Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case

Abstract

The present paper deals with the parabolic–elliptic Keller–Segel equation in the plane in the general framework of weak (or “free energy”) solutions associated to initial datum with finite mass M, finite second moment and finite entropy. The aim of the paper is threefold: (1) We prove the uniqueness of the “free energy” solution on the maximal interval of existence [0,T*) with T* = ∞ in the case when M ≦ 8π and T* 8π. The proof uses a DiPerna–Lions renormalizing argument which makes it possible to get the “optimal regularity” as well as an estimate of the difference of two possible solutions in the critical L4/3 Lebesgue norm similarly to the 2d vorticity Navier–Stokes equation. (2) We prove the immediate smoothing effect and, in the case M < 8π, we prove the Sobolev norm bound uniformly in time for the rescaled solution (corresponding to the self-similar variables). (3) In the case M < 8π, we also prove the weighted L4/3 linearized stability of the self-similar profile and then the universal optimal rate of convergence of the solution to the self-similar profile. The proof is mainly based on an argument of enlargement of the functional space for semigroup spectral gap.