Archive for January, 2013

The relativistic Boltzmann equation

January 3, 2013

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 g and the scattering angle \theta. 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 g and a dependence on \theta which cannot be determined explicitly. It has a non-integrable singularity in \theta at \theta=0. It has been observed that properties of solutions of the Boltzmann equation determine two different regimes for the exponent of g. The cases between an inverse square and an inverse fifth power force between particles are known as soft potentials. The exponent of g varies from -4 to -1. Cases with powers of the force more negative than -5 are known as hard potentials. As the power in the force ranges from -5 to -\infty the exponent of g varies from -1 to zero. When the exponent of g is equal to -1  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 g 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.