Acta Applicandae Mathematicae

, Volume 119, Issue 1, pp 43–55 | Cite as

A Note on the Subcritical Two Dimensional Keller-Segel System

  • Jose A. CarrilloEmail author
  • Li Chen
  • Jian-Guo Liu
  • Jinhuan Wang


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).


Chemotaxis Critical mass Global existence Maximal density function 

Mathematics Subject Classification (2000)

35K55 35B33 


  1. 1.
    Blanchet, A., Dolbeault, J., Perthame, B.: Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Equ. Conf. 44, 32 (2006) (electronic) Google Scholar
  2. 2.
    Blanchet, A., Calvez, V., Carrillo, J.A.: Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model. SIAM J. Numer. Anal. 46, 691–721 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Blanchet, A., Carrillo, J.A., Masmoudi, N.: Infinite time aggregation for the critical Patlak-Keller-Segel model in ℝ2. Commun. Pure Appl. Math. 61, 1449–1481 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Blanchet, A., Carlen, E.A., Carrillo, J.A.: Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model. arXiv:1009.0134v1
  5. 5.
    Carlen, E., Loss, M.: Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on S n. Geom. Funct. Anal. 2, 90–104 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Carlen, E., Carrillo, J.A., Loss, M.: Hardy-Littlewood-Sobolev inequalities via fast diffusion flows. Proc. Natl. Acad. Sci. USA 107, 19696–19701 (2010) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Delort, J.M.: Existence de nappes de tourbillon en dimension deux. J. Am. Math. Soc. 4, 386–553 (1991) MathSciNetCrossRefGoogle Scholar
  8. 8.
    DiPerna, R., Majda, A.: Concentrations in regularizations for 2D incompressible flow. Commun. Pure Appl. Math. 40, 301–345 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dolbeault, J., Perthame, B.: Optimal critical mass in the two dimensional Keller-Segel model in ℝ2. C.R. Acad. Sci. Paris, Ser. I 339, 611–616 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dolbeault, J., Schmeiser, C.: The two-dimensional Keller-Segel model after blow-up. Discrete Contin. Dyn. Syst. 25, 109–121 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dunford, N., Schwartz, J.T.: Linear Operators: General Theory. Wiley-Interscience, New York (1988) zbMATHGoogle Scholar
  12. 12.
    Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14, 2nd edn. Am. Math. Soc., Providence (2001) Google Scholar
  14. 14.
    Liu, J.G., Xin, Z.P.: Convergence of vortex methods for weak solution to the 2-D Euler equations with vortex sheet data. Commun. Pure Appl. Math. 48, 611–628 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Liu, J.G., Xin, Z.P.: Convergence of point vortex method for 2-D vortex sheet. Math. Comput. 70, 565–606 (2001) MathSciNetGoogle Scholar
  16. 16.
    Majda, A.J.: Remarks on weak solution for vortex sheets with a distinguished sign. Indiana Univ. Math. J. 42, 921–939 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Perthame, B.: Transport Equations in Biology. Birkhäuser, Basel (2007) zbMATHGoogle Scholar
  18. 18.
    Poupaud, F.: Diagonal defect measures, adhesion dynamics and Euler equation. Methods Appl. Anal. 9, 533–562 (2002) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Jose A. Carrillo
    • 1
    Email author
  • Li Chen
    • 2
  • Jian-Guo Liu
    • 3
  • Jinhuan Wang
    • 2
    • 4
  1. 1.ICREA and Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra (Barcelona)Spain
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingP.R. China
  3. 3.Department of Physics and Department of MathematicsDuke UniversityDurhamUSA
  4. 4.Department of MathematicsLiaoning UniversityShenyangP.R. China

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