Local Existence and Continuation Criterion for Solutions of the Spherically Symmetric Einstein-Vlasov-Maxwell System

Abstract

Using the iterative scheme we prove the local existence and uniqueness of solutions of the spherically symmetric Einstein-Vlasov-Maxwell system with small initial data. We prove a continuation criterion to global in-time solutions.

The Einstein–Vlasov–Maxwell (EVM) system with spherical symmetry

“…Then the following inequalities hold on [0, σ(d)]: Using once again Theorem 3.2 and Lemma 3.1 in [7], one deduces the Corollary 3.2 With the assumption in Theorem 3.2, let • g ∈ C 2 (R 6 ). Then the following …”

“…Before doing so, we recall the local existence theorem we proved in [7] on which our present result relies. The constraint equations are obtained setting t = 0 in (2.2), (2.3) and (2.5).…”

“…Next as the second step, we derive an estimate for the time evolution of f − g along the characteristics Z f = (X f , V f ) corresponding to f (see [7], Proposition 3.1). Using the fact that f is constant along these characteristics, the mean value theorem and since µ − λ ≤ 1, µ + λ ≤ 1 both for f and g, one obtains the following estimate:…”

“…In [7], the authors established with the help of an iterative scheme one local existence theorem and a result on the continuation criterion of solutions is deduced. Note that these results are valid only for the Cauchy problem with small initial data.…”

The Newtonian Limit for Perfect Fluids

“…In the special case of spherical symmetry, the situation improves and there are some global results available on the Newtonian limit [26,32]. However, because spherically symmetric systems do not generate gravitational radiation, these results do not shed light on the "far zone" problem for post-Newtonian expansions where radiation plays a crucial role and the ǫ ց 0 limit must be analyzed in the region "close" to future null infinity.…”