Abstract. The main tool in the analysis is the free energy, a conserved quantity. of stationary solutions which satisfy the above stability condition

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1 Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions Jurgen att, Philip J. Morrison, & Gerhard Rein Abstract Rigorous results on the stability of stationary solutions of the Vlasov-Poisson system are obtained in the contexts of both plasma physics and stellar dynamics. It is proven that stationary solutions in the plasma physics (stellar dynamics) case are linearly stable if they are decreasing (increasing) functions of the local, i. e., particle, energy. The main tool in the analysis is the free energy, a conserved quantity of the linearized system. In addition, an appropriate global existence result is proven for the linearized Vlasov-Poisson system and the existence of stationary solutions which satisfy the above stability condition is established. Introduction The evolution considered in this paper is governed by the Vlasov-Poisson t f x f ;@ x v f = 4U =4 ; + + (t x):= f(t x v)dv where t denotes time, x IR 3 position, and v IR 3 velocity. For = ; the system describes a collisionless plasma of electrons, which move in an electrostatic eld that arises self-consistently from the electron spatial charge density (t x) and a xed ion background with spatial charge density + = + (x). The case where = and + is set to zero describes a collisionless ensemble of self-gravitating point masses, e.g., stars in a galaxy or galaxies in a galactic cluster. In this case (t x) represents the spatial mass density. The function f = f(t x v) denotes the phase space density of either the electrons or stars, while ;U or U denotes the electrostatic or gravitational potential, respectively.

2 The initial-value problem for this system, where the initial phase space density f( x v)= f(x v) is prescribed, is now well understood, and the existence of global, classical solutions for C data with appropriate decay at innity is established [,, 7, 3, 35]. However, the above rigorous results provide only limited information about the qualitative behavior of the solutions. The purpose of the present investigation is to clarify the question of stability of certain stationary solutions. Two main stability concepts have to be distinguished: A stationary solution is nonlinearly stable if solutions of the nonlinear Vlasov-Poisson system remain arbitrarily close to the stationary solution in some norm for all times, provided the Vlasov-Poisson solutions start suciently close to the stationary solution. The stationary solution is linearly stable if the solutions of the nonlinear system are replaced by the solutions of the corresponding linearized problem in the above \denition." Obviously, a global existence result at least for initial data close to the steady state under consideration is an integral part of both stability concepts. If the solution is written as f +f(t), where f is the distribution function of the steady state, and the term that is quadratic in f(t) is neglected, one obtains the linearized Vlasov-Poisson t f x f ;@ x v f x U v f 4U f =4 f f (t x):= f(t x v)dv where the steady state (f U ) satises the stationary Vlasov-Poisson system: x f ;@ x v f = 4U =4( + + ) (x):= f (x v)dv: Stability conditions are often expressed in terms of how f depends on the local or particle energy E(x v):= v +U (x). Since E and, for spherical symmetry under which U (x)=u (jxj), also F := jxvj are constant along the characteristics of the stationary Vlasov equation, it is natural to represent f (x v)='(e)orf (x v)='(e F) with some function '. The present work is restricted to the rst case.

3 There exists a large number of investigations of both linear and nonlinear stability, cf. [,,,, 3, 6, 9,,, 36, 37, 38]. The general and long standing opinion seems to be that both in the plasma physics and in the stellar dynamics cases a steady state is stable if ' is a decreasing function of the energy. Although these results are physically appealing and plausible, a distinction must be made between these results and rigorous mathematics. (Note, too, that certain conclusions drawn for anisotropic spherical systems are admittedly contradictory [3, p. 38].) Concerning nonlinear stability we mention the following rigorous results: In [9] it is shown, for the plasma physics case with spatial periodicity, that spatially homogeneous steady states are nonlinearly stable if ' is decreasing the analogous result for the relativistic Vlasov-Maxwell system is shown in [5]. oth results are based on using the total energy of the system as a Lyapunov function, cf. also [8]. In [34] it is shown by Rein how to put the so-called energy-casimir method as described in [] into a rigorous mathematical setting for the plasma physics case, thereby obtaining a nonlinear stability result in this case. In [38] an attempt to prove nonlinear stability using the energy-casimir method for the stellar dynamics case was made however, shortcomings of this work were pointed out in [4]. (ounds used in [38] require that solutions of the Vlasov-Poisson system have uniformly bounded gradients, a property that has not to date been established. Also, it is not clear whether steady states that are \regular" in the sense of Denition exist.) As for linear stability, the only rigorous result we areaware of is [8] where the eect of Landau damping is established for a homogeneous steady state in the one-dimensional case. It is important topoint out that in innite dimensional dynamical systems such as the Vlasov-Poisson system, the relationship between the two concepts of stability is in general not clear. The present investigation is intended to ll part of the gap between what is done in more physically motivated papers and what is mathematically established. We proceed as follows: In the next section the basic assumptions on the steady states under consideration are collected. In Section 3 we prove a global existence and uniqueness result for the linearized Vlasov-Poisson system. Section 4 introduces a conserved quantity of the linearized system, the dynamical free energy, for solutions with initial conditions whose support lies inside the the compact support of the steady state f. This is preparatory for Section 5 where we derive our stability results intermsofaweighted L -norm induced by the free energy: For the plasma physics case we obtain linear stability if' < on the support of ', inthe stellar dynamics case we obtain linear stability if' > on its support, and 3

4 since the steady state has to have nite total mass this necessitates a jump discontinuity of'. In Section 6 we show that there exist steady states which satisfy our assumptions. This is necessary since in the plasma physics case the existence of steady states in the above situation has not yet been demonstrated we refer to [6, 4,5,33] for related results. In the stellar dynamics case the polytropes of the form '(E)=(E ;E) + with ; <<, which are investigated in [7] and [8], do not satisfy our assumptions, but their approximations ~'(E)='(E) ]; E ](E) withe <E do. Since the restriction on the support of the initial distribution may seem unphysical we show in an appendix that it is a natural assumption within the framework of a linearized theory. In the stellar dynamics case the result that f is stable if ' is increasing on its support and has a jump discontinuity at its boundary may also seem unphysical. However, our analysis then at least shows that the results of a linearized theory have tobetaken with care. This in turn should be a valuable insight for the plasma physics and stellar dynamics communities, since there stability questions are very often approached by linearization. Assumptions on the stationary solutions We consider stationary solutions (f U ) of the Vlasov-Poisson system such that f (x v)=' v +U (x) x vir 3 where ' satises the assumptions (') ' L loc (IR) ', (') E := inf fe IR j '(E ) = a.e. for E >Eg]; [, ('3) ' C (]; E [) with ' L loc (]; E ]), and U satises the assumptions (U) U C (IR 3 ), (U) U is bounded U min := inf xir 3 U (x) <E, (U3) the set := n o (x v) IR 6 j v +U (x) E measure zero. is bounded, has 4

5 Here ' L loc (IR) means that ' jk L (K) for every compact interval K IR, and L loc (]; E ]) is dened analogously. Conditions (') and (U) imply that the energy levels of the distribution function f vary between the values U min and E.Together with (U3) this means that f has phase-space support in the bounded set. In particular, the steady state has nite radius, i. e., there exists a radius R > such that f (x v)= for jxj >R, and by (') it has nite mass or charge: f (x v)dvdx vol() sup '(E) < : IR 6 Condition ('3) implies the estimates IR 3 j' (E(x v))jdv =4 and, with z := (x v) IR 6, E U (x) E[U min E ] q j' (E)j (E ;U (x))de E q 4 (E ;U min ) j' (E)jdE U min < x IR 3 E q j' (E(z))jdz< (4) (E ;U min ) j' (E)jdE IR 6 U min R r dr < which are needed in the proof of global existence for the linearized Vlasov- Poisson system in Section 3. We write (f U ) S if (f U ) is a stationary solution of the Vlasov- Poisson system satisfying the above assumptions. In Section 6 we show that stationary solutions of this type exist both in the plasma physics and in the stellar dynamics cases. Throughout the paper constants which depend only on the steady state under consideration such as the above integrals and which may change from line to line are all denoted by C. 3 Global existence for the linearized Vlasov- Poisson system Let (f U ) S and let t 7! f +f(t) be a solution of the Vlasov-Poisson system with initial condition f + f. If the term which is quadratic in f(t) in 5

6 the Vlasov equation is neglected, we arrive at the linearized Vlasov-Poisson system for the (small) perturbation t f x f ;@ x v f x U v f (3:) 4U f =4 f (3:) f (t x):= f(t x v)dv (3:3) together with the initial condition f() = f. Assuming that U f vanishes at innity, we obtain f (t y) U f (t x)=; jx;yj dy x IR3 : (3:4) Consider the system of characteristics corresponding to (3.) _x = v _v = ;@ x U (x): (3:5) Due to the regularity ofu there exists for every t IR and z =(x v) IR 6 a unique global solution ( t z)=(x V )( t x v) of (3.5) with (t t z)=z. The mapping is continuously dierentiable in all variables and (s t ): IR 6! IR 6 is a measure preserving dieomorphism for all s t IR. Using the ow, we can write (3.) in the form d ds f(s (s t z)) x U f (s)@ v f ((s t z)) s t IR z IR 6 which upon integration yields f(t z)= f(( t z))+ x U f (s)@ v f ((s t z))ds t z IR 6 : Since for steady states of class S wehave f (x v)=' v +U (x) and since the energy E(x v)= v +U (x) isinvariant under the characteristic ow, this relation becomes f(t z)= f(( t z))+' (E(z)) t This motivates the following x U f (s)v ((s t z))ds t z IR 6 : (3:6) Denition 3. Let f L (IR 6 ). A function f :[ [IR 6! IR is a solution of the linearized Vlasov-Poisson system with initial value f if and only if 6

7 (i) f C([ [ L (IR 6 )), (ii) f C([ [ L (IR 3 )), (iii) U f C([ [ C b (IR3 )), (iv) f satises (3:6) for t and (v) f() = f. Here f and U f are dened by(3:3) and (3:4), respectively, and C b (IR3 ) denotes the space of continuously dierentiable functions which are bounded together with their rst derivatives. Note that a solution in this sense satises the linearized Vlasov-Poisson system classically if f and U f are suciently regular. Theorem 3. Let (f U ) S and let f L (IR 6 ) beapointwise-dened, IRvalued function such that f ( ) satises conditions (i) and (ii) of Denition 3.. Then there exists a unique solution of the linearized Vlasov-Poisson system with initial value f. Proof : We construct a converging sequence of iterates in the set M := n g :[ [IR 6! IR j g satises (i), (ii), (v) of Denition 3. o : Obviously, f := f ( ) M. Let g M. Then (i) implies that g C([ [ L (IR 3 )) and the well-known estimates, (cf. [4]) yield (iii) for U g. Dene ku g (t)k () =3 k g (t)k =3 k g(t)k =3 (3:7) k@ x U g (t)k 3() =3 k g (t)k =3 k g (t)k =3 (Tg)(t z):=f (t z)+' (E(z)) x U g (s)v ((s t z))ds t 7

8 and zero otherwise. The estimate t ' x U g (s)v ((s t z))ds dz IR 6 j' (E(z))jdz supjvj C t z t k g (s)k =3 k g (s)k =3 ds k@ x U g (s)k ds shows that (Tg)(t) L (IR 6 ) recall that constants denoted by C may depend on the steady state under consideration. Let <t then k(tg)(t);(tg)()k kf (t);f ()k +C k@ x U g (s)k ds + j' dz x U g (s)v ((s t x U g (s)v ((s z)) ds! for! t since f satises condition (i) and (@ x U g (s)v)((s z)) is uniformly continuous on [ t]. The case >t is analogous, and thus Tg satises condition (i). Next observe that t j g (t x)j j f (t x)j+c j' (E(z))jdv k@ x U g (s)k ds j f (t x)j+c t t k@ x U g (s)k ds which implies that g (t) L (IR 3 ) for t. Furthermore, for <t, k g (t); g ()k k f (t); f ()k +C +C sup z! for! x U g (s)v t k@ x U g (s)k ds ds ((s t x U g (s)v ((s z)) where the rst term converges by the assumption on f, the second converges by (iii) and the last by the same argument asabove. Since the case >tis analogous, we have (ii), and (v) being obvious we have shown that TgM, i. e., T maps the set M into itself. Now letg g M then the above estimates show that k(tg )(t);(tg )(t)k C t k g (s); g (s)k =3 kg (s);g (s)k =3 ds 8

9 and k Tg (t); Tg (t)k C t k g (s); g (s)k =3 kg (s);g (s)k =3 ds: Hence, if we dene f n+ := Tf n n, it follows that there exist functions f C([ [ L (IR 6 )) and C([ [ L (IR 3 )) such that f n (t)! f(t) in L (IR 6 ), t-locally uniformly on [ [ and n (t)! (t) inl (IR 3 ), t-locally uniformly on [ [. Since fn (t)! f (t) inl (IR 3 )wehave = f and fn (t)! f (t) inl (IR 3 )\L (IR 3 ), t-locally uniformly on [ [. This implies that U fn (t)! U f (t) incb (IR3 ), t-locally uniformly on [ [. Passing to the limit in the relation f n+ (t z)=f (t z)+' (E(z)) we obtain t f(t z)= f(( t z))+' x U fn (s)v ((s t z))ds t x U f (s)v ((s t z))ds t after redening f on a set of measure zero. Since condition (v) is clear, f is a solution in the sense of Denition 3.. Uniqueness of the solution is obvious. Corollary 3.3 The solution f obtained intheorem 3. has the following properties for t : (a) f(t z)= f(( t z)) for z=, inparticular, if f vanishes outside, then so does f(t). (b) d dt f(t (t z)) = ' x U f (t)v ((t z)) (c) f(z)dz = f(t z)dz. (d) If f has compact support or vanishes suciently rapidly at innity then U f (t x)f(t z)dz = ; j@ x U f (t x)j dx : 4 9

10 Proof : (a) is obvious, (b) follows from replacing z in (3.6) by (t z)and dierentiating the resulting equation, (c) follows by integrating (3.6) with respect to z and using the fact that the ow preserves measure and that the x U f (s x)v is odd in v. Note that the set is invariant with respect to (x v) 7! (x ;v). (d) If f (t) is, in addition, Holder-continuous, then we have U f (t x)f(t z)dz = U f (t x) f (t x)dx = lim U f (t x) f (t x)dx = r! jxjr 4 lim U f (t x)4u f (t x)dx r! jxjr = " # 4 lim U f (t x)@ x U f (t x)n(x)d!(x); j@ x U f (t x)j dx r! jxj=r jxjr = ; j@ x U f (t x)j dx 4 if the decay ofu f (t x)@ x U f (t x) at spatial innity issuch that the boundary term vanishes if f has compact support then U f (t x)@ x U f (t x)=o(jxj ;3 ). In case f (t) is not Holder-continuous we can use a mollication of f (t) to get the result. 4 Conservation of free energy Theorem 4. Let (f U ) S and let f L (IR 6 ) be asintheorem 3. with f(z)= for z= and f (z) j' dz < : (4:) (E(z))j Then and f (t z) F(t):=; ' (E(z)) dz + f (t z) j' dz < t (E(z))j U f (t x)f(t z)dz = F() t : (4:) Here the quotient is to be understood as ( g (z) for ' ' (E(z)) = (E(z))= and g(z) 6= for ' (E(z)) arbitrary and g(z)=:

11 Proof : Assume that f (t z) j' (E(z))j dz < for some t, then with Corollary 3.3, f (t (t z)) f(t (t z))f(t (t z F(t) )) =; ' dz ; (E(z)) jx(t z);x(t z dz dz )j f (z) t d = ; ' (E(z)) dz ; ds f (s (s z)) f(z) f(z ) ' dsdz ; (E(z)) jx;x dz dz j t d f(s (s z))f(s (s z )) ; ds jx(s z);x(s z dsdz dz )j t = F(); f(s (s x U f (s)v ((s z))dsdz t X(s z);x(s z ) + jx(s z);x(s z )j (V (s z);v 3 (s z )) ; = F(); + t t t t f(s (s z))f(s (s z ))dsdz dz ' x U f (s)v ((s z)) f(s (s z )) jx(s z);x(s z )j dz dsdz f(s z)@ x U f (s x)vdzds x;x jx;x j 3 (v ;v )f(s z)f(s z )dz dz ds + ' (E(z))@ x U f (s x)vu f (s x)dz ds t = F(); f(s z)@ x U f (s x)vdzds t + f(s z)@ x U f (s x)vdzds t + ' (E(z))@ x U f (s x)vu f (s x)dz ds = F() the last integral vanishes because the integrand is odd in v and invariant under the mapping v 7!;v. Retracing all the steps of the argument, we observe that all the integrals exist by the boundedness of the x U f (s x)v

12 on and the integrability of' E, and Fubini's Theorem applies. Remark: The energy expression of (4.), restricted to monotonically decreasing (nonvanishing) stationary phase-space densities, was rst obtained in [6] in a plasma physics model more general than that of the Vlasov- Poisson system. Imposing condition (4.) allows one to consider stationary solutions of compact support by restricting the class of initial conditions. In the Appendix we comment on the restriction that f hastovanish outside the support of f. 5 Linear stability Theorem 5. Let (f U ) S, assume that ' (E) > for U min E<E, and dene the weighted L -norm g kgk ' := (z) ' (E(z)) dz : Then (f U ) is linearly stable in the following sense: For every f as in Theorem 4. with kfk =3 k k =3 the corresponding solution f of the linearized Vlasov-Poisson system satises the estimate f kf(t)k ' c kfk ' +kfk ' t where the constant c depends only on the stationary solution (f U ). Proof : kf(t)k ' = f (t z) ' (E(z)) dz j@ x U f (t x)j dx = ;F(t); 4 ;F() = kfk ' ; kfk ' + f (z) ' (E(z)) dz ; U f (x) f(z) dz f(z) q p ' ' (E(z))U (E(z)) = j' (E(z))jU (x)dz kfk ' f kfk ' +() =3 kfk =3 k k =3 f f (x)dz = j' (E(z))jdz kfk '

13 where the last estimate follows from (3.7). Thus, the proof is complete, with Remarks: c := () =3 j' (E(z))jdz = :. Using [38, Lemma ] to estimate the potential energy corresponding to f in the above proof we obtain the alternative stability estimate kf(t)k ' c kfk +k fk ' t for all initial data as in Theorem 4., where c again depends on the stationary solution (f U ).. If <c ; j' (E)jc + < on ]; E [ then the norm kk ' is equivalent to the usual L norm, and we obtain the stability estimate kf(t)k c 3 k fk t for all initial data as in Theorem The stellar dynamics case where ' (E) > is of particular interest. This result requires the jump discontinuity in' and the restricted class of initial conditions f described in Theorem 4.. It is natural to question the physical relevance of and the sensitivity to these assumptions. One would expect collisions, i. e., the eect of short-range interactions, to smooth out the jump discontinuity in' and produce a transition region where ' (E) > (and large). In this way collisions can provide a mechanism for the onset of instability. 6 Examples In this section we establish the existence of a large class of stationary solutions (f U ) S. Among these there are steady states satisfying the stability condition of Theorem 5., i. e., ' (E) < in the plasma physics case and ' (E) > in the stellar dynamics case. Any f of the form f (x v)='(e)=' v +U (x) (6:) 3

14 automatically satises Vlasov's equation, since the energy E is constant along characteristics. Therefore, the stationary Vlasov-Poisson system is reduced to the semilinear Poisson equation 4U (x)=4( + (x)+h ' (U (x)) x IR 3 where h ' (u):= ' v +u dv: Here weinvestigate spherically symmetric solutions of this problem, i. e., solutions of r U r (r) =4(+ (r)+h ' (U (r)) r> (6:) where h ' (u):= r ' w + F r +u df dw (6:3) r := jxj w:= xv jxj F:= jxvj = x v ;(xv). The distribution function f is then a function of r w F, and (r)=h ' (U (r)) is a function of r, i.e., the whole steady state is spherically symmetric. For the rest of this section let ' satisfy the conditions (') and (') from Section and assume in addition ('4) Case (S) (stellar dynamics case): '(E ):=lim E%E '(E) exists and '(E ) >, Case (P) (plasma physics case): + C([ [) +, r + L ([ [), and there exist constants r > and + > such that + (r) + r [ r ]. Theorem 6. Let the assumptions ('), ('), and('4) be satised. Then there exists a constant <E such that for ] E [ the problem (6:) has a unique solution U C ([ [) with U () =, where h ' is dened by (6:3). U is strictly increasing, U () =, E < lim r! U (r) <, andthere exists R > such that U (R )=E and U(R ) >. Consequently, if ' in addition satises ('3) and if f is dened by(6:) and := h ' U,then (f U ) S, C ([ [), and (r)= for r R, (r) > for r<r. Under the further assumption that ' (E) > for E<E, the steady state (f U ) is stable in the sense of Theorem 5.. For the proof of this result the following lemma is useful: 4

15 Lemma 6. Let ' satisfy the assumptions (') and ('). Then (6:3) de- nes a function h ' C (IR), h ' (u)= for u E,and h ' (u) =4 p u '(E) p E ;ude h '(u) =; p 4 de '(E) p uir: E ;u u The proof of this lemma is an easy application of Lebesgue's theorem on dominated convergence and therefore omitted. Proof of Theorem 6.: Local existence and uniqueness of the solution for arbitrary IR followby the contraction mapping principle, applied to the following reformulation of the problem: U (r)= 4 r r s + (s)+h ' + s U ()d ds: Let U C ([ R[) be the solution, extended to its maximal interval of existence [ R[, and (r):=h ' (U (r)). Then U C (] R[), U (r) has a limit for r!, U () =, and this implies the regularity assertions for U and. For the rest of the proof, we have to treat the two cases (P) and (S) separately. Case (P): Take <E such that (i) h ' (u) < + =foru ] E [, (ii) E ; < 3 + r, and let ] E [. Then by (i) U (r) >, and U is strictly increasing on [ r ]\[ R[ in fact, because h ' is decreasing, U (r) = 4 r > 4 r r r = 3 + r: s ( + (s);h ' (U (s))ds! s + ; + Since h ' (u)= for u E this implies that R>r, and by condition (ii), ds U (r ) >+ 3 + r >E : 5

16 Thus there exists R ] r [withu (R )=E and U (r) >E (<E ) for r> R (<R ) which implies the assertions on. Case (S): First of all we notethatforany <E the potential U is strictly increasing on [ R[. Either U (r) <E for r [ R[ in which case U is bounded and thus exists globally, oru (r) E for r R and some R in which case vanishes for r R, and U again exists globally. To prove that actually the latter case holds, we rely on the analysis in [5]. The existence of R follows if we show that the (possibly innite) limit L := lim r! U (r) >E : Assume L<E. Then the monotonicity of U and h ' implies that h ' (U (r)) h ' (L) for r, and thus, U (r) 4 r r s h ' (L)ds = 4 3 h '(L)r r : ut this means that U (r)! for r!,acontradiction. Thus it remains to show that the assumption L = E leads to a contradiction as well. To this end, dene y(r):=e ;U (r) r then (i) y(r) > andy (r) < for r, (ii) y lim y(r)= lim ry(r) >, and lim (r) <, r! r! r! r (iii) (r y (r)) = ;H(r y(r)) r>, where H(r y):=4r h ' (E ;y). Here the assertions in (i) follow from the strict monotonicity ofu, and the rst assertion in (ii) is our assumption L = E. The second assertion in (ii) follows by L'Hospital's rule: Finally, lim r(l;u L;U (r) (r)) = lim r! r! =r y (r) r = lim r! 4 = ; U (r) = ; 4 r r 3 r r U =lim (r) r! =r s (s)ds =4 s (s)ds > : s (s)ds! 4 3 () = ; 4 3 h '() < for r!. Condition (iii) is Equation (6.), rewritten for y. Obviously, H satises the relation 6

17 (iv) r H(r y)=h(r y) r y IR. The assumption ('4) in case (S) implies the existence of constants <E and <c <c such that c '(E) c for E [ E [ : Take ] E [ then <y(r) E ;<E ; or <E ;y(r) <E for r. y H(r y(r)) = ;4r h '(E ;y(r)) = p (4) r E E ;y(r) q (4) p r c y(r) =(4) p c r qy(r) de '(E) p E ;E +y(r) and H(r y(r)) (4) p c r E E ;y(r) q E ;E +y(r)de =(4) p 3 c r y(r) 3= : For suciently close to E we can assume that c < c 3 : Then there exists a constant m ] 5[ such that c =c m=3i.e.,c 3 c m, which in view of the above estimates for H(r y(r)) y H(r y(r)) implies that (v) y(r)@ y H(r y(r)) mh(r y(r)) r. Conditions (i) to (v) now lead to the desired contradiction in the following way cf. also [5]: Dene q(r):= ry m+ (r) y r (r) 7

18 and then Q(r):=r y(r) ;m y (r) q (r) =3r y(r)y (r)+ m+ r 3 y (r) +ry(r)h(r y(r)) r Q (r) = 5;m r y (r) +(r@ r H ;H)y(r)+(y@ y H ;mh)ry (r) 5;m r y (r) r by (iv) and (v), and Thus lim Q(r)=: r! r y(r) ;m y (r) q (r)=q(r)= r Q (s)ds > which implies that q (r) > forr>, and we have shown that q is strictly increasing. The third assertion in (ii) yields and we conclude that Therefore, which implies that lim q(r)=;a A > r! ;A<q(r)= ry m+ (r) y r>: (r) (;A)y(r) ; m; < (;A) y(r) ; m; ;y() ; m; =(;A) < ;m r r ; m; y(s) ; m+ y (s)ds sds= ;m r r> 4 ry(r) <Cr m;5 m; r> for some constant C>. ut this contradicts the second assertion in (ii). Thus, the only remaining possibility isl>e which implies that U (R )= E for some R > also in the stellar dynamics case. In both cases U is 8

19 strictly increasing and U (r) r; for r>r so that E < lim r! U (r) <. Furthermore, U (R ) > so z E(z)=( x r U (r) v) 6= for z =(x This shows is a C -submanifold of IR 6 and is of measure zero. Since U C ([ [) and U () =, U is C when interpreted as a function on IR 3, and the proof is complete. Appendix: Some comments on the class of admissible perturbations The purpose of this section is to show that the restriction that f vanish outside the support of f is a natural one within the present, linearized setting. A class of perturbations that one certainly wishes to include are perturbations caused by external forces. For example, one could think of f as representing some stationary galaxy which is perturbed by some other, distant galaxy. The particle orbits are then given by a perturbed characteristic system _x = v _v = ;@ x P where the (time-dependent) potential P of the perturbed force eld can be thought of as the sum of the self-consistent potential caused by the (perturbed) f, and the potential of the outside force. Let P =(X P V P )be the ow of the perturbed characteristic system. The perturbed distribution function is then given by f P ( ) and a natural class of perturbations would be the set M := n f P ( );f j P C(]; [ C b (IR 3 )) ]; [ > o : It is quite obvious that this set contains perturbations whose support is not contained in the support of f. However, in a linearized theory it is consistent to approximate the above perturbations to rst order in, that is to take as perturbations the tangent vectors to all curves 7! f P ( ) at = where P C(]; [ Cb (IR3 )). In more geometric language: We replace the \manifold" M by its \tangent space" at f.now d d j = f ( P ( z)) x f (z) d P ( z)+@ v f (z) d j =X d V P ( z) d j = 9

20 = x f (z)v +@ v f (z)@ x P ( x) =[g f ](z) where [ ] denotes the usual Poisson bracket, and note that P ( z) satises g(z):= v +P ( x) d d P ( x P ( x)) d + d V P ( xp ( X P ( z)) d d X P ( z) d: Therefore, the set [g f ] j g(z)= v +P (x) P Cb (IR 3 ) is natural as the set of admissible perturbations in a linearized setting, and obviously these functions have support in the support of f. If we take f C c (IR 6 ), then the curve 7! f P ( ) is dierentiable as a curve withvalues in any L p (IR 6 ), p. In general, we do not require this regularity off, and the corresponding curve need not be dierentiable. However, neither do we require the form [g f ]for the initial perturbation, but keep only the restriction on the support which makes sense without any dierentiability assumption on f. Perturbations of the form [g f ]were called dynamically accessible perturbations in [9, 3, 3]. (See also [3, 7, 3].) For such perturbations condition (4.) turns into the following condition on g: [g E] (z)j' (E(z))jdz < and the singularity in (4.) due to the vanishing of ' outside disappears. A second argument justifying our assumption on the support of the perturbation goes as follows. Let us for the moment return to the original nonlinear system and write the solution as F = f +f. Ifwe insert this into

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