J. math. fluid. mech. 99 (9999), 1–24 1422-6928/99000-0, DOI 10.1007/s00009-003-0000 c  2009 Birkh¨ auser Verlag Basel/Switzerland Journal of Mathematical Fluid Mechanics On spherically symmetric motions of a viscous compressible barotropic and selfgravitating gas Bernard Ducomet, ˇ S´arkaNeˇ casov´a and Alexis Vasseur SUMMARY. We consider the Cauchy problem for the equations of spherically symmetric motions in R 3 , of a selfgravitating barotropic gas, with possibly non monotone pressure law, in two different situations: in the first one we suppose that the viscosities μ(ρ), and λ(ρ) are density-dependent and sat- isfy the Bresch-Desjardins condition, in the second one we consider constant densities. In the two cases, we prove that the problem admits a global weak solution, provided that the polytropic index γ satisfy γ> 1. Mathematics Subject Classification (2000). 76N10,35Q30. Keywords. spherically symmetric motion, selfgravitating gas, non monotone pressure law, density-dependent viscosities. 1. Introduction We consider the Navier-Stokes-Poisson system in R 3 for a compressible isentropic gas with density-dependent viscosities            ∂ t ρ + div(ρv)=0, ∂ t (ρv) + div(ρv ⊗ v) − div (2µ(ρ)D(v)) −∇ (λ(ρ)divv)+ ∇P − ρ∇Φ=  0, ΔΦ = −4πGρ. (1.1) Here ρ is the density, v is the velocity, Φ is the Newtonian gravitational potential (G> 0 is the Newton constant), P is the pressure, D is the strain tensor with D(v)=1/2(∇v + t ∇v).