2d$, i.e. lower bounds on the Laplacian of a suitable notion of pressure in any dimension. We show that these estimates entail $L^{\infty}$ bounds on the density, depending on its initial mass, up to the critical mass case for $d \in \{ 2, 3 \}$. We deduce from these results the global existence of smooth solutions in two cases: first, when the initial data is merely a measure but has sufficiently small mass; and second, when the initial free energy is bounded, and the mass is subcritical or critical. Our argument requires a careful study of the subsolutions of the Liouville and Lane-Emden equations arising in the model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent Elbar, Charles Fernández-Jiménez, Alejandro Santambrogio, Filippo Analysis of PDEs 35Q92, 35K57, 35B33, 49J20 We prove Li-Yau and Aronson-Bénilan type estimates for the parabolic-elliptic Keller-Segel system with critical exponent $m=2-\frac 2d$, i.e. lower bounds on the Laplacian of a suitable notion of pressure in any dimension. We show that these estimates entail $L^{\infty}$ bounds on the density, depending on its initial mass, up to the critical mass case for $d \in \{ 2, 3 \}$. We deduce from these results the global existence of smooth solutions in two cases: first, when the initial data is merely a measure but has sufficiently small mass; and second, when the initial free energy is bounded, and the mass is subcritical or critical. Our argument requires a careful study of the subsolutions of the Liouville and Lane-Emden equations arising in the model. |
| title | A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent |
| topic | Analysis of PDEs 35Q92, 35K57, 35B33, 49J20 |
| url | https://arxiv.org/abs/2512.17772 |