The notation follows the one already introduced with the
exception that the momenta are now denoted by
*v*
instead of
*p*
. This has become a standard notation in this field.
*E*
and
*B*
are the electric and magnetic fields, and
is the relativistic velocity

where
*c*
is the speed of light. The charge density
and current
*j*
are given by

Equation (12) is the relativistic Vlasov equation and (13, 14) are the Maxwell equations.

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

Here , and the constant magnetic field has been set to zero, since a constant field has no significance in this discussion. This system makes sense for any initial data, without symmetry constraints, and is called the relativistic Vlasov-Poisson equation. Another special case of interest is the classical limit, obtained by letting in (12, 13, 14), yielding:

where . See Schaeffer [80] for a rigorous derivation of this result. This is the (nonrelativistic) Vlasov-Poisson equation, and corresponds to repulsive forces (the plasma case). Taking means attractive forces and the Vlasov-Poisson equation is then a model for a Newtonian self-gravitating system.

One of the fundamental problems in kinetic theory is to find out whether or not spontaneous shock formations will develop in a collisionless 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 (see [30, 35, 10, 11]). 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

i.e. to bound
*Q*
(*t*) by a continuous function so that
*Q*
(*t*) will not blow up in finite time. That such a control is
sufficient for obtaining global existence of smooth solutions
follows from well-known results in the different cases
(see [34,
42,
12,
30]). For the full three-dimensional relativistic Vlasov-Maxwell
system, this important problem of establishing whether or not
solutions will remain smooth for all time is open. In two space
and three momentum dimensions, Glassey and Schaeffer [31] have shown that
*Q*
(*t*) can be controlled, which thus yields global existence of smooth
solutions in that case (see also [32]).

The relativistic and nonrelativistic 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 nonrelativistic case Batt [12] gave an affirmative solution 1977 in the case of spherically symmetric data. Pfaffelmoser [55] (see also Schaeffer [82]) was the first one to give a proof for general smooth data. He obtained the bound

where could be taken arbitrarily small. This bound was later improved by different authors. The sharpest bound valid for and has been given by Horst [43] and reads

In the case of repulsive forces () Rein [59] has found the sharpest estimate by using a new identity for the Vlasov-Poisson equation, discovered independently by Illner and Rein [45] and by Perthame [54]. Rein's estimate reads

Independently and about the same time as Pfaffelmoser gave his
proof, Lions and Perthame [50] used a different method for proving global existence. To some
extent their method seems to be more generally applicable to
attack problems similar to the Vlasov-Poisson equation but which
are still quite different (see [3], [48]). On the other hand, their method does not give such strong
growth estimates on
*Q*
(*t*) as described above. For the relativistic Vlasov-Poisson
equation (with
), Glassey and Schaeffer [30] showed that if the data are spherically symmetric,
*Q*
(*t*) can be controlled, which is analogous to the result by Batt
mentioned above (we also mention that the cylindrical case has
been considered in [33]). If
, it was also shown in [30] that blow-up occurs in finite time for spherically symmetric
data with negative total energy. This system, however, is
unphysical in the sense that it is not a special case of the
Einstein-Vlasov system. Quite surprisingly, for general smooth
initial data none of the techniques discussed above for the
nonrelativistic Vlasov-Poisson equation apply in the relativistic
case. This fact is annoying since it has been suggested that an
understanding of this equation may be necessary for understanding
the three-dimensional relativistic Vlasov-Maxwell equation.
However, the relativistic Vlasov-Poisson equation lacks the
Lorentz invariance; it is a hybrid of a classical Galilei
invariant field equation and a relativistic transport
equation (17). Only for spherical symmetric data is the equation a
fundamental physical equation. The classical Vlasov-Poisson
equation is on the other hand Galilean invariant. In [1] a different equation for the field is introduced that is
observer independent among Lorentz observers. By coupling this
equation for the field to the relativistic Vlasov equation, the
function
*Q*
(*t*) may be controlled as shown in [1]. This is an indication that the transformation properties are
important in studying existence of smooth solutions (the
situation is less subtle for weak solutions, where energy
estimates and averaging are the main tools, see [44] and [24]). Hence, it is unclear whether or not the relativistic
Vlasov-Poisson equation will play a central role in the
understanding of the Lorentz invariant relativistic
Vlasov-Maxwell equation.

We refer to the book by Glassey [29] for more information on the relativistic Vlasov-Maxwell system and the Vlasov-Poisson equation.

The Einstein-Vlasov System/Kinetic Theory
Håkan Andréasson
http://www.livingreviews.org/lrr-2002-7
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