August  2022, 15(4): 651-662. doi: 10.3934/krm.2021028

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 Published  August 2022 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 and Related Models, 2022, 15 (4) : 651-662. doi: 10.3934/krm.2021028
References:
[1]

J. BattW. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal., 93 (1986), 159-183.  doi: 10.1007/BF00279958.

[2]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.

[3]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia/PA 1996. doi: 10.1137/1.9781611971477.

[4]

R. T. Glassey and J. Schaeffer, On symmetric solutions of the relativistic Vlasov-Poisson system, Comm. Math. Phys., 101 (1985), 459-473.  doi: 10.1007/BF01210740.

[5]

R. Glassey and J. Schaeffer, Time decay for solutions to the linearized Vlasov equation, Transport Theory Statist. Phys., 23 (1994), 411–453. doi: 10.1080/00411459408203873.

[6]

R. Glassey and J. Schaeffer, The "two and one-half-dimensional" relativistic Vlasov Maxwell system, Comm. Math. Phys., 185 (1997), 257-284.  doi: 10.1007/s002200050090.

[7]

R. Kurth, A global particular solution to the initial value problem of stellar dynamics, Quart. Appl. Math., 36 (1978/79), 325-329.  doi: 10.1090/qam/508777.

[8]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford University Press, Oxford 1998.

[9]

J. Moser and E. J. Zehnder, Notes on Dynamical Systems, American Mathematical Society, Providence/RI 2005. doi: 10.1090/cln/012.

[10]

T. Ramming and G. Rein, Oscillating solutions of the Vlasov-Poisson system–a numerical investigation, Phys. D, 365 (2018), 72-79.  doi: 10.1016/j.physd.2017.10.013.

[11]

G. Rein, Stability of spherically symmetric steady states in galactic dynamics against general perturbations, Arch. Ration. Mech. Anal., 161 (2002), 27-42.  doi: 10.1007/s002050100167.

show all references

References:
[1]

J. BattW. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal., 93 (1986), 159-183.  doi: 10.1007/BF00279958.

[2]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.

[3]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia/PA 1996. doi: 10.1137/1.9781611971477.

[4]

R. T. Glassey and J. Schaeffer, On symmetric solutions of the relativistic Vlasov-Poisson system, Comm. Math. Phys., 101 (1985), 459-473.  doi: 10.1007/BF01210740.

[5]

R. Glassey and J. Schaeffer, Time decay for solutions to the linearized Vlasov equation, Transport Theory Statist. Phys., 23 (1994), 411–453. doi: 10.1080/00411459408203873.

[6]

R. Glassey and J. Schaeffer, The "two and one-half-dimensional" relativistic Vlasov Maxwell system, Comm. Math. Phys., 185 (1997), 257-284.  doi: 10.1007/s002200050090.

[7]

R. Kurth, A global particular solution to the initial value problem of stellar dynamics, Quart. Appl. Math., 36 (1978/79), 325-329.  doi: 10.1090/qam/508777.

[8]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford University Press, Oxford 1998.

[9]

J. Moser and E. J. Zehnder, Notes on Dynamical Systems, American Mathematical Society, Providence/RI 2005. doi: 10.1090/cln/012.

[10]

T. Ramming and G. Rein, Oscillating solutions of the Vlasov-Poisson system–a numerical investigation, Phys. D, 365 (2018), 72-79.  doi: 10.1016/j.physd.2017.10.013.

[11]

G. Rein, Stability of spherically symmetric steady states in galactic dynamics against general perturbations, Arch. Ration. Mech. Anal., 161 (2002), 27-42.  doi: 10.1007/s002050100167.

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