The relativistic Boltzmann equation

In a previous post I wrote about the Einstein-Boltzmann system and some recent work on that subject by Ho Lee and myself. One of the things we realized as a consequence of this work is that the known local existence theorem for the Einstein-Boltzmann system requires very restrictive assumptions on the collision kernel. Now we have looked in more detail at other kinds of collision kernel which are closer to what is desirable from the point of view of the physical applications. As a result of this we have written a paper which is concerned with the hard potential type of collision kernel. The subject of the paper is the Boltzmann equation in special relativity or on a homogeneous and isotropic background. This is intended to prepare the ground for similar work on the coupled Einstein-Boltzmann system. The main results are global existence theorems for spatially homogeneous solutions of the Boltzmann equation without any small data restriction. They are analogous to results obtained previously by Norbert Noutchegueme and collaborators for a more restrictive type of collision kernel.

The collision kernel is a function of the relative momentum and the scattering angle . In the case of the classical (i.e. non-relativistic) Boltzmann equation a type of collision kernel which has often been studied is that arising from a power-law interaction between particles. The corresponding collision kernel has a power-law dependence on and a dependence on which cannot be determined explicitly. It has a non-integrable singularity in at . It has been observed that properties of solutions of the Boltzmann equation determine two different regimes for the exponent of . The cases between an inverse square and an inverse fifth power force between particles are known as soft potentials. The exponent of varies from to . Cases with powers of the force more negative than are known as hard potentials. As the power in the force ranges from to the exponent of varies from to zero. When the exponent of is equal to   there are simplifications in some calculations and this has led to this case being popular among theorists. It is called Maxwell molecules. The limit where the exponent of tends to zero corresponds to the case of collisions of hard spheres.

Given the importance of the distinction between the soft potential and hard potential cases in the theory of the classical Boltzmann equation it is natural to look for an analogous distinction in the relativistic case. This was done by Dudynski and Ekiel-Jezewska. The analogy seems to be not at all straightforward. This work was carried further by Robert Strain and collaborators, who were able to apply these concepts and obtain a variety of global existence results. In their work the data are not required to be symmetric but are assumed to be close to data for known solutions. Our work is at the opposite extreme with a very strong symmetry assumption (spatially homogeneous) but no smallness requirement. It is modelled on theorems for the classical Boltzmann equation due to Mischler and Wennberg in Annales IHP (Analyse non lineaire) 16, 467. There is an analogue of Maxwell molecules in the relativistic case called Israel molecules but the analogy is not simple. My global conclusion from my experience with this problem is that there are a lot of interesting and challenging open problems around in the study of the relativistic Boltzmann equation and the Einstein-Boltzmann system.

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The Einstein-Boltzmann system

The Boltzmann equation provides a description of the dynamics of a large number of particles undergoing collisions, such as the molecules of a gas. The classical Boltzmann equation belongs to Newtonian physics. It has a natural relativistic generalization. The Boltzmann model is adapted to capture the effects of short-range forces acting on short time scales during collisions. The model can be extended to also include the effects of long-range forces generated collectively by the particles. If the forces are gravitational and the description is made fully relativistic then the system of equations obtained is the Einstein-Boltzmann system. In any of the cases mentioned up to now the Boltzmann equation is schematically of the form . The term on the left is a transport term giving the rate of change of the function , the density of particles, along a vector field on phase space. The vector field is in general determined by the long-range forces. The term on the right is the collision term which, as its name suggests, models the effect of collisions. It is an integral term which is quadratic in . The function itself is a function of variables representing time, position and velocity (or momentum).

How is the collision term obtained? It is important to realize that it is in no sense universal – it contains information about the particular interaction between the particles due to collisions. This can be encoded in what is called the scattering kernel. In the classical case it is possible to do the following. Fix a type of interaction between individual particles and solve the corresponding scattering problem. Each specific choice of interaction gives a scattering kernel. Once various scattering kernels have been obtained in this way it is possible to abstract from the form of the kernels obtained to define a wider class. A similar process can be carried out in special relativity although it is more complicated. Any scattering kernel which has been identified as being of interest in special relativity can be taken over directly to general relativity using the principle of equivalence. Concretely this means that if the Boltzmann collision term is expressed in terms of the components of the momenta in an orthonormal frame
then the resulting expression also applies in general relativity.

For a system of evolution equations like the Einstein-Boltzmann system one of the most basic mathematical questions is the local well-posedness of the initial value problem. For the EB system this problem was solved in 1973 by Daniel Bancel and Yvonne Choquet-Bruhat (Commun. Math. Phys. 33, 83) for a certain class of collision terms. The physical interpretation of the unknown in the Boltzmann equation as a number density means that it should be non-negative. In the context of the initial value problem this means that it should be assumed that is initially non-negative and that it should then be proved that the corresponding solution is non-negative. In the existence proofs for many cases of the Boltzmann equation the solution is obtained as the limit of a sequence of iterates, each of which are by construction non-negative. The convergence to the limit is strong enough that that the non-negativity of the iterates is inherited by the solution. In the theorem of Bancel and Choquet-Bruhat the solution is also constructed as the limit of a sequence of iterates but no attention is paid to non-negativity. In fact that issue is not mentioned at all in their paper. To prove non-negativity of solutions of the EB system it is enough to prove the corresponding statement for solutions of the Boltzmann equation on a given spacetime background. The latter question has been addressed in papers of Bichteler and Tadmon. On the other hand it is not easy to see how their results relate to those of Bancel and Choquet Bruhat. This question has now been investigated in a paper by Ho Lee and myself . The result is that with extra work the desired posivity result can be obtained under the assumptions of the theorem of Bancel and Choquet-Bruhat. While working on this we obtained some other insights about the EB system. One is that the assumptions of the existence theorem appear to be very restrictive and that treating physically motivated scattering kernels will probably require more refined approaches. In the almost forty years since the local existence theorem there have been very few results on the initial value problem for the EB system (with non-vanishing collision term). We hope that our paper will set the stage for further progress on this subject.

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 , the number density of particles with position and velocity at time . A regular solution is one for which the function is smooth (or at least ). 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 dependence on the velocity variables. For fixed the support of in 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 is the phase space defined by the application and a stationary solution depending on the parameters. What needs to be shown is that for suitable values of the parameters the solution which starts at converges to for large values of the independent variable. In more detail, when one of the parameters is fixed to be positive and sufficiently small there exists a value of the other paramater 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 . (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 and another for large values of . 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.

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 to be general. For convenience I also introduce a parameter which can take the values and . The equations are and where . Here is time and and denote position and velocity variables, each belonging to . is the Laplacian. Because of their most frequent applications the cases and 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 .

Local existence in the Cauchy problem is known. For it is furthermore known that the local solution extends to a global in time solution, independently of the sign of . (The first proofs by two different methods were given by Pfaffelmoser and Lions/Perthame around 1991.) When and 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 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 the potential and kinetic energies satisfy an inequality of the form 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 then the power one half is replaced by the power one. Thus in a sense is the critical case. For the global existence problem for the Vlasov-Poisson system with is open. For 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 where . 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 .

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 is bounded any solution with and whether in the case , as is known to be the case for small initial data.