Stationary Spherically Symmetric Solutions of the Vlasov–Poisson System in the Three-Dimensional Case

Abstract

We consider the three-dimensional stationary Vlasov–Poisson system of equations with respect to the distribution function of the gravitating matter $f = {{f}_{q}}(r,u)$, the local density $\rho = \rho (r)$, and the Newtonian potential $U = U(r)$, where $r: = {\text{|}}x{\text{|}}$, $u: = {\text{|}}v{\text{|}}$ ($(x,v) \in {{\mathbb{R}}^{3}} \times {{\mathbb{R}}^{3}}$ are the space–velocity coordinates), and f is a function q of the local energy $E: = U(r) + \tfrac{{{{u}^{2}}}}{2}$. For a given function $p = p(r)$, we obtain sufficient conditions for p to be “extendable.” This means that there exists a stationary spherically symmetric solution $({{f}_{q}},\rho ,U)$ of the Vlasov–Poisson system depending on the local energy E such that ρ = p.