Batt, Jürgen; Jörn, Enno; Li, Yi
(2019):
Stationary Solutions of the Flat VlasovPoisson System.
In: Archive for Rational Mechanics and Analysis, Vol. 231, No. 1: pp. 189232

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Abstract
The stationary solutions are triples (f,., U) of three functions: the distribution function f = f (x, v), the potential U = U (x) and the local density. =. (x), x, v. R2, which are linked by the VlasovPoisson system. We prove the existence of wide classes of spherically symmetric stationary solutions with the property that. depends on  x = r and f on the energy E := U (x) + v2 2. First we answer the question of which given functions. are the local density of a stationary solution (inverse problem). Our result is (up to technicalities) that every. = 0 which is strictly decreasing on an interval [ 0, R) and zero on its complement (R similar to 8) belongs to this class. Second, we ask: which given functions q induce distribution functions f of the form f = q (E0 E) (E0 similar to 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.