The Vlasov-Poisson system

The Vlasov-Poisson system is a system of partial differential equations which comes up in mathematical physics. I have been involved quite a bit with these equations and related systems for many years now. In this post I want to reflect a little on what is and is not known about solutions of this system. One of the things which has stimulated me to think more about these questions just now is a lecture course on kinetic equations which I am giving at the Free University in Berlin. Because of the physics motivation the Vlasov-Poisson system is usually studied in three space dimensions. Here I will allow the space dimension n to be general. For convenience I also introduce a parameter \gamma which can take the values +1 and -1. The equations are \partial_t f+v\cdot \nabla_x f+\gamma\nabla_x U\cdot\nabla_v f=0 and \Delta U=\rho where \rho=\int f dv. Here t is time and x and v denote position and velocity variables, each belonging to {\bf R}^n. \Delta is the Laplacian. Because of their most frequent applications the cases \gamma=1 and \gamma=-1 are often called the plasma physics case and the stellar dynamics case respectively. A natural problem here is the initial value problem (Cauchy problem) with data prescribed for f.

Local existence in the Cauchy problem is known. For n\le 3 it is furthermore known that the local solution extends to a global in time solution, independently of the sign of \gamma. (The first proofs by two different methods were given by Pfaffelmoser and Lions/Perthame around 1991.) When n\ge 4 and \gamma=-1 there are large classes of smooth initial data for which global existence fails. More specifically, these equations have a conserved energy and when this energy is negative the corresponding smooth solution breaks down after finite time. The easiest way to realize that n=4 might be important is to look at scaling properties of the equations. For a discussion of the significance of scaling properties in general see Terry Tao’s post on the Navier-Stokes equations. In the case n=3 the potential and kinetic energies satisfy an inequality of the form |{\cal E}_{\rm pot}|\le C{\cal E}_{\rm kin}^{\frac12} and this plays an important role in the global existence proof. The essential feature is that the power on the right hand side is less than one. If similar arguments are carried out in the case n=4 then the power one half is replaced by the power one. Thus in a sense n=4 is the critical case. For n\ge 4 the global existence problem for the Vlasov-Poisson system with \gamma=1 is open. For n=4 it is a critical problem and might be solvable in a not too distant future. Similar remarks might be made about the relativistic Vlasov-Poisson system with massless particles in three space dimensions which is given by \partial_t f+\hat v\cdot \nabla_x f+\nabla_x U\cdot\nabla_v f=0 where \hat v=\frac{v}{|v|}. The analogue of this last system plays an important role in the recent work of Lemou, Méhats and Raphaël on the nature of singularities in solutions of the relativistic Vlasov-Poisson system with \gamma=-1.

Other open questions concern the behaviour of solutions of the Vlasov-Poisson system at late times. There are various results on this but they seem to be far from an exhaustive understanding of the asymptotics. Interesting questions include whether the density \rho is bounded any solution with \gamma=-1 and whether \|\rho\|_{L^\infty}=O(t^{-3}) in the case \gamma=1, as is known to be the case for small initial data.

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2 Responses to “The Vlasov-Poisson system”

  1. Martin Lo Says:

    Dear Hydrobates,
    Where can I find a good derivation and physical explanation for the Vlasov-Poisson equation? I’m particularly interested in the stellar dynamics case. I’m doing naive direct n-body simulations and would like to calculate the time-varying potential of some of the most interest simulations in order to further analyze it using dynamical systems theory.
    Thanks.
    Martin

    • hydrobates Says:

      Dear Martin,

      A very nice reference for the Vlasov-Poisson system in the stellar dynamics case is the book ‘Galactic Dynamics’ by Binney and Tremaine. (They use the name collisionless Boltzmann equation rather than Vlasov equation)

      Regards,
      Alan

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