or equivalently
These are the conservation equations for relativistic particle dynamics. In the classical case these equations read The functionThe main result concerning the existence of
solutions to the classical Boltzmann equation is a theorem by
DiPerna and Lions
[36]
that proves existence, but not uniqueness, of renormalized
solutions (i.e. solutions in a weak sense, which are even
more general than distributional solutions). An analogous result
holds in the relativistic case, as was shown by Dudyńsky and
Ekiel-Jezewska
[37]
. Regarding classical solutions, Illner and Shinbrot
[58]
have shown global existence of solutions to the nonrelativistic
Boltzmann equation for small initial data (close to vacuum). At
present there is no analogous result for the relativistic
Boltzmann equation and this must be regarded as an interesting
open problem. There is however a recent result
[74]
for the homogeneous relativistic Boltzmann equation where global
existence is shown for small initial data and bounded scattering
kernel. When the data are close to equilibrium (see below),
global existence of classical solutions has been proved by
Glassey and Strauss
[48]
in the relativistic case and by Ukai
[115]
in the nonrelativistic case (see also
[108]
and
[69]).
The collision operator
may be written in an obvious way as
where
and
are called the gain and loss term, respectively. If the loss
term is deleted the gain-term-only Boltzmann equation is
obtained. It is interesting to note that the methods of proof for
the small data results mentioned above concentrate on
gain-term-only equations, and once that is solved it is easy to
include the loss term. In
[5]
it is shown that the gain-term-only classical and relativistic
Boltzmann equations blow up for initial data not restricted to a
small neighbourhood of trivial data. Thus, if a global existence
proof for unrestricted data will be given it will necessarily use
the full collision operator.
The gain term has a nice regularizing property
in the momentum variable. In
[2]
it is proved that given
and
with
, then
The regularizing theorem has many applications.
An important application is to prove that solutions tend to
equilibrium for large times. More precisely, Lions used the
regularizing theorem to prove that solutions to the (classical)
Boltzmann equation, with periodic boundary conditions, converge
in
to a global Maxwellian,
as time goes to infinity. This result had first been obtained by Arkeryd [10] by using non-standard analysis. It should be pointed out that the convergence takes place through a sequence of times tending to infinity and it is not known whether the limit is unique or depends on the sequence. In the relativistic situation, the analogous question of convergence to a relativistic Maxwellian, or a Jüttner equilibrium solution,
had been studied by Glassey and Strauss [48, 49] . In the periodic case they proved convergence in a variety of function spaces for initial data close to a Jüttner solution. Having obtained the regularizing theorem for the relativistic gain term, it is a straightforward task to follow the method of Lions and prove convergence to a local Jüttner solution for arbitrary data (satisfying the natural bounds of finite energy and entropy) that is periodic in the space variables. In [2] it is next proved that the local Jüttner solution must be a global one, due to the periodicity of the solution.
For more information on the relativistic
Boltzmann equation on Minkowski space we refer to
[41, 34, 110]
and in the nonrelativistic case we refer to the excellent review
paper by Villani
[116]
and the books
[41
, 24]
.
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