A special case in three dimensions is obtained by considering spherically-symmetric initial data. For such
data it can be shown that the solution will also be spherically symmetric, and that the magnetic field has to
be constant. The Maxwell equation then implies that the electric field is the gradient of a
potential
. Hence, in the spherically-symmetric case the relativistic Vlasov–Maxwell system takes the
form
One of the fundamental problems in kinetic theory is to find out whether or not spontaneous shock formations will develop in a collision-less gas, i.e., whether solutions to any of the equations above will remain smooth for all time, given smooth initial data.
If the initial data are small this problem has an affirmative solution in all cases considered
above [81, 86, 32
, 33]. For initial data unrestricted in size the picture is more involved. In order to obtain
smooth solutions globally in time, the main issue is to control the support of the momenta
The relativistic and non-relativistic Vlasov–Poisson equations are very similar in form. In particular, the equation for the field is identical in the two cases. However, the mathematical results concerning the two systems are very different. In the non-relativistic case, Batt [34] gave an affirmative solution in 1977 in the case of spherically-symmetric data. Pfaffelmoser [133] was the first one to give a proof for general smooth data. A simplified version of the proof is given by Schaeffer in [168]. Pfaffelmoser obtained the bound
where can be taken as arbitrarily small. This bound was later improved by different
authors. The sharpest bound valid for
and
has been given by Horst [97] and
reads
In the case of repulsive forces () Rein [137] has found a better estimate by using a new identity for
the Vlasov–Poisson system, discovered independently by Illner and Rein [98] and by Perthame [132]. Rein’s
estimate reads
Independently, and at about the same time as Pfaffelmoser gave his proof, Lions and Perthame [113] used a
different method for proving global existence. Their method is more generally applicable, and the two
studies [5] and [105] are examples of problems in related systems, where their method has been successful.
On the other hand, their method does not give such strong growth estimates on as described above.
For the relativistic Vlasov–Poisson equation, Glassey and Schaeffer [81
] showed in the case
that if
the data are spherically symmetric,
can be controlled, which is analogous to the result by Batt
mentioned above. Also in the case of cylindrical symmetry they are able to control
; see [84]. If
it was shown in [81] that blow-up occurs in finite time for spherically-symmetric data with
negative total energy. More recently, Lemou et al. [111
] have investigated the structure of the
blow-up solution. They show that the blow-up is determined by the self-similar solution of the
ultra-relativistic gravitational Vlasov–Poisson system. It should be pointed out that the relativistic
Vlasov–Poisson system is unphysical since it lacks the Lorentz invariance; it is a hybrid of a
classical Galilei invariant field equation and a relativistic transport equation (17
), cf. [3]. In
particular, in the case
, it is not a special case of the Einstein–Vlasov system. Only for
spherically-symmetric data, in the case
, is the equation a fundamental physical equation.
The results mentioned above all concern classical solutions. The situation for weak solutions is
different, in particular the existence of weak solutions to the relativistic Vlasov–Maxwell system is
known [70, 139].
We also mention that models, which take into account both collisions and the electric and magnetic fields generated by the particles have been investigated. Classical solutions near a Maxwellian for the Vlasov–Maxwell–Boltzmann system are constructed by Guo in [90]. A similar result for the Vlasov–Maxwell–Landau system near a Jüttner solution is shown by Guo and Strain in [180].
We refer to the book by Glassey [79] and the review article by Rein [141] for more information on the
relativistic Vlasov–Maxwell system and the Vlasov–Poisson system.
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
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