Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 189–232 | Cite as

Stationary Solutions of the Flat Vlasov–Poisson System

  • Jürgen BattEmail author
  • Enno Jörn
  • Yi Li


The stationary solutions are triples \({\left(f, \rho, U\right)}\) of three functions: the distribution function \({f = f \left(x, v\right)}\), the potential \({U = U \left(x\right)}\) and the local density \({\rho = \rho \left(x\right)}\), \({x, v \in \mathbb{R}^2}\), which are linked by the Vlasov–Poisson system. We prove the existence of wide classes of spherically symmetric stationary solutions with the property that \({\rho}\) depends on \({\left|x\right| = r}\) and f on the energy \({E:= U\left(x\right) + \frac{v^2}2}\). First we answer the question of which given functions \({\rho}\) are the local density of a stationary solution (inverse problem). Our result is (up to technicalities) that every \({\rho \geq 0}\) which is strictly decreasing on an interval \({\left[0, R\right)}\) and zero on its complement (\({R \leqq \infty}\)) belongs to this class. Second, we ask: which given functions q induce distribution functions f of the form \({f=q \left(-E_0 -E \right)}\) (\({E_0 \geqq 0}\)) of a stationary solution? (direct problem). This question is answered for many q which are positive for positive and vanish for negative arguments in an approximative and constructive way which is based on numerical methods.


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Authors and Affiliations

  1. 1.Mathematisches Institut der Universität MünchenMunichGermany
  2. 2.Department of MathematicsCalifornia State UniversityNorthridgeUSA

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