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 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
[42, 47, 12
, 13]). 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 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
[14]
gave an affirmative solution 1977 in the case of spherically
symmetric data. Pfaffelmoser
[80]
(see also Schaeffer
[107]) 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
[55]
and reads
In the case of repulsive forces () Rein
[83]
has found the sharpest estimate by using a new identity for the
Vlasov-Poisson equation, discovered independently by Illner and
Rein
[57]
and by Perthame
[79]
. Rein’s estimate reads
Independently and about the same time as
Pfaffelmoser gave his proof, Lions and Perthame
[66]
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, 61]). 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
[42
]
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
[45]
. If
it was also shown in
[42]
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
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
[56, 35
, 86]). 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 [41] for more information on the relativistic Vlasov-Maxwell system and the Vlasov-Poisson equation.
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