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The present paper deals with the parabolic-elliptic Keller-Segel equation inthe plane in the general framework of weak (or ''free energy") solutionsassociated to initial datum with finite mass $M$, finite second moment andfinite entropy. The aim of the paper is threefold: (1) We prove the uniquenessof the ''free energy" solution on the maximal interval of existence $[0,T^*)$with $T^*=\infty$ in the case when $M\le8\pi$ and $T^*< \infty$ in the casewhen $M> 8\pi$. The proof uses a DiPerna-Lions renormalizing argument whichmakes possible to get the ''optimal regularity" as well as an estimate of thedifference of two possible solutions in the critical $L^{4/3}$ Lebesgue normsimilarly as for the $2d$ vorticity Navier-Stokes equation. (2) We proveimmediate smoothing effect and, in the case $M < 8\pi$, we prove Sobolev normbound uniformly in time for the rescaled solution (corresponding to theself-similar variables). (3) In the case $M < 8\pi$, we also prove weighted$L^{4/3}$ linearized stability of the self-similar profile and then universaloptimal rate of convergence of the solution to the self-similar profile. Theproof is mainly based on an argument of enlargement of the functional space forsemigroup spectral gap.

Uniqueness and long time asymptotic for the Keller-Segel equation: The parabolic-elliptic case