Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions
| dc.contributor.author | Blanchet, Adrien | |
| dc.contributor.author | Dolbeault, Jean | |
| dc.contributor.author | Perthame, Benoit | |
| dc.date.accessioned | 2021-07-16T14:24:44Z | |
| dc.date.available | 2021-07-16T14:24:44Z | |
| dc.date.issued | 2006-04-06 | |
| dc.description.abstract | The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial mass, there is global existence of solutions and for large initial mass blow-up occurs. In this paper we complete this picture and give a detailed proof of the existence of weak solutions below the critical mass, above which any solution blows-up in finite time in the whole Euclidean space. Using hypercontractivity methods, we establish regularity results which allow us to prove an inequality relating the free energy and its time derivative. For a solution with sub-critical mass, this allows us to give for large times an "intermediate asymptotics'' description of the vanishing. In self-similar coordinates, we actually prove a convergence result to a limiting self-similar solution which is not a simple reflect of the diffusion. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 33 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | Blanchet, A., Dolbeault, J., & Perthame, B. (2006). Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions. <i>Electronic Journal of Differential Equations, 2006</i>(44), pp. 1-33. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/13917 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University-San Marcos, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2006, San Marcos, Texas: Texas State University-San Marcos and University of North Texas. | |
| dc.subject | Keller-Segel model | |
| dc.subject | Existence | |
| dc.subject | Weak solutions | |
| dc.subject | Free energy | |
| dc.subject | Entropy method | |
| dc.subject | Logarithmic Hardy-Littlewood-Sobolev inequality | |
| dc.subject | Critical mass | |
| dc.subject | Aubin-Lions compactness method | |
| dc.subject | Hypercontractivity | |
| dc.subject | Large time behavior | |
| dc.subject | Time-dependent rescaling | |
| dc.subject | Self-similar variables | |
| dc.subject | Intermediate asymptotics | |
| dc.title | Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions | |
| dc.type | Article |