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.

中文翻译：

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三维情况下Vlasov-Poisson系统的平稳球对称解

摘要

考虑重力物质\（f = {{{f} _ {q}}（r，u）\），局部密度\（\ rho = \ rho（r）\）和牛顿势\（U = U（r）\），其中\（r：= {\ text {|}} x {\ text {|}} \），\ （u：= {\ text {|}} v {\ text {|}} \）（\（（x，v）\ in {{\ mathbb {R}} ^ {3}} \ times {{\ mathbb {R}} ^ {3}} \）是空速坐标），和˚F是一个函数q局部能量的ë\（：= U（R）+ \ tfrac {{{{U】^ {2 }}}} {2} \）。对于给定的函数\（p = p（r）\），我们获得了p的充分条件是“可扩展的”。这意味着根据局部能量E，存在一个Vlasov-Poisson系统的平稳球对称解\（（{{f（f）_ {q}}，rho，U）\），因此ρ= p。