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 * < ∞ in the case when M > 8π. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the "optimal regularity" as well as an estimate of the difference of two possible solutions in the critical L 4/3 Lebesgue norm similarly as for the 2d vorticity Navier-Stokes equation.(2) We prove immediate smoothing effect and, in the case M < 8π, we prove 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 weighted L 4/3 linearized stability of the self-similar profile and then 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.

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