  We establish new convergence results, in strong topologies, for solutions of
the parabolic-parabolic Keller--Segel system in the plane, to the corresponding
solutions of the parabolic-elliptic model, as a physical parameter goes to
zero. Our main tools are suitable space-time estimates, implying the global
existence of slowly decaying (in general, nonintegrable) solutions for these
models, under a natural smallness assumption.
