Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions
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...
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Texas State University
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doaj-b8c14c6be93944ea87b112a4c117c60e2020-11-25T01:07:36ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912006-04-01200644132Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutionsAdrien BlanchetJean DolbeaultBenoit PerthameThe 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. http://ejde.math.txstate.edu/Volumes/2006/44/abstr.htmlKeller-Segel modelexistenceweak solutionsfree energyentropy methodlogarithmic Hardy-Littlewood-Sobolev inequalitycritical massAubin-Lions compactness methodhypercontractivitylarge time behaviortime-dependent rescalingself-similar variablesintermediate asymptotics. |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Adrien Blanchet Jean Dolbeault Benoit Perthame |
spellingShingle |
Adrien Blanchet Jean Dolbeault Benoit Perthame Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions Electronic Journal of Differential Equations Keller-Segel model existence weak solutions free energy entropy method logarithmic Hardy-Littlewood-Sobolev inequality critical mass Aubin-Lions compactness method hypercontractivity large time behavior time-dependent rescaling self-similar variables intermediate asymptotics. |
author_facet |
Adrien Blanchet Jean Dolbeault Benoit Perthame |
author_sort |
Adrien Blanchet |
title |
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions |
title_short |
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions |
title_full |
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions |
title_fullStr |
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions |
title_full_unstemmed |
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions |
title_sort |
two-dimensional keller-segel model: optimal critical mass and qualitative properties of the solutions |
publisher |
Texas State University |
series |
Electronic Journal of Differential Equations |
issn |
1072-6691 |
publishDate |
2006-04-01 |
description |
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. |
topic |
Keller-Segel model existence weak solutions free energy entropy method logarithmic Hardy-Littlewood-Sobolev inequality critical mass Aubin-Lions compactness method hypercontractivity large time behavior time-dependent rescaling self-similar variables intermediate asymptotics. |
url |
http://ejde.math.txstate.edu/Volumes/2006/44/abstr.html |
work_keys_str_mv |
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