Electron. J. Diff. Eqns.,
Vol. 2006(2006), No. 44, pp. 1-33.
Two-dimensional Keller-Segel model: Optimal critical mass
and qualitative properties of the solutions
Adrien Blanchet, Jean Dolbeault, Benoit Perthame
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.
Submitted February 28, 2006. Published April 6, 2006.
Math Subject Classifications: 35B45, 35B30, 35D05, 35K15, 35B40, 35D10, 35K60.
Key Words: 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.
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Adrien Blanchet
CEREMADE (UMR CNRS no. 7534), Université Paris Dauphine
Place de Lattre de Tassigny, 75775 Paris Cédex 16,
France
and
CERMICS, ENPC, 6-8 avenue Blaise Pascal
Champs-sur-Marne, 77455 Marne-la-Vallée Cédex 2, France
email: blanchet@ceremade.dauphine.fr
http://www.ceremade.dauphine.fr/~blanchet/
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Jean Dolbeault
CEREMADE (UMR CNRS no. 7534), Université Paris Dauphine
Place de Lattre de Tassigny, 75775 Paris Cédex 16,
France
Phone: (33) 1 44 05 46 78, Fax: (33) 1 44 05 45 99
email: dolbeaul@ceremade.dauphine.fr
http://www.ceremade.dauphine.fr/~dolbeaul/
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Benoît Perthame
DMA (UMR CNRS no. 8553), Ecole Normale Supérieure
45 rue d'Ulm, 75005 Paris Cédex 05, France
email: Benoit.Perthame@ens.fr
http://www.dma.ens.fr/users/perthame
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