Self-similar solutions of the Einstein-Vlasov system

The Einstein-Vlasov system describes the evolution of a collisionless gas in general relativity. The unknown in the Vlasov equation is a function f(t,x,v), the number density of particles with position x and velocity v at time t. A regular solution is one for which the function f is smooth (or at least C^1). These equations can be used to model gravitational collapse in general relativity, i.e. the process by which a concentration of matter contracts due to its own weight. I concentrate here on the case that the configuration is spherically symmetric since it is already difficult enough to analyse. It has been known for a long time that a solution of this system corresponding to a sufficiently small concentration of matter does not collapse. The matter spreads out at late times, with the matter density and the gravitational field tending to zero. More recently it has been proved that there is a class of data for which a black hole is formed. In particular singularities occur in these equations. It is of interest to know whether singularities can occur which are not contained in black holes. This is the question of cosmic censorship.

One way of trying to prove cosmic censorship involves investigating whether general initial data give rise to solutions which are global in a certain type of coordinate system. In spherical symmetry this has been looked at for the Schwarzschild coordinates. Here we can ask whether solutions are always global in Schwarzschild time. This statement is consistent with the presence of a black hole since in that case it can happen that the coordinate system being considered only covers a region outside the black hole. For general smooth spherically symmetric initial data this global existence question is open. A number of people, including myself, have put a lot of time and effort into proving global existence for this problem but this enterprise has not yet been successful. In view of the fact that research on this subject seems to be stuck it makes sense to think about trying to prove the opposite statement, in other words to prove that there are data for which global existence fails. This might in principle lead to either a positive or a negative answer to the global existence problem. An investigation of this type is being carried out by Juan Velázquez and myself and we have just written a paper on this. I will now explain what we were able to prove.

A type of matter which is frequently studied in general reletivity is dust, a perfect fluid with zero pressure. It has unpleasant mathematical properties and a strong tendency to form singularities even in the absence of gravity. Solutions of the Einstein-dust system can be interpreted as solutions of the Einstein-Vlasov system which instead of being regular have a Dirac \delta dependence on the velocity variables. For fixed (t,x) the support of f in v is a single point. In a smooth solution this support is three-dimensional. In the paper we look at a class of solutions where the support is two-dimensional. These solutions are self-similar. The construction of solutions of the type considerd in the paper can be reduced to the study of certain solutions of a four-dimensional dynamical system depending on two parameters. There is a point P_0 is the phase space defined by the application and a stationary solution P_1 depending on the parameters. What needs to be shown is that for suitable values of the parameters the solution which starts at P_0 converges to P_1 for large values of the independent variable. In more detail, when one of the parameters y_0 is fixed to be positive and sufficiently small there exists a value of the other paramater \theta for which this statement holds. The proof is a shooting argument. This uses a family of initial data for a three-dimensional dynamical system depending on a parameter q_0. (The relation of the three-dimensional system to the original four-dimensional one is too complicated to be described here.) It is proved that the solution has one type of asymptotic behaviour for small values of q_0 and another for large values of q_0. It is shown that there must be at least one intermediate value of the parameter for which the asymptotics is of the type required to construct the solution of interest of the Einstein-Vlasov system. Shooting arguments are rather common when constructing solutions of ODE numerically. Here we have an example where similar ideas can be used to obtain an existence proof.


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