A functional framework for the Keller-Segel system: logarithmic   Hardy-Littlewood-Sobolev and related spectral gap inequalities
  This note is devoted to several inequalities deduced from a special form of the logarithmic Hardy-Littlewood-Sobolev, which is well adapted to the characterization of stationary solutions of a Keller-Segel system written in self-similar variables, in case of a subcritical mass. For the corresponding evolution problem, such functional inequalities play an important role for identifying the rate of convergence of the solutions towards the stationary solution with same mass. 
Jean Dolbeault (CEREMADE), Juan Campos Serrano (CEREMADE, DIM)
math.AP