## Duke Mathematical Journal

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

#### Abstract

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

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

Dates
First available in Project Euclid: 14 February 2013

https://projecteuclid.org/euclid.dmj/1360874856

Digital Object Identifier
doi:10.1215/00127094-2019931

Mathematical Reviews number (MathSciNet)
MR3024094

Zentralblatt MATH identifier
1307.26027

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

#### Citation

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

#### References

• [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.