Spherically Symmetric Equilibria for Self-Gravitating Kinetic or Fluid Models in the Nonrelativistic and Relativistic Case---A Simple Proof for Finite Extension

We consider a self-gravitating collisionless gas as described by the Vlasov--Poisson or Einstein--Vlasov system or a self-gravitating fluid ball as described by the Euler--Poisson or Einstein--Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cutoff energy $E_0$, and in the Euler case by the corresponding growth condition on the equation of state $p=P(\rho)$ at $\rho=0$. These purely local conditions are slight generalizations to known such conditions. [PUBLICATION ABSTRACT]