# American Institute of Mathematical Sciences

doi: 10.3934/krm.2021028
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## A second look at the Kurth solution in galactic dynamics

 Universität Köln, Institut für Mathematik, Weyertal 86-90, D-50931 Köln, Germany

Received  March 2021 Revised  June 2021 Early access August 2021

The Kurth solution is a particular non-isotropic steady state solution to the gravitational Vlasov-Poisson system. It has the property that by means of a suitable time-dependent transformation it can be turned into a family of time-dependent solutions. Therefore, for a general steady state
 $Q(x, v) = \tilde{Q}(e_Q, \beta)$
, depending upon the particle energy
 $e_Q$
and
 $\beta = \ell^2 = |x\wedge v|^2$
, the question arises if solutions
 $f$
could be generated that are of the form
 $f(t) = \tilde{Q}\Big(e_Q(R(t), P(t), B(t)), B(t)\Big)$
for suitable functions
 $R$
,
 $P$
and
 $B$
, all depending on
 $(t, r, p_r, \beta)$
for
 $r = |x|$
and
 $p_r = \frac{x\cdot v}{|x|}$
. We are going to show that, under some mild assumptions, basically if
 $R$
and
 $P$
are independent of
 $\beta$
, and if
 $B = \beta$
is constant, then
 $Q$
already has to be the Kurth solution.
This paper is dedicated to the memory of Professor Robert Glassey.
Citation: Markus Kunze. A second look at the Kurth solution in galactic dynamics. Kinetic & Related Models, doi: 10.3934/krm.2021028
##### References:

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