Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 189–232

# Stationary Solutions of the Flat Vlasov–Poisson System

Article

## Abstract

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

• Jürgen Batt
• 1
• Enno Jörn
• 1
• Yi Li
• 2
1. 1.Mathematisches Institut der Universität MünchenMunichGermany
2. 2.Department of MathematicsCalifornia State UniversityNorthridgeUSA