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Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities

Authors: Vincent Calvez and José Antonio Carrillo
Journal: Proc. Amer. Math. Soc. 140 (2012), 3515-3530
MSC (2010): Primary 35B40, 92C17
Published electronically: February 23, 2012
MathSciNet review: 2929020
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Abstract: We analyze the rate of convergence towards self-similarity for the subcritical Keller-Segel system in the radially symmetric two-dimensional case and in the corresponding one-dimensional case for logarithmic interaction. We measure convergence in the Wasserstein distance. The rate of convergence towards self-similarity does not degenerate as we approach the critical case. As a byproduct, we obtain a proof of the Logarithmic Hardy-Littlewood-Sobolev inequality in the one-dimensional and radially symmetric two-dimensional cases based on optimal transport arguments. In addition we prove that the one-dimensional equation is a contraction with respect to Fourier distance in the subcritical case.

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  • 1. L. A. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, Birkhäuser, 2005. MR 2129498 (2006k:49001)
  • 2. F. Barthe, Inégalités de Brascamp-Lieb et convexité, C. R. Math. Acad. Sci. Paris 324 (1997), no. 8, 885-888. MR 1450443 (98a:26022)
  • 3. F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), no. 2, 335-361. MR 1650312 (99i:26021)
  • 4. W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), no. 1, 213-242. MR 1230930 (94m:58232)
  • 5. P. Biler, G. Karch, P. Laurençot, T. Nadzieja, The $ 8\pi $-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci. 29 (2006), no. 13, 1563-1583. MR 2249579 (2007f:35156)
  • 6. P. Biler, T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles I, Colloq. Math. 66 (1994), no. 2, 319-334. MR 1268074 (95b:35223)
  • 7. A. Blanchet, V. Calvez, J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal. 46 (2008), no. 2, 691-721. MR 2383208 (2009a:35113)
  • 8. A. Blanchet, E. Carlen, J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Funct. Anal., to appear.
  • 9. A. Blanchet, J. A. Carrillo, P. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations 35 (2009), no. 2, 133-168. MR 2481820 (2010e:35149)
  • 10. A. Blanchet, J. A. Carrillo, N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $ \mathbb{R}^2$, Comm. Pure Appl. Math. 61 (2008), no. 10, 1449-1481. MR 2436186 (2009d:35132)
  • 11. A. Blanchet, J. Dolbeault, M. Escobedo, J. Fernández, Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model, J. Math. Anal. Appl. 361 (2010), no. 2, 533-542. MR 2568716 (2010m:35190)
  • 12. A. Blanchet, J. Dolbeault, B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations (2006), no. 44, 32 pp. (electronic). MR 2226917 (2007e:35277)
  • 13. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375-417. MR 1100809 (92d:46088)
  • 14. V. Calvez, B. Perthame, M. Sharifi Tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, Stochastic analysis and partial differential equations, 45-62, Contemp. Math., 429, Amer. Math. Soc., Providence, RI, 2007. MR 2391528 (2009d:60272)
  • 15. E. Carlen, J.A. Carrillo, M. Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows, Proc. Natl. Acad. Sci. USA 107 (2010), no. 46, 19696-19701. MR 2745814
  • 16. E. Carlen, M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $ S\sp n$, Geom. Funct. Anal. 2 (1992), no. 1, 90-104. MR 1143664 (93b:58170)
  • 17. J. A. Carrillo, G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma 6 (2007), 75-198. MR 2355628 (2008i:82094)
  • 18. D. Cordero-Erausquin, B. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), no. 2, 307-332. MR 2032031 (2005b:26023)
  • 19. J. Dolbeault, B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in $ \mathbb{R}^2$, C. R. Math. Acad. Sci. Paris 339 (2004), no. 9, 611-616. MR 2103197 (2005h:35189)
  • 20. B. Düring, D. Matthes, G. Toscani, A Boltzmann-type approach to the formation of wealth distribution curves, Riv. Mat. Univ. Parma 1 (2009), 199-261. MR 2597795
  • 21. R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998), no. 1, 1-17. MR 1617171 (2000b:35258)
  • 22. N. I. Kavallaris, P. Souplet, Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk, SIAM J. Math. Anal. 40 (2008/09), no. 5, 1852-1881. MR 2471903 (2010a:35121)
  • 23. E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability,
    J. Theor. Biol. 26 (1970), no. 3, 399-415.
  • 24. E. H. Lieb, M. Loss, Analysis, Vol. 14, Amer. Math. Soc, Providence, 1997. MR 1415616 (98b:00004)
  • 25. R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995), no. 2, 309-323. MR 1369395 (97d:49045)
  • 26. R. J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), no. 1, 153-179. MR 1451422 (98e:82003)
  • 27. T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995), no. 2, 581-601 MR 1361006 (96j:35249)
  • 28. F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), no. 1-2, 101-174. MR 1842429 (2002j:35180)
  • 29. C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953), 311-338. MR 0081586 (18:424f)
  • 30. C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, Vol. 58, Amer. Math. Soc, Providence, RI, 2003. MR 1964483 (2004e:90003)

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Additional Information

Vincent Calvez
Affiliation: École Normale Supérieure de Lyon, UMR CNRS 5669 “Unité de Mathématiques Pures et Appliquées”, 46 allée d’Italie, F-69364 Lyon Cedex 07, France

José Antonio Carrillo
Affiliation: Institució Catalana de Recerca i Estudis Avançats and Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain

Received by editor(s): July 16, 2010
Received by editor(s) in revised form: April 12, 2011
Published electronically: February 23, 2012
Communicated by: Walter Craig
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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