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. © 2013 Duke University Press.