## Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation.(English)Zbl 1307.26027

The authors obtain some stability results for a Galiardo-Nirenberg-Sobolev inequality in the plane and for the logarithmic Hardy-Littlewood-Sobolev inequality. Finally, they prove a quantitative result for the critical mass Keller-Segel system.
Reviewer: Emil Popa (Sibiu)

### MSC:

 26D15 Inequalities for sums, series and integrals 26D10 Inequalities involving derivatives and differential and integral operators 49M20 Numerical methods of relaxation type
Full Text:

### 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. · Zbl 1090.35002 [2] Th. Aubin, Problèmes isopérimétriques et espaces de Sobolev , J. Differential Geom. 11 (1976), 573-598. · Zbl 0371.46011 [3] D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of diffusion semigroups , in preparation. · Zbl 1376.60002 [4] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality , Ann. of Math. (2) 138 (1993), 213-242. · Zbl 0826.58042 [5] G. Bianchi and H. Egnell, A note on the Sobolev inequality , J. Funct. Anal. 100 (1991), 18-24. · Zbl 0755.46014 [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. · Zbl 1237.35155 [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. · Zbl 1107.35063 [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. · Zbl 1256.42028 [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. · Zbl 0754.47041 [10] E. A. Carlen and M. Loss., Sharp constant in Nash’s inequality , Int. Math. Res. Not. IMRN 1993 , 213-215. · Zbl 0822.35018 [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. · Zbl 1185.46025 [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. · Zbl 1048.26010 [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. · Zbl 1112.35310 [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. · Zbl 1056.35076 [15] A. Figalli, F. Maggi, and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities , Invent. Math. 182 (2010), 167-211. · Zbl 1196.49033 [16] A. Figalli, F. Maggi, and A. Pratelli, Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation , · Zbl 1286.46035 [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. · Zbl 1121.46029 [18] N. Fusco, F. Maggi, and A. Pratelli, The sharp quantitative isoperimetric inequality , Ann. of Math. (2) 168 (2008), 941-980. · Zbl 1187.52009 [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. · Zbl 0901.49012 [21] F. Otto, The geometry of dissipative evolution equations: the porous medium equation , Comm. Partial Differential Equations 26 (2001), 101-174. · Zbl 0984.35089 [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. · Zbl 0201.38704 [23] G. Talenti, Best constants in Sobolev inequality , Ann. Mat. Pura Appl. (4) 110 (1976), 353-372. · Zbl 0353.46018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.