REFINED ASYMPTOTICS FOR THE SUBCRITICAL KELLER-SEGEL SYSTEM AND RELATED FUNCTIONAL INEQUALITIES

We analyze the rate of convergence towards self-similarity for the subcritical Keller-Segel system in the radially symmetric two-dimensional case and in the corresponding one-dimensional case for logarithmic interaction. We measure convergence in the Wasserstein distance. The rate of convergence towards self-similarity does not degenerate as we approach the critical case. As a byproduct, we obtain a proof of the Logarithmic Hardy-Littlewood-Sobolev inequality in the one-dimensional and radially symmetric two-dimensional cases based on optimal transport arguments. In addition we prove that the one-dimensional equation is a contraction with respect:, to Fourier distance in the subcritical case.