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)

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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.

Equilibrium states in galactic dynamics can be described as stationary solutions of the Vlasov-Poisson system, which is the non-relativistic case, or of the Vlasov-Einstein system, which is the relativistic case. To obtain spherically symmetric stationary solutions the distribution function of the particles (stars) on phase space is taken to be a function Φ(E,L) of the particle energy and angular momentum. We give a new condition on Φ which guarantees that the resulting steady state has finite mass and compact support both for the non-relativistic and the relativistic case. The condition is local in the sense that only the asymptotic behaviour of Φ for E → E 0 needs to be prescribed, where E 0 is a cut-off energy above which no particles exist.

The Vlasov-Einstein system describes a self-gravitating, collisionless gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with spherical, plane, or hyperbolic symmetry and prove that for small initial data solutions exist up to a spacetime singularity which is a curvature and a crushing singularity. An important tool in the analysis is a local existence result with a continuation criterion saying that solutions can be extended as long as the momenta in the support of the phase-space distribution of the matter remain bounded.

The present paper completes our earlier results on nonlinear stability of stationary solutions of the Vlasov-Poisson system in the stellar dynamics case. By minimizing the energy under a mass-Casimir constraint we construct a large class of isotropic, spherically symmetric steady states and prove their nonlinear stability against general, i. e., not necessarily symmetric perturbations. The class is optimal in a certain sense, in particular, it includes all polytropes of finite mass with decreasing dependence on the particle energy. 1