Stable models of elliptical galaxies

Transcription

1 Mon. Not. R. Astron. Soc. 344, (003) Stable models of elliptical galaxies G. Rein 1 and Y. Guo 1 Faculty of Mathematics and Physics, University of Bayreuth, Bayreuth, Germany Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 091, USA Accepted 003 June 13. Received 003 June 13; in original form 00 March 18 ABSTRACT We construct stable axially symmetric models of elliptical galaxies. For these models the particle density on phase space is a function of the particle energy and the third component of the angular momentum and is decreasing in the former. They are obtained as minimizers of suitably defined energy Casimir functionals, and this implies their non-linear dynamical stability against axisymmetric perturbations. Since our analysis proceeds from a rigorous but purely mathematical point of view, it should be interesting to determine if any of our models match observational data in astrophysics. The main purpose of this paper is to initiate some exchange of information between the astrophysics and the mathematics communities. Key words: galaxies: elliptical and lenticular, cd galaxies: general galaxies: kinematics and dynamics galaxies: structure. 1 INTRODUCTION Consider a large ensemble of mass points (stars) that interact only by the gravitational field that they create collectively. Such a collisionless, self-gravitating gas is used to model galaxies or globular clusters (cf. Fridman & Polyachenko 1984; Binney & Tremaine 1987; and references therein). The time evolution of the density f = f (t, x, v) 0 of the stars in phase space is governed by the following non-linear system of partial differential equations: t f + v x f x U v f = 0, U = 4πρ, lim U(t, x) = 0, x ρ(t, x) = f (t, x,v)dv. Here t R denotes time, x,v R 3 denote position and velocity respectively, ρ is the spatial mass density of the stars, and U is the gravitational potential that the stars induce collectively. Collisional or relativistic effects are neglected. This system, which Sir J. H. Jeans introduced at the beginning of the last century for the above modelling purposes, is sometimes in the astrophysics literature referred to as the collisionless Boltzmann Poisson system. Following the mathematics convention, we call it the Vlasov Poisson system. In the present paper we are interested in the steady states of this system. Besides their very existence, questions of interest are the possible shapes of such steady states, in particular their symmetry type, and whether or not they are dynamically stable. We approach these questions from a mathematics point of view, in particular, only such information as can be extracted from the system stated above is to enter our arguments. Our ideal reader is an astrophysicist interested in these same questions. We find it deplorable that there is gerhard.rein@uni-bayreuth.de little communication between astrophysicists and mathematicians investigating these problems, and the present paper is an attempt to change this. This will certainly cause some initial confusion about concepts, such as stability, which in the mathematics community need to have a rigorous meaning in order to be able to prove something, and this meaning will in general not encompass all that the astrophysics community connects with this concept. Thus we are trying to pay attention to the different uses of these concepts, without compromising their mathematical rigour in our analysis. Axially symmetric steady states of the the Vlasov Poisson system were obtained in Rein (000) as perturbations of spherically symmetric ones via the implicit function theorem. From an astrophysics point of view, axisymmetric models of elliptical galaxies have been investigated in Prendergast & Tomer (1970), Wilson (1975) and Toomre (198), but as to their stability no rigorous results are known to us. In the present paper we follow the variational approach developed in Guo (1999, 000), Guo & Rein (1999a,b, 001); and Rein (1999, 00a,b). We obtain axially symmetric steady states as minimizers of appropriately defined energy Casimir functionals. These steady states are shown to be non-linearly stable. The question we would like to raise is this: Do our mathematical constructions yield suitable models for real-world galaxies, or, if not, how can the mathematical approach be modified to do so? If U 0 = U 0 (x) is a stationary potential then any function f 0 = φ(e) of the particle energy E = E(x,v) = 1 v + U 0 (x) (1.1) solves the Vlasov equation with the potential U 0, since E is a conserved quantity along particle trajectories. So if U 0 is the potential induced by f 0, then this is a steady state of the Vlasov Poisson system; at this point it must be emphasized that to obtain a selfconsistent model by this approach, i.e., to make sure that U 0 is indeed the potential induced by f 0, is mathematically non-trivial. C 003 RAS

2 Stable models of elliptical galaxies 197 Any steady state obtained by this Ansatz is necessarily spherically symmetric, as follows from a result of Gidas, Ni & Nirenberg (1979). In the present paper we make f 0 depend on an additional invariant of the particle trajectories. If U 0 is axially symmetric, i.e. invariant under all rotations about, say, the x 3 -axis, then the corresponding component of angular momentum, P = P(x,v) = x 1 v x v 1, (1.) is such an additional invariant. To obtain steady states with a number density f 0 that depends on E and P and that are non-linearly stable, we proceed as follows. The total energy of a state f with induced spatial density ρ f, H( f ):= 1 v f (x,v)dv dx 1 (1.3) ρ f (x)ρ f (y) dx dy, x y is a conserved quantity along solutions of the Vlasov Poisson system. In addition, for any (reasonable) function Q = Q(f, P) of the two variables f 0 and P R, the quantity C( f ):= Q( f (x,v), P(x,v)) dv dx, (1.4) which we will refer to as a Casimir functional, is conserved along axially symmetric solutions; note that both f and P and thus also Q( f, P) are constant along the particle trajectories and that the latter induce a volume-preserving flow on phase space. The basic idea of our approach is to specify conditions on Q which guarantee that the energy Casimir functional H C ( f ):= H( f ) + C( f ) attains a minimum on the set of all non-negative, axially symmetric functions f subject to the constraint that the mass of f is prescribed: f (x,v)dv dx = M, with M > 0 given. The corresponding Euler Lagrange equation then shows that any minimizer is a steady state of the form f 0 = φ(e, P) with φ determined by Q. In order to conclude that this steady state is non-linearly stable, it is essential that the energy Casimir functional is conserved along solutions. However, this is the case only if P is constant along particle orbits, which is true in general only if the solution is axially symmetric. Thus, the stability result that we derive is restricted to axially symmetric perturbations of the steady state. The fact that C( f ) is not conserved along general solutions makes the terminology Casimir functional questionable, but we stick to it for lack of a better alternative. The situation is similar to the one of anisotropic, spherically symmetric steady states with f 0 depending on the particle energy and the modulus of the angular momentum, x v, where stability against spherically symmetric perturbations holds (cf. Guo & Rein 1999b), but the general case is open. For isotropic steady states where f 0 depends only on the particle energy, no such restriction is necessary (cf. Guo & Rein 001; Rein 00b). We want to stress the fact that we are by no means the first to use energy Casimirs in this context. There are quite a number of references in the astrophysics literature where this concept appears, and the concept as such is of course even older. However, in these references no non-linear stability results are proven, as is explicitly stated at the end of the second paragraph on page 105 of Kandrup (1990). In Kandrup (1990) the author then proceeds to use energy Casimir techniques to give a geometric structure to and understand better the linearized stability problem for the Vlasov Poisson system. Since the latter can be viewed as an infinite-dimensional Hamiltonian system, a linearized analysis cannot lead to a non-linear stability result. This is like the case of centre equilibria in a Hamiltonian system. In general no dynamical stability assertion can be deduced from a linearized analysis, and some Lyapunov-type methods are needed. Thus even the celebrated Antonov stability theorems for polytropes are at best only valid in the linearized sense. Recently, we were able to verify the validity of Antonov s theorems in a dynamical, non-linear sense (Guo & Rein 001). The paper proceeds as follows. In the next section we briefly discuss stability concepts from a mathematics point of view, describe the basic strategy of the energy Casimir method, and formulate and discuss our main results. In Section 3 we discuss various examples and some further properties of the steady states that we obtain. In particular we show that they are compactly supported, symmetric and decreasing in x 3 in addition to being axially symmetric, and they can have vanishing or non-vanishing average velocity field. Readers who are interested in the details of our proofs can find these in an Appendix. We conclude the introduction with some references to the mathematical literature. In addition to our work mentioned above, the stability of spherically symmetric steady states of the Vlasov Poisson system in the present stellar dynamics case is also investigated in Aly (1989), Wan (1999) and Wolansky (1999). Global classical solutions to the initial-value problem for the Vlasov Poisson system were first established in Pfaffelmoser (199, cf. also Schaeffer 1991, Rein 1997). For the plasma physics case where the sign in the Poisson equation is reversed, the stability problem is better understood, and we refer to Rein (1994), Guo & Strauss (1995a,b) and Braasch, Rein & Vukadinović (1999). A rather general condition that guarantees finite mass and compact support of steady states, but not their stability, is established in Rein & Rendall (000). MAIN RESULTS To put our results into perspective, we start with some general remarks on stability concepts. For a more detailed such discussion, we refer the reader to Holm et al. (1985). Consider a non-linear dynamical system, which for short we write as ẋ = A(x), where A is some non-linear operator on the possibly infinitedimensional state space X where solutions t x(t) take their values; in the Vlasov Poisson case think of X as being the space of all density functions f and A representing the non-linear integrodifferential operator constituted by the system. Let x 0 be a steady state, i.e. A(x 0 ) = 0. (i) The appropriate stability approach is one where we can deal with our problem without simplifying or linearizing it in any way. The steady state is called non-linearly stable or Lyapunov stable if sufficiently small initial perturbations of the steady state launch solutions that stay arbitrarily close to the steady state for future times. More precisely, one must find two norms x, x of elements x X of the state space such that for any (small) number ɛ>0 there always exists another (small) number δ>0 such that every solution x(t) with x(0) x 0 <δsatisfies x(t) x 0 <ɛfor all t 0. Some comments are in order: It is desirable but not always possible to use the same norm for measuring the initial perturbation and the one at time t; only on finite-dimensional spaces are all norms equivalent.

3 198 G. Rein and Y. Guo Sometimes one may not obtain norms but more general tools with which to measure the deviations. From a physics point of view it is desirable to have a rule on how to obtain the δ if ɛ is given, for example an estimate like x(t) x 0 C x(0) x 0, t 0 with some explicit constant C. Unfortunately, mathematicians are not always (not often?) able to provide this. A global existence result which guarantees that solutions exist for all time t 0 at least for initial data close to the steady state is a necessary prerequisite and integral part for all the above. (ii) Since assessing non-linear stability in the sense above is difficult, the problem is often approached via linearization. By Taylor expansion one can compute the linearization A lin of A at x 0 ; formally A lin is the (Frechet) derivative of A at x 0. Applying the definition for Lyapunov stability given above with A lin instead of A yields the concept of linearized stability. A related but different concept is spectral stability: one considers solutions of the linearized equation of the form e λt x this is what is done in Antonov s analysis. If the real parts of all the possible eigenvalues λ are strictly negative, then one calls the steady state spectrally stable. Again, some comments are in order: In general there is no guarantee that the linearized problem has solutions of the form e λt x. If it does, then in order to draw any conclusions there must be sufficiently many such solutions to get the general solution of the linearized problem by superposition. Even if this is true, there is simply no general result that allows any conclusion on the behaviour of the original, non-linear system, unless the system is finite-dimensional! For a conservative system with some sort of Hamiltonian structure (such as the Vlasov Poisson system), the best to expect as far as spectral stability is concerned may be that all the eigenvalues are purely imaginary (due to inherent symmetries of the spectrum). In this case no stability follows for the non-linear system, not even in the finite-dimensional case. Assume that one can establish the existence of a growing mode, i.e. of a solution of the form e λt x where λ has positive real part. Then again there is no general result saying that x 0 is now also non-linearly unstable, unless the system is finitedimensional. However, such a growing mode is a valuable first step towards proving a non-linear instability result. How difficult it still may be to get to a mathematically rigorous nonlinear instability result from there is illustrated by Guo & Strauss (1995a,b) The upshot of all this is that one should be very careful to draw conclusions about stability for non-linear, infinite-dimensional systems from linearization, in particular, if they are Hamiltonian. (iii) It is important for the present context to recall that in the mathematics literature one distinguishes between the concepts of stability and asymptotic stability. The latter means that a solution starting close to the steady state not only remains close to it but converges to it in some norm. This concept is clearly the appropriate one if one wants determine to which steady state a system will settle down as time proceeds. However, the Vlasov Poisson system that we use as a model here is clearly a conservative system, whereas for asymptotic stability a dissipative mechanism is necessary. The question to which steady state a galaxy settles down after an initial dynamical period is certainly very interesting from both a mathematics and a physics point of view, but it cannot be dealt with within the framework of a conservative model. We refer to Lynden-Bell (1967) and Tremaine, Henon & Lynden-Bell (1986) for astrophysics investigations of this issue. We now proceed to the Vlasov Poisson system and first have a look at the so-called energy Casimir method in this context. Let us fix some notation. For an integrable function f = f (x, v) 0, we define its induced spatial density ρ f (x) := f (x,v)dv, x R 3, and its induced gravitational potential ρ f (y) U f (x) := x y dy, x R3. Next we define the kinetic and potential energy of a state f : E kin ( f ):= 1 v f (x,v)dv dx, E pot ( f ):= 1 U f (x) dx = 1 ρ f (x)ρ f (y) dx dy. x y It is natural to try to relate steady states to minimizers (or more generally critical points) of the total energy (1.3), H( f ) = E kin ( f ) + E pot ( f ), since such minimizers are expected to be stable. However, for any fixed state f 0, [ ] 1 H( f ) = H( f 0 ) + v + U 0 (x) ( f f 0 )dv dx 1 U f U 0 dx, which shows that the total energy has no critical points to begin with: no matter how we choose f 0, the linear part in the expansion above does not vanish. This is the point where the Casimir functional comes into play. Suppose we have (a candidate for) a steady state of the form f 0 = φ(e, P) with particle energy E and P defined as in (1.1) and (1.). A judicious choice of a function Q = Q( f, P) can make the linear part in the expansion of the energy Casimir functional H C ( f ) = H( f ) + C( f ) vanish at f 0, where C( f )isdefined as in (1.4). For brevity we use the notation Q ( f, P) := Q ( f, P); f Q will never be differentiated with respect to P. Then H C ( f ) = H C ( f 0 ) + [E + Q ( f 0, P)]( f f 0 )dv dx + 1 Q (ξ, P)( f f 0 ) dv dx 1 U f U 0 dx, (.1) with some ξ between f 0 (x, v) and f (x, v). We want to choose Q such that Q (φ(e, P), P) = E. For this to be possible, φ should be strictly decreasing as a function of E (at least on the support of f 0 ) and Q ( f, P) increasing in f so that Q (ξ, P) > 0. Then the quadratic part in (.1) is the difference of two positive, quadratic terms, and therefore seems to be indefinite. Assume for the moment that we were looking at the plasma physics case of the Vlasov Poisson system where the sign in the Poisson equation is reversed.

4 Stable models of elliptical galaxies 199 Then we would obtain the sum of the two positive quadratic terms instead of their difference, and, up to some technicalities, a nonlinear stability result would follow (cf. Rein 1994). However, in the present stellar dynamics case, the method seems to fail, since it is the quadratic part in (.1) that is to measure the deviation of a perturbed solution from the steady state. It is precisely this difficulty that was also observed in (Kandrup 1990, p. 105) and led the author to the statement that [quote] these energy Casimir techniques no longer imply non-linear stability. Indeed, to obtain a non-linear stability result in spite of these difficulties, some non-trivial additional arguments seem necessary, and we have developed the following variational approach. We try to find conditions on the function Q such that the energy Casimir functional H C has a minimizer under a mass constraint, i.e. over the set of non-negative integrable functions with prescribed mass M > 0: F M := { f L 1 (R 6 ) f 0, f (x,v)dv dx = M, } C( f ) + E kin ( f ) <. The idea is, first of all, that such a minimizer will be a steady state of the form f 0 = φ(e, P) with φ determined by Q, and secondly that its minimizing property will allow us to get around the difficulty with the indefinite quadratic part in the expansion (.1), thereby concluding stability of the minimizer. For the function Q determining the Casimir functional (1.4) we make the following assumptions. Assumptions on Q. The function Q is non-negative and continuous in f 0 and P R with Q = Q/ f continuous and Q continuous for f > 0 and P R, Q(0, P) = 0 = Q (0, P) for P R, and with positive constants C, C, F, F >0, and 0 < k, k < 3/: (Q1) Q( f, P) Cf 1+1/k, f F, P R, (Q) Q ( f, P) > 0, f > 0, P R, (Q3) for every f 0, Q( f, P) is increasing in P ],0[ and decreasing in P ]0, [, (Q4) Q( f,0) C f 1+1/k, f F. The first two assumptions are essential whereas the assumptions (Q3) and (Q4) are technical and can be modified in various ways (cf. the next section). The conditions above look much simpler if Q does not depend on P; in particular, (Q3) can then be dropped. This simpler Ansatz leads to a stability analysis of isotropic, spherically symmetric steady states, which we carried out in Guo & Rein (001) and Rein (00b). Also for this simpler case no rigorous non-linear results were previously known. For any η 0 and P R the equation Q ( f, P) = η can be solved uniquely for f, since Q > 0. Therefore there exists a unique function φ = φ(η, P) such that Q (φ(η, P), P) = η, η 0, P R, (.) and we extend φ by 0 for negative values of η. The steady states that we shall obtain will be of the form f 0 (x,v) = φ(e 0 E, P), where the particle energy E is defined in terms of the induced potential U 0 as in (1.1) and E 0 0 is some cut-off energy. Thus f 0 is a decreasing function of the particle energy E and vanishes for energies larger than E 0, the latter being a necessary condition to ensure finite total mass of the steady state (cf. Rein & Rendall 000). Eventually, we need to restrict ourselves to axially symmetric functions in the set F M, { FM S := f F M f (Ax, Av) = f (x,v), x,v R 3, } A any rotation about the x 3 -axis, but we should point out that if Q does not depend on P then no symmetry assumptions need to be made in the choice of the set F M nor anywhere else. We now state our first main result as follows. Theorem 1. For every M > 0 there is a minimizer f 0 FM S of the energy Casimir functional H C, i.e. H C ( f ) H C ( f 0 ), f FM S. Let U 0 denote the potential induced by f 0. Then f 0 is a function of the particle energy E and angular momentum P as defined in (1.1) and (1.), f 0 (x,v) = φ(e 0 E, P) where φ is defined in terms of Q as explained above, cf. (.). The parameter E 0 plays the role of a Lagrange multiplier and is given by E 0 := 1 M [E + Q ( f 0, P)] f 0 dv dx. In particular, f 0 is an axially symmetric steady state of the Vlasov Poisson system with total mass M. Sketch of the proof. In the sequel g p = ( g p ) 1/p denotes the usual L p -norm over R 3 or R 6 as the case may be. Step 1. The energy Casimir functional H C is bounded from below on the set FM S. Therefore one can choose a minimizing sequence ( f i ) i N in FM S so that H C( f i ) inf F S H C, i. M Step. Along any minimizing sequence, f i 1+1/k and ρ i 1+1/n are bounded where k is from (Q1) and n := k + 3/. This allows us to extract a subsequence (again denoted by f i ) which converges to some f 0 FM S in the so-called weak sense. Step 3. f 0 is a natural candidate for a minimizer. To prove that it is a minimizer, we try to pass to the limit in H C along the weakly convergent subsequence obtained above. The crucial part is to show that for the induced potentials we obtain convergence U i U 0 strongly in L (R 3 ), i.e. the potential energy E pot ( f i ) converges. The details of these steps are given in the Appendix. Once the existence of a minimizer is established, the fact that, for all f FM S, H C ( f ) H C ( f 0 ) together with the constraints f 0 = M and f 0 0 imply that on the set where f 0 > 0 the Euler Lagrange relation Q ( f 0, P) + E = E 0 holds with some Lagrange multiplier E 0 note that the left-hand side represents the linear part in (.1) and E > E 0 where f 0 = 0. This proves the desired form of f 0 ; for the details of the latter arguments we refer to Guo & Rein (1999b) or Guo & Rein (001). Remark 1. If one is interested only in the existence of a steady state, a more direct approach is to prescribe some suitable function f 0 (x, v) = φ(e 0 E, P). It then remains to prove that the semilinear Poisson equation [ U 0 = 4π φ E 0 1 ] v U 0, P(x,v) dv (.3) has a suitable solution. Besides the very existence of a solution to (.3), the non-trivial problem here is to decide which solutions have finite total mass and compact support in space. The first requirement for φ from the point of view of our theorem is that φ is a strictly

5 1300 G. Rein and Y. Guo increasing function of E 0 E so that a corresponding function Q and the Casimir functional can be defined. For the existence question alone, this monotonicity condition, which in astrophysics textbooks appears as a stability condition, is not necessary. A mathematical approach that makes no such monotonicity assumption can be found in Rein & Rendall (000). The point is that the minimizer that we obtain is not just any old steady state, but a non-linearly stable one. So let us now investigate the dynamical stability of a minimizer f 0 as obtained in Theorem 1. For states f with f = M we rewrite the expansion (.1) of the energy Casimir functional about f 0 in the form H C ( f ) H C ( f 0 ) = d( f, f 0 ) 1 U f U 0, (.4) where the distance d( f, f 0 ) between f and f 0 is defined by d( f, f 0 ):= [(E E 0 )( f f 0 ) + Q( f, P) Q( f 0, P)] dv dx. We are allowed to subtract the term E 0 ( f f 0 ) from the integrand since its integral vanishes. Since Q( f, P) as a function of f is convex, the integrand can be estimated from below by [(E E 0 ) + Q ( f 0, P)]( f f 0 ). According to Theorem 1, this quantity is zero where f 0 > 0, while it equals (E E 0 ) f 0 where f 0 = 0. Thus we see that d( f, f 0 ) 0, f F M. We seem to be back to the difficulty that the quadratic term in the expansion (.4) is indefinite. However, from step 3 of the sketch of the proof of Theorem 1, we now know that along a minimizing sequence the term U f U 0, which has the wrong sign, tends to 0. This allows us to prove the following non-linear stability result as follows. Theorem. For every ɛ>0there is a δ>0such that for any solution t f (t) of the Vlasov Poisson system with f (0) FM S continuously differentiable, compactly supported, and reflection symmetric, i.e. f (0, x 1, x, x 3, v 1, v, v 3 ) = f (0, x 1, x, x 3, v 1, v, v 3 ), the estimate d( f (0), f 0 ) + 1 U f (0) U 0 <δ implies that, for all t 0, d( f (t), f 0 ) + 1 U f (t) U f0 <ɛ, provided f 0 is unique or at least isolated in the reflection-symmetric subset of FM S. Proof. Since after what was said above the proof is simple and instructive, we include it here. First, we point out that if we shift a minimizer in the x 3 -direction we obtain another minimizer. Moreover, we in general do not know whether the minimizers are unique up to such shifts. As discussed in the next section, a minimizer must, a posteriori, have the additional symmetry property f 0 ( x1, x, x 3 x 3,v 1,v, v 3 ) = f 0 ( x1, x, x 3 +x 3,v 1,v,v 3 ), x,v R 3, for some x3 R. Without loss of generality we take x 3 = 0 and refer to this as reflection symmetry. This symmetry propagates along solutions of the time-dependent problem, and, by restricting ourselves to data with this symmetry, we can at least ignore the non-uniqueness of the minimizer due to shifts. Now assume the assertion of the theorem were false. Then there exist ɛ>0, t n > 0, and initial data f n (0) as specified in the theorem such that, for all n N, d( f n (0), f 0 ) + 1 U fn(0) U 0 < 1 (.5) n but d( f n (t n ), f 0 ) + 1 U fn(t n) U 0 ɛ. (.6) By (.5) and (.4), lim H C( f n (0)) = H C ( f 0 ). n Since H C is conserved along classical, axially symmetric solutions as launched by f n (0), since mass is conserved, and since the symmetry propagates, lim H C( f n (t n )) = H C ( f 0 ) and f n (t n ) F S M, n N, n i.e. ( f n (t n )) is a reflection-symmetric minimizing sequence for H C in FM S. Up to a subsequence we may therefore assume by step 3 of the sketch of the proof of Theorem 1 or by Theorem A1 respectively that U fn(tn) U f0 0. (.7) It is at this point that the uniqueness or isolation of the minimizer f 0 is used. Note that, owing to the reflection symmetry of the minimizing sequence and of any minimizer, the shifts along the x 3 -axis in the statement of Theorem A1 must be zero in the present situation. Since lim n H C ( f n (t n )) = H C ( f 0 ) we conclude by (.7) and (.4) that d( f n (t n ), f 0 ) 0, n, and we arrive at a contradiction to (.6). It should be noted here that, for non-negative, continuously differentiable, and compactly supported initial data, the Vlasov Poisson system has smooth solutions that exist for all time (cf. Schaeffer 1991; Pfaffelmoser 199; Rein 1997). Remark. If the minimizer is not isolated, i.e. if arbitrarily close to it with respect to the distance used in the theorem there are other minimizers that do not result by a simple translation in space, then we obtain a somewhat weaker stability result in the sense that any solution that starts close to the set of all minimizers stays close to this set for all future times (cf. Guo & Rein 001, theorem 4, for the precise formulation). For the case Q( f ) = f 1+1/k, which corresponds to the isotropic polytropes f 0 (x, v) = (E 0 E) k, the minimizer is unique, i.e. there is only one polytropic steady state for prescribed k and mass M. Numerical evidence suggests that the minimizers are not unique in general: the potential of a minimizer satisfies the semi-linear Poisson equation (.3). This equation can be solved numerically, in particular if there is no dependence on P and hence U 0 is spherically symmetric and (.3) reduces to an ordinary differential equation. One can then check which of the numerically found solutions have a given mass M and minimize the energy Casimir functional. In Rein (003 remark 3(b)) an example is presented for a closely related problem where two distinct minimizers are found. This example can easily be adapted to the present situation. On the other hand, no numerical evidence for minimizers that are not isolated has been found along these lines. For the isotropic case it is shown in Schaeffer (00) that Theorem remains correct without the assumption of isolatedness.

6 Stable models of elliptical galaxies 1301 Remark 3. We showed that d( f, f 0 ) 0 for f F M, but in addition d( f, f 0 ) = 0 if and only if f = f 0. One may think of d( f, f 0 ) as a weighted L -difference of f and f 0, and if Q is bounded away from zero the more explicit estimate d( f, f 0 ) C f f 0 can be obtained. Remark 4. The restriction f (0) F M for the perturbed initial data would be acceptable from a physics point of view. A perturbation of a given galaxy, say by the gravitational pull of some outside object, results in a perturbed state that is just a rearrangement of the original state in particular, its mass remains unchanged. In general, symmetry properties of the steady state will be destroyed by such a perturbation. However, we need that f (0) is axially symmetric in order that C be conserved along the resulting axially symmetric solution f (t). So while this symmetry restriction for the perturbation seems necessary within the present mathematical framework, it is quite undesirable from a physics point of view. If the steady state does not depend on P, and thus is isotropic, then no symmetry restrictions are necessary anywhere in our results, and the stability is with respect to quite general perturbations (cf. Guo & Rein 001; Rein 00b). Remark 5. Our stability result is of the non-linear Lyapunov type discussed above, but it does suffer from the defect that, given ɛ, it does not say how δ needs to be chosen. The reason for this is that our proof is by contradiction and relies on a compactness result. Remark 6. To conclude this section, we stress the fact that the input of our method is the function Q which fixes the Casimir functional and needs to satisfy the assumptions specified above. The output of the method is a steady state of the Vlasov Poisson system which is non-linearly stable against axisymmetric perturbations and whose distribution function φ as a function of the particle energy E and third component of angular momentum P is determined by Q. Alternatively, one can start with φ, compute the corresponding Q via (.), check if Q satisfies the assumptions, and conclude that a selfconsistent model with the prescribed φ exists and is non-linearly stable. As is argued in the astrophysics literature, realistic models for many galaxies must depend on an additional third integral besides E and P. If such a third integral J like P is given as a function on phase space (not depending on the potential), and if it is preserved along particle trajectories of time-dependent solutions with some symmetry assumption that is preserved by these solutions, then we can extend our method to Casimir s given by Q = Q( f, P, J) under the assumption that the scaling arguments in Lemma A3 go through. Unfortunately, the non-classical third integral J that appears in the astrophysics literature for potentials of Stäckel form (cf. de Zeeuw 1985; de Zeeuw & Lynden-Bell 1985; Bishop 1986; Dejonghe & de Zeeuw 1988) is not of this type. In particular, J does depend on the gravitational potential, so it is more like the particle energy than like P, and the form of the potential is not specified by a symmetry assumption which like spherical or axial symmetry would be preserved for time-dependent solutions. It seems to us that from a mathematics point of view both the existence and the stability of such models is an open, challenging problem calling for a new line of investigation. Moreover, it should be noted that in the astrophysics literature axially symmetric steady states are usually constructed as follows. Given a suitable pair of gravitational potential and mass density obeying the Poisson equation, one tries to find possible integrals of the particle motion in addition to the particle energy. Then one tries to produce (numerically or analytically) a distribution function as a function of these integrals that gives rise to the given mass density. The latte function is rarely known explicitly. An axisymmetric model depending only on the two classical integrals E and P for which the distribution function is given (rather) explicitly is considered in Batsleer & Dejonghe (1993). While our major stability condition, the convexity of the corresponding function Q, seems to hold, we have not been able to verify the remaining assumptions for this model. 3 EXAMPLES AND FURTHER PROPERTIES In this section we investigate some additional properties of the resulting steady states. Then we give examples for functions Q that satisfy our assumptions or possible variations of them. We start by some general comments on how to extract information about the resulting steady states. In our approach we prescribe Q. The potential of the resulting steady state then satisfies the equation (.3), and by analysing the solutions of this equation one may obtain additional information about the steady state. Since (.3) is a non-linear partial differential equation, detailed information on the steady state, for example the density profile along the major and minor axes, can (probably) only be obtained by solving this equation numerically. This would be a non-trivial, separate investigation. However, there are a couple of questions that one cannot or should not try to settle numerically. Before one starts to compute a solution of (.3) numerically, one must know that a solution actually exists. If a solution exists, the resulting steady state may or may not have finite mass and compact support. A priori, one needs to solve (.3) on the whole space, and whether ρ vanishes for large distances from the origin or just becomes very small can be difficult to distinguish numerically. Under the assumptions on Q, Theorem 1 not only answers the existence question in the affirmative but guarantees a finite value for the total mass M of the steady state. Below we will indicate how to prove that ρ vanishes outside some bounded spatial region. Regularity and boundary condition Since, by Theorem 1, f 0 is a function of the quantities E and P, which are constant along particle trajectories, we are justified to call f 0 a steady state provided U 0 is sufficiently regular to allow for the definition of particle trajectories to begin with. Now U 0 = 4πρ 0 on R 3, and from the very construction of f 0 certain integrability properties of ρ 0 follow (cf. the Appendix). These imply corresponding properties of (generalized) derivatives of U 0. Since on the other hand ρ 0 is a functional of U 0 (cf. equation 3.1 below), this regularity can be bootstrapped and U 0 becomes twice continuously differentiable with lim x U 0 (x) = 0. For the technical details, which include a use of the so-called Sobolev embedding theorem, we refer the reader to Guo & Rein (001, theorem 3). Finite mass and compact support By construction, the steady states have finite mass M > 0, which we prescribe by our mass constraint. To continue, we note that, by Theorem 1 and a change of variables, ρ 0 (x) = f 0 (x,v)dv E0 [E U 0 (x)] = π (3.1) φ(e 0 E, r(x)p) dp de U 0 (x) [E U 0 (x)] if U 0 (x) < E 0, and ρ 0 = 0 else; here r(x) := x1 + x. By the discussion above, lim x U 0 (x) = 0. Thus the cut-off energy E 0 cannot be positive, since otherwise we get infinite mass at spatial

7 130 G. Rein and Y. Guo infinity. Since U 0 < 0 everywhere, (3.1) shows that ρ 0 will have compact support if and only if E 0 < 0. An additional, sufficient condition that implies this is fq ( f, P) 3Q( f, P), f 0, P R, (3.) which holds for example for Q( f, P) = f 1+1/k g(p) with 1/ k < 3/ and some function g. To see this we rewrite the formula for E 0 from Theorem 1: [ ] E 0 = 1 M E kin ( f 0 ) + E pot ( f 0 ) + f 0 Q ( f 0, P) dv dx Now we note that for solutions of the Vlasov Poisson system, d x v f = E kin ( f ) + E pot ( f ), dt. (3.3) so that for a steady state the right-hand side is zero. Thus by (3.), E kin ( f 0 ) + E pot ( f 0 ) + E kin ( f 0 ) + E pot ( f 0 ) + 3 = 3E kin ( f 0 ) + 3E pot ( f 0 ) + 3C( f 0 ) = 3H C ( f 0 ) < 0 f 0 Q ( f 0, P) dv dx Q( f 0, P) dv dx by Lemma A3(a). An alternative condition which also implies that E 0 < 0 and therefore guarantees compact support is the following: φ(e 0 E, P) C 1 (E 0 E) k 1 }, E, P R, φ(e 0 E, P) C (E 0 E) k (3.4), E E 0, P R, with constants C 1, C > 0 and 0 < k 1, k < 3/. Note that this corresponds to conditions on Q that are compatible with the conditions on Q. For the details, see Rein (00b, theorem 3). In the spherically symmetric case our approach provides an explicit bound on the radius of the steady state (cf. Guo & Rein 1999a). In this case one can decide whether U 0 crosses the cut-off energy level E 0 by direct examination of the semi-linear Poisson equation (.3), since this equation reduces to an ordinary differential equation with respect to the radial variable (cf. Rein & Rendall 000). However, in the axially symmetric case this is not true, and the corresponding analysis of the partial differential equation (.3) would be much more difficult. Symmetry By construction the steady states that we obtain are axially symmetric, and we want to show that they are in general not spherically symmetric. To see this, we take Q such that φ(e 0 E, P) is not constant in P on any neighbourhood of P = 0. We claim that neither ρ 0 nor U 0 are spherically symmetric in this case. Indeed, if ρ 0 were spherically symmetric, the same would be true for U 0.IfU 0 were spherically symmetric, we take some x R 3 with r(x) = x1 + x 0 small and take A SO(3) such that Ax = (0, 0, x ), hence r(ax) = 0 r(x) butu 0 (Ax) = U 0 (x). Inserting this into the formula (3.1) for ρ 0, we see that in general ρ 0 (Ax) ρ 0 (x) soρ 0 is not spherically symmetric. Indeed, this is more than just saying that f 0 is not spherically symmetric. To see this, take a spherically symmetric steady state f 0 with induced density ρ 0, potential U 0 and particle energy E = 1/ v + U 0 (x). Let g 0 (x, v) = ψ(e, P) be any function that is odd in P, and such that f 0 + g 0 0. Since P is not invariant under general rotations, neither is f 0 + g 0, but the fact that ψ is odd in P implies that ρ f0 +g 0 = ρ 0 and the same holds for the potential. So this trivial construction gives an axially symmetric steady state where the phase space density is not spherically symmetric but the macroscopic quantities ρ 0 and U 0 are. Our steady states are in general not of this trivial type. Next we show that any minimizer is symmetric and decreasing in x 3. Given a minimizer f 0 we can perform a symmetric decreasing rearrangement of f 0 with respect to x 3 while keeping all other variables fixed. Denote this rearrangement by f 0 ;as to the rearrangement concept, we refer the reader to Lieb & Loss (1996, chapter 3). The rearrangement does not change the kinetic energy, nor the Casimir functional, since P does not depend on x 3. By Lieb & Loss (1996, theorem 3.7) it can only decrease the potential energy. But since f 0 already minimizes H C, f 0 minimizes H C as well and the potential energy remains unchanged under the rearrangement. By Lieb & Loss (1996, theorem 3.9) this can happen only if f 0 (x, v) = f 0 (x + (0, 0, T v), v) for some possibly v-dependent shift T v in the x 3 -direction. Since both f 0 and f 0 are minimizers, they are both of the form stated in Theorem 1, so f 0 (x,v) = φ(e 0 1/ v U f0 (x), P) and f0 (x,v) = φ(e0 1/ v U f 0 (x), P). The explicit form of E 0 now implies that E 0 = E 0 (cf. equation 3.3), hence U f 0 (x) = U f 0 (x+t v ) and the translation T v is independent of v. Hence the minimizer f 0 is a translation in the x 3 -direction of f 0, which by definition is symmetric and decreasing in x 3, and without loss of generality we can assume that f 0 is reflection symmetric in x 3. The corresponding symmetry in the variable v 3 follows from the form that f 0 must have due to Theorem 1. Stationary versus static solutions If instead of the Vlasov Poisson system we consider a selfgravitating fluid as described by the Euler Poisson system, then every so-called static solution, i.e. every steady state with vanishing velocity field, is spherically symmetric (cf. Lichtenstein 1933). This turns out to be false for the steady states of the Vlasov Poisson system that we obtain. By definition the velocity field equals j 0 /ρ 0 on the support of ρ 0 where the mass current density j 0 is given by j 0 (x) = v f 0 (x,v)dv = π E0 U 0 (x) [E U 0 (x)] pφ(e 0 E, r(x)p) dp dee t (x) [E U 0 (x)] with e t (x) := ( x, x 1,0)/r(x); on the x 3 -axis the velocity field vanishes. If φ is even in P then j 0 vanishes identically, so there exist static solutions that are not spherically symmetric among the ones we obtain. On the other hand, if, say, φ(η, P) >φ(η, P) for all η 0 and P > 0, the velocity field does not vanish and corresponds to an average rotation about the axis of symmetry in the anticlockwise direction. Now we present some examples for functions Q that satisfy our assumptions or possible variations of them. Examples A simple class of examples for functions Q that satisfy the assumptions (Q1) (Q4) is given by Q( f, P) = f 1+1/k g(p), f 0, P R, (3.5) where 0 < k < 3/ and g is positive, continuous, bounded, bounded away from zero, increasing on ], 0[, and decreasing on ]0, [.

8 Stable models of elliptical galaxies 1303 In this case, f 0 (x,v) = C(E 0 E) k g(p) k on its support. Examining the proof of Lemma A3 in the Appendix, which is the only place where the assumptions (Q3) and (Q4) enter, one can see that these assumptions can be replaced by (Q3 ) Q(λ f, P) λ 1+1/k Q( f, P), f 0, 0 λ 1, P R, with 1/ k < 3/, (Q4 ) Q( f, P) C f 1+1/k, f F, P P 0, with constants C > 0, F > 0, P 0 > 0 and 0 < k < 3/. Examples that satisfy the assumptions (Q1), (Q), (Q3 ) and (Q4 ) but not the original ones are provided by (3.5) with 1/ k < 3/ and g, which has all the properties stated above except the monotonicity. Less regular dependence on P When examining the Appendix the reader may notice that we do at no place make use of the fact that all the functions under consideration can be taken to be axially symmetric. This is because exploiting axial symmetry in the estimates for the potential is technically quite unpleasant. However, it is possible and even necessary if one wishes to study examples of the form Q( f, P) = f 1+1/k P l/k, f 0, P R, (3.6) which for l 0 do not satisfy the assumptions on Q that we stated above. The corresponding steady states would take the form f 0 (x,v) = C(E 0 E) k P l, which is analogous to the classical, spherically symmetric polytropes, except that the square of the third component of the angular momentum replaces the square of the modulus of the angular momentum. Clearly, (3.6) satisfies the crucial convexity condition (Q), provided k > 0. The scaling arguments in Lemma A3 go through, provided l > 1, 0 < k < l + 3. The place where significant changes become necessary are the estimates in Lemma A1 and Lemma A6. One cannot control f 1+1/k and ρ f 1+1/n in terms of E kin ( f ), C( f ) and M. Instead, one can control the weighted norms P l/k (x,v) f 1+1/k (x,v)dv dx, r l/n (x)ρ 1+1/n f (x) dx, where r(x) = x1 + x. It seems reasonable to expect that the weight factor can be dealt with in the estimates for the potential energy if one makes proper use of the axial symmetry. ACKNOWLEDGMENTS The research of YG is supported in part by an NSF Grant and a Sloan Fellowship. GR acknowledges support by the Wittgenstein 000 Award of P. A. Markowich. Furthermore, the authors acknowledge valuable help and criticism of an unknown referee, which hopefully have made the paper more accessible to our intended audience. REFERENCES Aly J. J., 1989, MNRAS, 41, 15 Batsleer P., Dejonghe H., 1993, A&A, 71, 104 Batt J., Faltenbacher W., Horst H., 1986, Arch. Rat. Mech. Anal., 93, 159 Binney J., Tremaine S., 1987, Galactic Dynamics. Princeton Univ. Press, Princeton, NJ Bishop J. L., 1986, ApJ, 305, 14 Braasch P., Rein G., Vukadinović J., 1999, SIAM J. Appl. Math., 59, 831 Evans L., 1998, Partial Differential Equations. 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It may be useful if the reader at this point recalls the basic steps in the sketch of the proof of Theorem 1 stated above. First we collect some estimates for ρ f and U f induced by an element f F M. These estimates make no use of symmetry and rely on the assumption (Q1). Their main purpose is to see that the energy Casimir functional H C is bounded from below on the constraint set F M, and to establish certain bounds along minimizing sequences of H C. Constants denoted by C are positive, may depend only on Q and M, and their value may change from line to line. Lemma A1. Let n := k + 3/ < 3. Then for any F M the following hold: (a) f 1+1/k (x,v)dv dx C[1 + C( f )],

9 1304 G. Rein and Y. Guo (b) ρ 1+1/n f (x) dx C[1 + E kin ( f ) + C( f )], (c) U f L 6 (R 3 ) with U f L (R 3 ), the two forms of E pot ( f ) stated in Section are equal, and U f dx C ρ f 6/5 C[1 + E kin( f ) + C( f )] n/3. Proof Part (a) follows from (Q1). Splitting the v integral in the definition of ρ f into small and large v ranges and optimizing the split yields (b). The Hardy Littlewood Sobolev inequality (Lieb & Loss 1996, theorem 4.3) and interpolation (Evans 1998, appendix B, h) together with (b) imply (c). For more details, see Guo & Rein (1999b) or Guo & Rein (001). The estimates above have an immediate and important consequence, as follows. Lemma A. The energy Casimir functional H C is bounded from below on F M, i.e. h M := inf H C ( f ) >, f F M and along any minimizing sequence of H C in F M the quantities E kin ( f ), C( f ), f 1+1/k and ρ f 1+1/n are bounded. Proof By Lemma A1, for f F M, H C ( f ) E kin ( f ) + C( f ) C[1 + E kin ( f ) + C( f )] n/3, and since n < 3 the assertions follow. Lemmas A1 and A remain valid if we replace F M by its axially symmetric subset FM S. Of course the infimum h S M := inf H C ( f ) f F S M of H C on this smaller set may be larger. It is a general result that any bounded sequence (g i ) L p with 1 < p < has a weakly convergent subsequence (g i j ), i.e. there exists some g 0 L p such that for any test function h L q where 1/p + 1/q = 1wehave gi j h g 0 h (cf. Lieb & Loss 1996, theorem.18). By the previous lemma, any minimizing sequence has a weakly convergent subsequence, and the remaining arguments are to improve this weak convergence in such a way that we eventually can pass to the limit in the energy Casimir functional H C. The assumptions (Q3) and (Q4) determine the behaviour of H C under scaling transformations which we use to show that h M is negative and to relate the h M for different values of M as follows. Lemma A3. (a) Let M > 0. Then < h M < 0. (b) For 0 < M M, h M ( M/M) 5/3 h M. Proof Given any function f,wedefine a rescaled function f (x,v) = af(bx, cv), where a, b, c > 0. Then f dv dx = a(bc) 3 f dv dx, (A1) i.e. f F M iff f F M where M = a(bc) 3 M.Next H C ( f ) = ab 3 c 5 E kin ( f ) + a b 5 c 6 E pot ( f ) + (bc) 3 Q(af(x,v), (bc) 1 P(x,v)) dv dx. (A) To prove (a) we fix a function f F 1 with f F and let a = (bc) 3 M so that f F M. Then by (Q3) and (Q4) and with positive constants C 1, C and C 3 that depend on f, H C ( f ) Mc E kin ( f ) + M be pot ( f ) + Ma 1 Q(af(x,v), 0) dv dx C 1 c C b + C 3 a 1/k, provided a 1 so that (Q4) can be applied; note that E pot ( f ) < 0. We want the negative term to dominate as b 0, so we let c = b γ/ for some γ>0. Then a = Mb 3(1 γ/), and H C ( f ) C 1 b γ C b + C 3 M 1/k b 3(1 γ/)/k. Since 0 < k < 3/, we can fix γ ]1, [ such that we have 3(1 γ/)/k > 1. For b > 0 sufficiently small, H C ( f ) will then be negative and a = Mb 3(1 γ/) < 1 as required. This proves part (a) of the lemma. To prove (b) we choose a = c = 1 and b = ( M/M) 1/3 so that the mapping F M F M, f is one-to-one and on to and b 1 1. By (A.), H C ( f ) = b 3 E kin ( f ) + b 5 E pot ( f ) + b 3 Q( f (x,v), b 1 P(x,v)) dv dx b 5 E kin ( f ) + b 5 E pot ( f ) + b 5 Q( f (x,v), P(x,v)) dv dx = b 5 H C ( f ); we multiplied the two positive terms by b 1 and used the monotonicity of Q in the variable P which we required in (Q3). By the choice of b and the definitions of h M and h M, the proof is complete. It is again obvious that the assertions of the lemma remain valid if we restrict ourselves to the axially symmetric functions in F M. The reader might also check that the scaling arguments work under the assumptions (Q3 ) and (Q4 ) from Section 3 as well. Next we provide a splitting estimate which will be used to show that along a minimizing sequence the mass cannot escape to infinity; here and in the following we denote, for 0 < R < S, B R :={x R 3 x R}, B R,S :={x R 3 R x < S}. Lemma A4. Let f F M. Then sup a R 3 a+b R f (x,v)dv dx [ ] 1 E pot ( f ) M RM R C ρ f 1+1/n R (5 n)/(n+1), R > 1. The proof follows from splitting the potential energy into three parts according to x y 1/R, 1/R < x y < R and R x y ; for the details we refer to Guo & Rein (001) or Rein (00a, lemma 3). The splitting estimate above has the following important consequence for minimizing sequences. Lemma A5. Let ( f i ) F M be a minimizing sequence of H C. Then there exist δ 0 > 0, R 0 > 0, i 0 N, and a sequence of shift vectors (a i ) R 3 such that f i (x,v)dv dx δ 0, i i 0, R R 0. a i +B R