  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 $T^*=\infty$ in the case when $M\le8\pi$ and $T^*< \infty$ in the case
when $M> 8\pi$. 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 $M < 8\pi$, we prove Sobolev norm
bound uniformly in time for the rescaled solution (corresponding to the
self-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 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.
