A Note on the Subcritical Two Dimensional Keller-Segel System
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A Note on the Subcritical Two Dimensional Keller-Segel System Jose A. Carrillo · Li Chen · Jian-Guo Liu · Jinhuan Wang
Received: 30 August 2011 / Accepted: 24 November 2011 / Published online: 17 December 2011 © Springer Science+Business Media B.V. 2011
Abstract The existence of solution for the 2D-Keller-Segel system in the subcritical case, i.e. when the initial mass is less than 8π, is reproved. Instead of using the entropy in the free energy and free energy dissipation, which was used in the proofs (Blanchet et al. in SIAM J. Numer. Anal. 46:691–721, 2008; Electron. J. Differ. Equ. Conf. 44:32, 2006 (electronic)), the potential energy term is fully utilized by adapting Delort’s theory on 2D incompressible Euler equation (Delort in J. Am. Math. Soc. 4:553–386, 1991). Keywords Chemotaxis · Critical mass · Global existence · Maximal density function
J.A. Carrillo is partially supported by the project MTM2011-27739-C04 DGI-MCI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya. L. Chen is partially supported by National Natural Science Foundation of China (NSFC) grant 10871112, 11011130029. The research of J.-G. Liu was partially supported by NSF grant DMS 10-11738. J. Wang is partially supported by Science Foundation of Liaoning Education Department grant L2010146 and China Postdoctoral Science Foundation grant 20110490409. J.-G. Liu wish to acknowledge the hospitality of Mathematical Sciences Center of Tsinghua University where part of this research was performed. J.A. Carrillo () ICREA and Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra (Barcelona) 08193, Spain e-mail: [email protected] L. Chen · J. Wang Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China L. Chen e-mail: [email protected] J. Wang e-mail: [email protected] J.-G. Liu Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA e-mail: [email protected] J. Wang Department of Mathematics, Liaoning University, Shenyang 110036, P.R. China
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Mathematics Subject Classification (2000) 35K55 · 35B33
1 Introduction The 2D Keller-Segel system ut = u − div(u∇c),
x ∈ R2 , t ≥ 0,
−c = u,
x ∈ R2 , t ≥ 0,
u(x, 0) = u0 (x),
x∈R ,
(1.1)
2
has widely been studied in the literature. Keller-Segel-type systems are used to describe the motion of biological cells or organisms in response to chemical gradients. The main feature of this system is that it can describe the mass aggregation phenomena in chemotaxis. The system (1.1) considered here is the simplest version of the Keller-Segel system in the parabolic-elliptic case, one can check more complete models in [12]. The sharp bound on the critical mass was given by Dolbeault and Perthame in [9]. It was announced there that if the initial mass is less than 8π then weak solutions exist globally, while in the case of initial mass larger than 8π , there must be a mass concentration. Later on, in [1] they completed the global existence of weak solution in the subcri
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