Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data

Abstract

In the above paper the authors proved a local existence result for the spherically symmetric Vlasov-Einstein system (Theorem 3.1). Unfortunately, the proof contains an error: To estimate J~n in the proof of Lemma 3.3 we had in mind to differentiate the relation (3.4) A(t,x,v) =f((x., Vn)(O,t,X,V)) with respect to t, and use the boundedness of the right-hand side of the characteristic system (3.3) and the "fact" that (X,, Vn)(S,t,x,v) is symmetric in s,t in the sense that (X~, V~)(O,t,x,v) as a function of t solves (3.3) with the signs of the right-hand side reversed. This "fact" is wrong, it would be correct only if (3.3) were autonomous. In the following we indicate the main arguments which have to be added to the analysis in the above paper in order to set things right. A detailed exposition of the arguments can be obtained from the first author. As a first step we prove Lemma 3.3. By (3.26) and (3.27) we have to bound I t lips()H~ and IlPn(t)ll~. Using the Vlasov equation to express J~ in tSn(t,x) = f g/1 + v2j~n(t,x,v)dv, integrating the term with 0~f~ by parts and using Lemma 3.2 we get l Ilpn(t)lloo, IIA(t)ll~ =< Cl(t)(1 + I[~f~(t)llo~), t c [0, T[, where C~(t) depends on the functions Z1,Z 2 introduced in Lemma 3.2. Differentiating (3.3) with respect to x and using a Gronwall argument yields the estimate t IOxy.+l(o,t,z)l + I~xVn+~(o,t,z)l ~: exp f Cl(S)(1 + II~xA(~)ll~)ds, 0

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

“…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].…”

“…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 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.…”

“…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.…”

“…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.…”