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

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.


Introduction
In [5], the authors prove the global existence solutions of spatial asymptotically flat spherically symmetric Einstein-Vlasov system. This provides a base for the mathematical study of gravitational collapse of collisionless matter; for related works see [2], [4], [7], [8]. That study concerns uncharged particles. We consider, under the same assumption of spherical symmetry, the case where the particles are charged. To describe the full physical situation, we must then couple the previous system to the Maxwell equations that determine the electromagnetic field created by the fast moving charged particles, and that reduces, in the spherically symmetric case, to its electric part.
It is appropriate at this point to examine the motivation for considering this particular problem which unlike the probleme [5] has no direct astrophysical applications, there are, however, two reasons why the problem is interesting. The first reason is that it extends the knowledge of the Cauchy problem for systems involving the Vlasov equation (which models collisionless matter) and it will be seen that it gives rise to new mathematical features compared to those cases studied up to now. The second reason is connected with the fact that it would be desirable to extend the work of [5] beyond spherically symmetric. In particular, it would be desirable from a physical point of view to include the phenomenon of rotation. Unfortunately, presently available techniques do not suffice to get away from spherical symmetry. In this situation it is possible to attempt to obtain further intuition by using the analogy between angular momentum and charge, summed up in John Wheeler's statements, "charge is poor man's angular momentum". Thus we study spherical systems with charge in hope that this will give us insight into non-spherical systems without charge. This strategy has recently been pursued in the case of a scalar field as matter model, with interesting results [1].
Due to the presence of electromagnetic field, the matter quantities are not compactly supported in the spatial variable as it is the case for uncharged particles. So, with this default of compactness, it becomes difficult to prove that the sequence of iterates we use is well defined and converges to a unique solution of the Cauchy problem. The interest of this work lies on the fact that, with a weak regularity condition on matter quantities, the authors prove a local existence theorem and a continuation criterion for solutions which may allow to study the global behaviour of such solutions. We are not aware that this has been done before.
In our specific case, we are led to a difficulty in solving the Cauchy problem by following [5]. Let us first recall the situation in [5] before seeing how it changes in the case of charged particles. In [5], using the assumption of spherical symmetry, the authors look for two metrics functions λ and µ, that depend only on the time coordinate t and the radial coordinate r, and for a distribution function f of the uncharged particles that depends on t, r and on the 3-velocity v of the particles; the metric functions λ, µ are subject to the Einstein equations with sources generated by the distribution function f of the collisionless uncharged particles which is itself subject to the Vlasov equation. They show that the Einstein equations to determine the unknown metric functions λ and µ, turn out to be two first order O.D.E. in the radial variable r, coupled to the Vlasov equation in f . Putting t = 0, and denoting by • λ(r) , • µ(r) and • f (r, v) the initial data for λ(t, r), µ(t, r) and f (t, r, v) respectively, the constraints equations on the initial data can be solved easily and they need just to prescribe an appropriate condition on • f (r, v) to obtain a unique local solution of the Cauchy problem.
In the case of charged particles, due to the presence of the electromagnetic field in the source terms of Einstein's equations, the initial value problem is not easy to solve. We consider the case of a spherically symmetric electric field E of the form E(t, r) = e(t, r) r r , where e(t, r) is an unknown scalar function and r the position vector in R 3 . We denote by • e the initial datum for e(t, r). The Einstein -Maxwell equations imply three constraints equations on the initial data, that are a singular first order O.D.E in the radial variable r. In [3], using singular O.D.E techniques, the authors describe one large class of functions • f for which the constraints equations on the initial data are solved. In this paper, we use the above result to define the iterates and we obtain sequences of iterates that converge to the unique local solution of the initial value problem. Moreover we prove the continuation criterion, i.e the control of momenta in supp f which may allow the extendability of the solution for all time, proving the extension of the results of [5] to the case of charged particles with the above indicated consequences.
The paper is organized as follows. In Sect. 2, we recall the general formulation of the Einstein-Vlasov-Maxwell system, from which we deduce the relevant equations in the spherically symmetric spatial asymptotically flat case. In Sect. 3, we establish some properties of the characteristics of the Vlasov equation and we show how to solve each equation when the others unknown are given. In Sect.4, we prove a local existence and uniqueness theorem of solutions for the system, together with a continuation criterion for such solutions.

Derivation of the relevant equations
We consider fast moving collisionless particles with charge q. The basic spacetime is (R 4 , g), with g a Lorentzian metric with signature (−, +, +, +). In what follows, we assume that Greek indices run from 0 to 3 and Latin indices from 1 to 3, unless otherwise specified. We also adopt the Einstein summation convention. The metric g reads locally, in cartesian coordinates (x α ) = (x o , x i ) ≡ (t,x): The assumption of spherical symmetry means that we can take g of the following form (Schwarzschild coordinates) [10] where µ = µ(t, r); λ = λ(t, r); t ∈ R; r ∈ [0, +∞[; θ ∈ [0, π]; ϕ ∈ [0, 2π]. The Einstein -Vlasov -Maxwell system can be written: with: dp 1 dp 2 dp 3 p 0 , p 0 = g 00 p 0 , where Γ α βγ denote the Christoffel symbols, and L X(F ) the Lie derivative. Here, x = (x α ) is the position and p = (p α ) is the 4-momentum of the particles. In the expressions above, f stands for the distribution function of the charged particles, F stands for the electromagnetic field created by the charged particles. Here (2.3) are the Einstein equations for the metric tensor g = (g αβ ) with sources generated by both f and F , that appear in the stress-energy tensor 8π(T αβ + τ αβ ). Equation (2.4) is the Vlasov equation for the distribution function f of the collisionless particles and (2.5) are the Maxwell equations for the electromagnetic field F , with source(current) generated by f through J = J(f ). One verifies (using the normal coordinates) that the conservation laws ∇ α (T αβ + τ αβ ) = 0 hold if f satisfies the Vlasov equation.

Preliminary results, conservation laws, and reduced systems
For given ρ and p, (2.6) and (2.8) determine (λ, µ) and the right hand side of (2.7) is known when k is given. So we can use this equation to defineλ as the time derivative of λ. Indeed we show later thatλ =λ . Therefore, we call auxiliary system to equations (2.7) and (2.10) the following equations: together with (2.6), (2.8) and (2.11). It appears clearly that a solution (λ, µ, e, f ) of the coupled system (2.6), (2.8), (2.11), and (3.1) that satisfies (2.9), (2.10), (2.12) andλ =λ is a solution of the full initial system. Before we do so, we make precise the regularity properties which we require of solution: for all t ∈ I.  Let us now consider the Vlasov equation (3.1) for prescribed functions λ, µ,λ and e.
Proposition 3.1 Let I ∈ R be an interval with 0 ∈ I, λ, µ,λ and e regular on Then a) F ∈ C 1 (I × R 6 ).
b) For every t ∈ I, z ∈ R 6 , the characteristics systeṁ c) For a nonnegative, spherically symmetric function where diṽ x is divergence in the Euclidian metric on R 3 and thus the quantity is conserved.
Proof : The crucial point in the proof of part a) is the regularity of F 2 at r = 0. Now the term x i r µ ′ (s, r) = ∂µ ∂x i (s, r) is continuously differentiable with respect tox ∈ R 3 and vanishes at r = 0 by virtue of the regularity of µ. The termλ xixj r 2 is continuously differentiable with respect tox, using the regularity ofλ. The continuously differentiability of both terms with respect to t atx = 0 follows from the fact thaṫ λ(t, 0) =μ ′ (t, 0) = 0, t ∈ I and the following expression shows that the term ex r is also continuously differentiable at r = 0, since by the regularity of e, we have: Therefore F 2 is continuously differentiable on I × R 6 . This implies local existence, uniqueness and regularity of Z(., t, z). Since X(., t, z) remains bounded on bounded sub-intervals of I. On the other hand, by regularity of λ, µ, e and which is bounded on every bounded sub-interval of I by the Gronwall lemma, the same is true for V (., t, z). Therefore, Z(., t, z) exists on I. The other assertions in b) are standard, or follow by uniqueness. Assertion c) is an immediate consequence of b) and the fact that according to (2.10), f remains constant along the trajectories. Now, to prove part d), we multiply (2.10) with e λ , integrate with respect to v and apply Gauss theorem to obtain (3.4). The conservation law in d) corresponds to conservation of number of particles. The term e λ comes from the fact that the coordinates v on the mass shell are not the canonical momenta corresponding tox, and proposition 3.1 is proved. We need the following result obtained by a direct computation to control certain derivatives of the unknown [6]: Then, where the coefficients of matrices a 1 , ..., a 5 are a regular functions of λ, µ, e and their derivatives.
Note that a 1 , a 2 , a 4 and a 5 are the same as in ( [6], lemma 2.3). But here, due to the presence of electromagnetic field, coefficients (a 3 (s,x, v)) i k of the matrix a 3 contents the corresponding terms in [6] plus an additional one that is: Next, we investigate field equations (2.6), (2.8) for given ρ, p and the Maxwell equation (2.11) for given M . (2.13), (2.15) and (2.17), replacing f , λ, e byf ,λ,ē respectively, and let: The solution is given by and Since 1 − 2m(t,r) r < 1, also µ ′ ≥ 0 and thus µ ≤ 0 due to the boundary condition at r = ∞. From (3.11) it follows that λ + µ is increasing in r, and since this function vanishes at r = ∞, λ + µ ≤ 0. On the other hand, we obtain (3.13) by integrating (2.11) on [0, r] and using e(t, 0) = 0, since λ ≥ 0, λ and M are bounded in r. Now, the differentiability properties of λ, µ and e which are part of definition of being regular are obvious. Then the proof is complete.
We now show that the reduced system mentioned above is equivalent to the full system. We also prove the following conservation law: is conserved.
Proof : Using the conservation law (3.3), we can deduce that each solution of equation (2.11) is a solution of (2.12). Also, differentiating the relation (2.13) of ρ with respect to t and using the Vlasov equation (2.10) we obtain, by Gauss theorem: (3.14') So, we will have the conservation law (3.14) if the second term in the right hand side of (3.14') vanishes. But we obtain the latter if (2.7) holds, and this can be established, by differentiating (3.7) with respect to t, using (3.14'), Gauss theorem and the Gronwall lemma. Then, since (3.14) holds, dM(t) dt = 0 and the A.D.M mass is conserved. Next, because (2.7) holds, we use once again the Vlasov equation and Gauss theorem to show that equation (2.9) holds as well and the proof is complete.
Remark 3.2 We consider the auxiliary system (2.6), (2.8), (2.11), (3.1) and (3.2), which we use in the proof of local existence result in the next section. Remark 3.3 Consider a regular solution (f, λ, µ,λ, e) of the auxiliary system. Then, since e is a solution of (2.12), we can conclude that e ∈ C 1 (I × R 3 ) and by the regularity of λ, g αβ ∈ C 1 (I × R 3 ), where the metric g is given in cartesian coordinates by: Proof : Let (λ, µ, f, e,λ) be a regular solution of (2.6), (2.8), (2.11), (3.1) and (3.2), by proposition 3.1, we have only to show thatλ =λ, and this is obtained by differentiating (3.7) w.r.t t, using (3.11) and Gauss theorem. Thus the proof is complete. Now, we give this result we use later, obtained by induction and integration and that is We end this section by recalling the constraints equations on the initial data. As the authors show in [3], these equations are:

Local existence and continuation of solutions
In this section we prove a local existence and uniqueness theorem for regular solutions of the initial value problem corresponding to the spatial asymptotically flat, spherically symmetric Einstein-Vlasov-Maxwell system, together with a continuation criterion for such solutions. The basic idea of the proof is to use for given small  [3], and proposition 3.2, to construct the iterates and show that these iterates converge to a solution on some interval of the coupled system. Here, compared to the situation met by the authors in [5], the main difficulties are the following: equation (2.6) does not define directly λ for given f as it is the case for Einstein-Vlasov system, and if we consider (2.7) to define λ, thenλ will become very unpleasant to control. The latter difficulty is solved by using the auxiliary system (2.6), (2.8), (2.10), (2.11), (3.1), (3.2) and apply proposition 3.4.

The construction of iterates
Let • f ∈ C ∞ (R 6 ) be nonnegative, compactly supported and spherically symmetric with • e ∈ C ∞ (R 3 ) be a regular solution of (3.17), (3.18) and (3.19). By proposition 3.4, it sufficient to solve the auxiliary system (2.6), (2.8), (2.10), (2.11), (3.1) and (3.2). Furthermore, it is sufficient to solve this system for t > 0, the proof for t < 0 would proceed in exactly the same way. Note that by [3], assumption (4.1) on • f ensures the existence of a local solution of the constraints equations, for law charge. We assume that supp We consider the following iterative scheme:  where, following proposition 3.1: with Z n (t, t, z) = z, and define i.e f n satisfies the auxiliary Vlasov equation: with f n (0) = • f , and: .v r f n (t,x, v)dv (4.10) Now, (3.7) can be use to define λ n as long as the right hand side is positive. Thus we define We deduce from (4.12) that: We also use the Vlasov equation (3.1) and Gauss theorem to obtain the analogous conservation law given by (3.3): N n r e λn+µn−1−λn−1 . We now prove that all the above expression make sense.
Proposition 4.1 For all n ∈ N, the functions λ n , µ n , f n , e n , p n , k n , N n , M n , λ n are well defined and regular, T n > 0, and µ n + λ n ≤ 0, λ n ≥ 0, µ n ≤ 0.
Note that the regularity ofλ n and e n follows from the identities: Proof: First of all, we define and by virtue of (4.16), and the fact that λ n ≥ 0, one has: Thus, where C > 0 denotes a constant which in the sequel may change its value from line to line and does not depend on n, t and • f , and where R n (t) = (1 + P n−2 (t) + Q n−2 (t)) 7 (1 + P n−1 (t) + Q n−1 (t)) 7 × (1 + P n (t) + Q n (t)) 14 (1 + P n+1 (t) + Q n+1 (t)) 7 .
We combine the estimates above with (4.13) and (4.15) to obtain, since λ n + µ n ≤ 0: Note that r =|x |≤ r 0 + t for f n (t,x, v) = 0. Next, we insert theses estimates into the characteristic system which yields: Integrating (4.26) on [0, t], one has: Thus, (1 + r 0 + s)R n (s)ds. (4.27) Next, we look for an inequality for Q n (t). We can write, using (4.12): we see that we need an estimate for the time derivative of m n+1 in (4.9). We calculateṁ n+1 (t, r), use (4.7)(to express ∂fn ∂t ) Gauss theorem, (4.18) and the Gronwall lemma to obtain the estimate; since λ n + µ n ≤ 0: (4.28) Next, we integrate (4.27 ′ ) on [0, t] using (4.28) and obtain, with thenP n ,Q n are increasing sequences and for all n one has P n ≤P n , Q n ≤Q n .
Using the above expression of R n , one deduces: Now fix n ∈ N and write (4.27) and (4.29) for every m, where m ≤ n.Taking the supremum over m ≤ n, yields and by the Gronwall lemma,P n+1 ,Q n+1 and hence P n , Q n are bounded on the domain [0, T 0 ], T 0 ≥ 0, of the maximal solution z 0 of z 0 (t) = u 0 + q 0 + C t 0 exp CΛ(s)(1 + z 0 (s)) 35 ds (4.30) It follows that, P n (t)+Q n (t) ≤ z 0 (t), n ∈ N, t ∈ [0, T 0 [∩[0, T n [, and by definition T n ≥ T 0 , n ∈ N and the proof is complete. Now, in the following C(t) denotes an increasing, continuous function on [0, T 0 [ which depends on z 0 , but not on n. From the estimates in proposition 4.2 we deduce: Next we need to know more about some bounds on certain derivatives. We do it by proving the following result: There exists a unique nonnegative function z 1 ∈ C 1 defined on some interval [0, T 1 [ such that: Proof: We have the following estimates: and by the regularity of k n , we have, using (4.9), Gauss theorem: Next, the definition of f n implies that and ∂ zŻn+1 (s, t, z) = ∂ z F n (s, Z n+1 (s, t, z)).∂ z Z n+1 (s, t, z). The derivative ∂ z F n (s, t, z) contains terms which are bounded by proposition 4.2, terms likeλ n r , µ ′ n r , λ ′ n r , e n , e ′ n and en r which are again bounded by proposition 4.2, and the terms µ ′′ n , λ ′ n . Thus for any characteristics Z n+1 (s, t, z) with z ∈ suppf n+1 (t), and for which therefore, by proposition 4.2, | V n+1 (s, t, z) |≤ z 0 (s). By the Gronwall lemma, one deduces from integration of (4.33) on [s, t], since Z n+1 (t, t, z) = z: and combining this with (4.32), we obtain the inequality: Let z 1 be the maximal solution of and therefore the quantitiesλ ′ n and µ ′′ n , can also be estimated in terms of z 1 on the time interval [0, T 1 [ uniformly in n. This completes the proof of proposition 4.3.

The convergence of iterates
Here we show that the above sequence of iterates which we constructed converges. We prove in the sequel this important result: The sequence of iterates (f n , λ n , µ n , e n ) converges.
Now, by the definition of e n , one has; distinguishing the cases r ≤ r 0 and r ≥ r 0 : (4.37) We find an estimate for e λn+1 − e λn . Using the definition (4.12) of e −2λn , we have e λn+1 − e λn = 2 r e 2λn+λn+1 m n+1 − m n 1 + e λn−λn+1 and since e n and λ n are bounded, we obtain: s | e λn e n − e λn−1 e n−1 | (t, s)ds. (4.38) Next, inserting (4.38) into (4.37) and distinguishing the cases r ≤ r 0 and r ≥ r 0 , using permutation of variables in the double integral that appears inside the obtained inequality and applying lemma 3.2 to obtain: (4.39) Thus, from −λ n ≤ 0 and e n (t) L ∞ ≤ C, we obtain: i! (4.40) and we deduce also, since λ n is bounded,the quantities satisfies (4.40) and by proposition 4.3, , and the estimate of the difference of two iterates of characteristics gives, since (Ż n+1 −Ż n )(s, t, z) = (F n − F n−1 )(s, t, z): ; note that | Z i | (s, t, z) ≤ z 0 (s), for i = n, n + 1, and s ∈ [0, δ]; i.e the characteristics run in the set on which we have bounded ∂ z F n . Gronwall's lemma implies, after integrating (4.41)on [0, t]: Thus, from we deduce, using once again Gronwall's lemma: Thus, by induction, we obtain the following estimate: where C depends on z 0 and not on n. Now, consider two integers m and n such that m > n. Then and the right hand side of inequality above goes to zero as m and n go to infinity, since the series ∞ n=0 C n (1+r0+δ) n n! converges. We conclude that f n (t) is a Cauchy sequence in the complete space L ∞ , for all t ∈ [0, δ], and since all the differences which appear in (4.40) can be written in the form (4.42) such that the same holds for all sequences of functions that appear in (4.40) and others. So, the proof of proposition 4.4 is now complete.

The local existence and uniqueness theorem
In this section, we use lemma 3.1 to show that the limit obtained in proposition 4.4 is regular and thus is a solution of the auxiliary system under consideration. We replace λ, µ,λ, e in that lemma by λ n , µ n ,λ n , e n and choose an arbitrary compact subinterval [0, δ] ⊂ [0, T 1 [ and U > 0. Here the essential result to proved is the following: In particular this holds for t = 0. But by the construction of f n and λ n and separation of L ∞ one has immediately: Since • e is a regular solution of constraint equation (3.19) we obtain, taking (3.13) at t = 0: e(0) = • e and the result for µ follows by using equations (2.8) and (3.18). We end the proof of theorem 4.1 by showing uniqueness.
Assume that we have two regular solutions (λ f , µ f , f, e f ), (λ g , µ g , g, e g ), with λ f (0) = λ g (0), µ f (0) = µ g (0), f (0) = g(0), e f (0) = e g (0). The estimates, which we applied to the difference of two consecutive iterates in proposition 4.4 can be applied in analogous fashion to the difference of f and g to obtain and using the Gronwall lemma, one concludes that f (t) = g(t), and then λ f (t) = λ g (t); µ f (t) = µ g (t), e f (t) = e g (t) as long as both solutions exists.
By proposition 4.4,λ =λ = −4πre λ+µ k, and thus λ L ∞ ≤ C, t ∈ [0, T [, which implies the estimate Q 0 ≤ Q * = sup{e 2λ(t,r) , t ∈ [0, T [, r ≥ 0} | ∂ z Z(0, t, z) |≤ C, for z ∈ suppf (t) and t ∈ [0, T [, since all coefficients in lemma 3.1 are bounded along the characteristics in suppf ; for the coefficient a 3 we observe that due to (2.9), H = 4πq, where H denotes the left hand side of (2.9), andq is bounded due to the bound on suppf (t,x, .). Thus These estimates imply that there exists δ > 0, independent of t 0 , such that (z 0 , z 1 ) and thus also the solutionf , exists on the interval [t 0 , t 0 + δ]. For t 0 close enough to T this solution extends the solution f beyond T , which is a contradiction. Thus if P * < ∞ then T = +∞ and this ends the proof of theorem 4.2. Using theorem 4.1 and theorem 5.1 we can prove the following essential result of this section: Theorem 5.2 (local existence, continuation criterion) Let • f ∈ C ∞ (R 6 ) be nonnegative, compactly supported and spherically symmetric such that (4.1) be satisfied. Let