Advanced Search
Article Contents
Article Contents

A second look at the Kurth solution in galactic dynamics

Abstract Full Text(HTML) Related Papers Cited by
  • 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.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
  • 加载中

Article Metrics

HTML views(506) PDF downloads(447) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint