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

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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

Permanent link to this document
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


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