### 3.2 Local existence and the continuation criterion

In [142] it is shown that for initial data satisfying (44) there exists a unique, continuously-differentiable
solution with on some right maximal interval . If the solution blows up in finite
time, i.e., if , then becomes unbounded as . Moreover, a continuation criterion is
shown that says that a local solution can be extended to a global one, provided can be bounded on
, where
This is analogous to the situation for the Vlasov–Maxwell system. A control of the -support immediately
implies that and are bounded in view of Equations (40, 41). In the Vlasov–Maxwell case the field
equations have a regularizing effect in the sense that derivatives can be expressed through spatial integrals,
and it follows [85] that the derivatives of can also be bounded if the -support is bounded. For the
Einstein–Vlasov system such a regularization is less clear, since, e.g., depends on
in a point-wise manner. However, in view of Equation (39) certain combinations of second
and first order derivatives of the metric components can be expressed in terms of the matter
component , which is a consequence of the geodesic deviation equation. This fact turns
out to also be sufficient for obtaining bounds on the derivatives of , cf. [142, 135, 156] for
details.
The local existence result discussed above holds for compactly-supported initial data. The
compact support condition in the momentum variables is in [12] replaced by the fall-off condition

We also refer to [23] where a subclass of non-compactly-supported data is treated.
Local existence of solutions in double null coordinates and in Eddington–Finkelstein coordinates is
established in [64], and [24] respectively.