3.2 Local existence and the continuation criterion

In [142Jump To The Next Citation Point] it is shown that for initial data satisfying (44View Equation) there exists a unique, continuously-differentiable solution f with f(0) = f0 on some right maximal interval [0,T ). If the solution blows up in finite time, i.e., if T < ∞, then ρ(t) becomes unbounded as t → T. Moreover, a continuation criterion is shown that says that a local solution can be extended to a global one, provided Q (t) can be bounded on [0,T ), where
Q (t) := sup{|v| : ∃(s,x) ∈ [0,t] × ℝ3 such that f(s,x,v ) ⁄= 0 }. (45 )
This is analogous to the situation for the Vlasov–Maxwell system. A control of the v-support immediately implies that ρ and p are bounded in view of Equations (40View Equation, 41View Equation). 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 f can also be bounded if the v-support is bounded. For the Einstein–Vlasov system such a regularization is less clear, since, e.g., μ r depends on p in a point-wise manner. However, in view of Equation (39View Equation) certain combinations of second and first order derivatives of the metric components can be expressed in terms of the matter component pT, which is a consequence of the geodesic deviation equation. This fact turns out to also be sufficient for obtaining bounds on the derivatives of f, cf. [142Jump To The Next Citation Point, 135, 156Jump To The Next Citation Point] for details.

The local existence result discussed above holds for compactly-supported initial data. The compact support condition in the momentum variables is in [12Jump To The Next Citation Point] replaced by the fall-off condition

5 ˚ sup 6(1 + |v |) |f(x,v)| < ∞. (46 ) (x,v)∈ℝ
We also refer to [23Jump To The Next Citation Point] 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 [64Jump To The Next Citation Point], and [24Jump To The Next Citation Point] respectively.


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