or equivalently
These are the conservation equations for relativistic particle dynamics. In the classical case the corresponding conservation equations read The function In [44] and [178] classical solutions to the relativistic Boltzmann equations are studied as
, and
it is proven that the limit as
of these solutions satisfies the classical Boltzmann equation. The
former work is more general since general initial data is considered, whereas the latter is concerned with
data near vacuum. The latter result is stronger in the sense that the limit, as
is shown to be
uniform in time.
The main result concerning the existence of solutions to the classical Boltzmann equation is a theorem
by DiPerna and Lions [71] that proves existence, but not uniqueness, of renormalized solutions. An
analogous result holds in the relativistic case, as was shown by Dudyński and Ekiel-Jeżewska [72],
cf. also [102]. Regarding classical solutions, Illner and Shinbrot [99] have shown global existence of
solutions to the non-relativistic Boltzmann equation for initial data close to vacuum. Glassey showed global
existence for data near vacuum in the relativistic case in a technical work [80
]. He only requires
decay and integrability conditions on the differential cross-section, although these are not fully
satisfactory from a physics point of view. By imposing more restrictive cut-off assumptions on
the differential cross-section, Strain [178] gives a different proof, which is more related to the
proof in the non-relativistic case [99] than [80] is. For the homogeneous relativistic Boltzmann
equation, global existence for small initial data has been shown in [126] under the assumption
of a bounded differential cross-section. For initial data close to equilibrium, global existence
of classical solutions has been proven by Glassey and Strauss [87
] using assumptions on the
differential cross-section, which fall into the regime “hard potentials”, whereas Strain [177
] has
shown existence in the case of soft potentials. Rates of the convergence to equilibrium are given
in both [87
] and [177]. In the non-relativistic case, we refer to [189, 172, 119] for analogous
results.
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 [14] it is shown that the gain-term-only classical and relativistic
Boltzmann equations blow up for initial data not restricted to a small neighborhood of trivial data. Thus, if
a global existence proof of classical solutions 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 [4] it is proven 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 was first obtained by Arkeryd [29] 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,
was studied by Glassey and Strauss [87, 88]. 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 global Jüttner solution for arbitrary initial data (satisfying the natural bounds of finite energy and entropy), which are periodic in the space variables, cf. [4]. We also mention that in the non-relativistic case Desvillettes and Villani [69] have studied the convergence rate to equilibrium in detail. A similar study in the relativistic case has not yet been achieved.
For more information on the relativistic Boltzmann equation on Minkowski space we refer
to [54, 68, 181, 79] and in the non-relativistic case we refer to [190, 79
, 53].
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
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