We study on the whole space
the compressible Euler system with damping coupled to the Poisson equation when
the damping coefficient tends towards infinity. We first prove a result of global
existence for the Euler–Poisson system in the case where the damping is large
enough, then, in a second step, we rigorously justify the passage to the limit to the
parabolic-elliptic Keller–Segel after performing a diffusive rescaling, and get an
explicit convergence rate. The overall study is carried out in “critical” Besov spaces,
in the spirit of the recent survey by R. Danchin devoted to partially dissipative
systems.