No 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)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 righthand 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)

We consider the Vlasov-Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular center and may have isotropic or anisotropic pressure, and also solutions, which have a Schwarzschild-singularity at the center. The paper is an extension of previous work, where only smooth, globally defined solutions with regular center and isotropic pressure were considered, cf. [8]

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

The stability features of steady states of the spherically symmetric Einstein-Vlasov system are investigated numerically. We find support for the conjecture by Zeldovich and Novikov that the binding energy maximum along a steady state sequence signals the onset of instability, a conjecture which we extend to and confirm for non-isotropic states. The sign of the binding energy of a solution turns out to be relevant for its time evolution in general. We relate the stability properties to the question of universality in critical collapse and find that for Vlasov matter universality does not seem to hold.