Über Familien sphärisch symmetrischer stationärer Lösungen des ...

ABSTRACT
The present thesis is concerned with an investigation of families of steady states
of the Vlasov-Poisson system. This nonlinear system of partial differential equations
can be used for the dynamical description of a closed large particle ensemble
in a six-dimensional phase space, only interacting by gravity.
For the construction of stationary solutions we use a well-known approach
by making an ansatz for the density on phase space as a function of the particle
energy. According to a more general result of Gidas, Ni and Nirenberg
[7] solutions of this type are always spherically symmetric, whereby the angular
momentum is conserved along solutions of the characteristic system of
the Vlasov equation. Therefore, up to technical details, any function of these
conserved quantities satisfies the Vlasov equation, and it remains to solve a
semilinear Poisson equation which, due to spherical symmetry, reduces to a
second-order integro-differential equation. Several papers in recent years have
adressed the question under which assumptions on the ansatz function related
solutions have finite mass and compact support (cf. Rein and Rendall [20]
or Heinzle, Rendall and Ugla [8]).
For the outlined approach we give a new and simple proof for the compactness
of the support of the resulting solutions. Our method easily applies to a whole
class of systems (Vlasov-Poisson, relativistic Vlasov-Poisson, Einstein-Vlasov,
Euler-Poisson and Einstein-Euler system) so that they can unlike before be
treated by the same method. Moreover, the conditions we give are slight generalizations
to such conditions known.
In general, the set of steady states related to an ansatz function can in a
natural way be seen as one-parameter family. The qualitative structure of the
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