Stability for the logarithmic Hardy-Littlewood-Sobolev Inequality with application to the Keller-Segel system

Abstract

We apply a duality method to prove an optimal stability theorem for the logarithmic Hardy-Littlewood-Sobolev inequality, and we apply it to the estimation of the rate of approach to equilibrium for the critical mass Keller-Segel system.

This version corrects various typos in the previous version, and includes a new final section using the same methods to give a sharp stability bound for a one dimensional inequality of log HLS the due to Osgood, Philips and Sarnak. This was added on the suggestion of Rupert Frank, whom I thank