Article Contents
Article Contents

# A second look at the Kurth solution in galactic dynamics

• 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.

Mathematics Subject Classification: Primary: 35Q83.

 Citation:

•  [1] J. Batt, W. 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. Gidas, W. 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.