Duke Mathematical Journal

Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation

Eric A. Carlen and Alessio Figalli

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev inequality, we deduce several different stability results for a Gagliardo–Nirenberg–Sobolev (GNS) inequality in the plane. Then, exploiting the connection between this inequality and a fast diffusion equation, we get stability for the logarithmic Hardy–Littlewood–Sobolev (Log-HLS) inequality. Finally, using all these estimates, we prove a quantitative convergence result for the critical mass Keller–Segel system.

Article information

Duke Math. J., Volume 162, Number 3 (2013), 579-625.

First available in Project Euclid: 14 February 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A45: Miscellaneous inequalities involving matrices
Secondary: 49M20: Methods of relaxation type


Carlen, Eric A.; Figalli, Alessio. Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation. Duke Math. J. 162 (2013), no. 3, 579--625. doi:10.1215/00127094-2019931. https://projecteuclid.org/euclid.dmj/1360874856

Export citation


  • [1] L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Lectures Math. ETH Zürich, Birkhäuser, Basel, 2005.
  • [2] Th. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976), 573–598.
  • [3] D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of diffusion semigroups, in preparation.
  • [4] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), 213–242.
  • [5] G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18–24.
  • [6] A. Blanchet, E. A. Carlen, and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model, J. Funct. Anal. 262 (2012), 2142–2230.
  • [7] M. Bonforte and J. L. Vázquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal. 240 (2006), 399–428.
  • [8] E. A. Carlen, J. A. Carrilo, and M. Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows, Proc. Natl. Acad. Sci. USA 107 (2010), 19696–19701.
  • [9] E. A. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on $S^{n}$, Geom. Funct. Anal. 2 (1992), 90–104.
  • [10] E. A. Carlen and M. Loss., Sharp constant in Nash’s inequality, Int. Math. Res. Not. IMRN 1993, 213–215.
  • [11] A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc. (JEMS) 11 (2009), 1105–1139.
  • [12] D. Cordero-Erausquin, B. Nazaret, and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), 307–332.
  • [13] M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9) 81 (2002), 847–875.
  • [14] J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in $\mathord{\mathbb{R}}^{2}$, C. R. Math. Acad. Sci. Paris 339 (2004), 611–616.
  • [15] A. Figalli, F. Maggi, and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math. 182 (2010), 167–211.
  • [16] A. Figalli, F. Maggi, and A. Pratelli, Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation, preprint.
  • [17] N. Fusco, F. Maggi, and A. Pratelli, The sharp quantitative Sobolev inequality for functions of bounded variation, J. Funct. Anal. 244 (2007), 315–341.
  • [18] N. Fusco, F. Maggi, and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168 (2008), 941–980.
  • [19] E. H. Lieb and M. Loss, Analysis, Grad. Stud. Math. 14, Amer. Math. Soc., Providence, 1997.
  • [20] R. J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), 153–179.
  • [21] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), 101–174.
  • [22] G. Rosen, Minimum value for $c$ in the Sobolev inequality $\|{\phi^{3}}\|\leq c\|{\nabla\phi}\|^{3}$, SIAM J. Appl. Math. 21 (1971), 30-32.
  • [23] G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353-372.