Nonlinear Stability of Newtonian Galaxies and Stars from a Mathematical Perspective 5 0 Gerhard Rein 0 Department of Mathematics 2 University of Bayreuth n 95440 Bayreuth, Germany a J 5 2 Abstract 1 Thestabilityofequilibriumconfigurationsofgalaxiesorstarsaretime v honored problems in astrophysics. We present mathematical results on 1 theseproblemswhichhaveinrecentyearsbeenobtainedbyYanGuoand 4 the author in the context of the Vlasov-Poisson and the Euler-Poisson 5 model. Based on a careful analysis of the minimization properties of 1 conservedquantities—thetotalenergyandso-calledCasimirfunctionals— 0 non-linear stability results are obtained for a wide class of equilibria. 5 0 / h 1 Introduction p - o Under suitable idealizing assumptions the time evolution of a galaxy can be r modeled by the Vlasov-Poissonsystem t s a ∂ f+v·∇ f−∇U·∇ f=0, t x v : v i △U=4πρ, lim U(t,x)=0, X |x|→∞ r a ρ(t,x)= f(t,x,v)dv. Z Heref=f(t,x,v)≥0denotesthedensityofthestarsinphasespace,t∈R, x,v∈ R3 stand for time, position, and velocity, ρ=ρ(t,x) is the induced spatial mass density, and U=U(t,x) is the gravitationalpotential of the galaxy. The problem we wish to address is the non-linear stability of stationary solutions of this system. Our approachwill automatically addressthis question fortheEuler-Poissonsystemaswell,whichdescribesaself-gravitatingfluidball, i.e., a barotropic star. The latter model is presented in Section 4. Bydefinition,agivensteadystatef isstableifforanyneighborhoodN off 0 0 thereexistsanotherneighborhoodM off suchthatanysolutionofthesystem 0 starting in M will remain in N for all times. This is the usual mathematical definitionofLyapunovstability. Inthecaseofaninfinitedimensionaldynamical 1 system such as the Vlasov-Poisson system the choice of the proper concept of “neighborhood” is a non-trivial part of the stability problem. We emphasize that no approach of the solution to the steady state is asserted—that would be the concept of asymptotic stability. Since the system as stated does not include dissipative effects an approach to a particular steady state is not to be expected in a strict sense; we do in the present notes not enter into the highly interesting questions of course graining, phase mixing, etc. The existence of global-in-time solutions of the system under consideration at least for initial data close to the steady state is an integral part of the stability assertion. For the Vlasov-Poisson system it was shown in [1] that every smooth, compactly supportedinitialdatumforf launchesauniqueglobal-in-timesmoothsolution; a fairly short proof due to J. Schaeffer is given in [2]. There is a vast astrophysics literature on the stability question. However, essentiallyallinvestigationsthatweareawareofproceedvialinearization. This approach suffers from at least two major difficulties: Firstly, there is no gen- eral theory for infinite dimensional dynamical systems as to how to pass from linearizedtonon-linearstability. Secondly,itiswellknownthatifλisaneigen- value, i.e., the linearized system has a solution of the form eλtg(x,v), then the sameistruefor−λ. Hencetheoptimalcaseregardingstabilityoccursifallsuch eigenvalues are purely imaginary,which is precisely the case where stability for the non-linear system does not follow, not even in finitely many dimensions. Wewillprovenon-linearstabilityofcertainsteadystatesbyidentifyingthem asminimizersofaconservedquantityintermsofwhichtheaboveneighborhoods N andM arethen defined. More precisely,for a statef=f(x,v)≥0we denote the induced spatial mass density and potential by ρ (y) f ρ (x):= f(x,v)dv, U (x):=− dy, f f |x−y| Z Z and we introduce the functionals 1 E (f):= |v|2f(x,v)dvdx, (1.1) kin 2 ZZ 1 1 E (f):= U (x)ρ (x)dx=− |∇U (x)|2dx, (1.2) pot f f f 2 8π Z Z the kinetic and the potential energy of the state f. The total energy H:=E +E kin pot is conserved along solutions of the Vlasov-Poisson system, but it is indefinite, andithasnocriticalpoints,i.e.,thelinearpartinanexpansionaboutanystate f with potential U does not vanish: 0 0 1 1 H(f)=H(f )+ |v|2+U (f−f )dvdx− |∇U −∇U |2dx. 0 0 0 f 0 2 8π ZZ (cid:18) (cid:19) Z However, according to Liouville’s theorem the characteristic flow correspond- ing to the Vlasov equation preserves phase space volume, and hence for any 2 reasonable function Φ the so-called Casimir functional C(f):= Φ(f(x,v))dvdx ZZ is conserved as well. If we expand the energy-Casimir functional H :=H+C C about an isotropic steady state 1 f (x,v)=φ(E), E=E(x,v):= |v|2+U (x), 0 0 2 we find that H (f)= H (f )+ (E+Φ′(f ))(f−f )dvdx C C 0 0 0 ZZ 1 1 − |∇U −∇U |2dx+ Φ′′(f )(f−f )2dvdx+.... f 0 0 0 8π 2 Z ZZ Atleastformally,wecanchooseΦsuchthatf isacriticalpointofH ,namely 0 C Φ′=−φ−1,providedφis invertible. Theessentialproblemnowisthe following: Inorderforthesteadystatetohavefinitetotalmassthefunctionφmustvanish above a certain cut-off energy. For φ−1 to exist φ should thus be decreasing, at least on its support. But then Φ′′ is positive and the quadratic part in the expansionindefinite. Sinceonewouldliketousethisquadraticpartfordefining the concept of distance or neighborhood, the method seems to fail. This state of affairs had been observed by various authors, with the proposed conclusion that the energy-Casimir method does not work for the stellar dynamics case of the Vlasov-Poissonsystem, cf. for example [3]. If the issue is the stability of a plasma, the sign of the source term in the Poissonequation and hence also the oneinfrontofthepotentialenergydifferenceintheexpansionaboveisreversed, and up to some technicalities stability follows, cf. [4]. The approachdeveloped by Yan Guo and the author to overcome this diffi- culty is as follows. Starting with a given function Φ which defines the Casimir functionalwe tryto minimize the energy-CasimirfunctionalH under the con- C straintthat only states with a prescribed totalmass M>0 are considered. Un- der suitable assumptions on Φ a minimizer does exist in spite of the fact that thequadratictermintheexpansionaboveisindefinite. Onecanthenshowthat such a minimizer is a non-linearly stable steady state. The exact statements of theseresultswillbegiveninthe nextsection—themainassumptionisthatΦis strictly convex which is equivalent to f (x,v)=φ(E) being strictly decreasing. 0 The crucial step is to prove the existence of a minimizer. Here we first construct out of the energy-Casimir functional H a reduced functional H C r whichisdefinedonthespaceofspatialmassdensitiesρinsuchawaythatthere is a one-to-one correspondence between the minimizers of the two functionals. This reduced functional is analyzed in Section 3. The original motivation for introducing it was purely mathematical: It is defined on a simpler space, and 3 the troublesome part of the original functional is the quadratic and negative definite potential energy, but the latter depends on the spatial mass density ρ f and not directly on f. The detour via the reduced functional has a beautiful pay-off: The minimizers of the reduced functional are stable steady states of theEuler-Poissonsystemwithamacroscopicequationofstatecorrespondingto the microscopic equation of state f =φ(E) induced by the Casimir functional. 0 Hence via this reduction procedure we obtain a non-linear proof of what is often referred to as Antonov’s First Law [5, p. 305]: A spherical stellar system with f =f (E) and df /dE<0 is stable if the barotropic star with the same 0 0 0 equilibrium density distribution is stable. This relation to the Euler-Poisson system, i.e., to the stability of gaseous stars, is investigated in Section 4. In Section 5 we give the main arguments leading to the existence of a min- imizer for the reduced functional. Mathematically, this is the essential and non-trivial part; it can be skipped without compromising the understanding of the rest. To keep the presentation reasonably simple we restrict ourselves mostly to spherically symmetric, isotropic steady states. However, the method has also been applied to non-isotropic steady states, to axially symmetric ones, and to disk-likesteadystates. Somecommentsontheseextensionstogetherwithother related results as well as open questions are collected in the last section. To conclude this introduction we should mention that none of the results presented here are new, although the way they are presented is new. The motivation for these notes is to collect in one place the main features of our method, and to present them in such a way that the readers can hopefully appreciate the ideas involved. For some details which are not so relevant for the main argument we refer to existing papers, but mainly our presentation is aimedto be self-contained. We include almostno referencesto the astrophysics literature. Thisisreallynotdoneoutofdisrespectbutduetothebeliefthatour method is essentially the first to address the full non-linear stability problem forthesystemsunderconsideration. Shouldthesenotesinspiresomecomments or criticism from the astrophysics community we would truly appreciate this. Acknowledgments: These notes are an expanded version of my presentation attheworkshop“NonlinearDynamicsinAstronomyandPhysics”inNovember 2004attheUniversityofFlorida. Itrulyappreciatedthekindinvitationtothis inspiring event, as well as the feedback I received there. The results reported here originate from my collaboration with Y. Guo, Brown University, whom I would like to thank for many stimulating discussions. 2 Nonlinear stability for the Vlasov-Poisson system—statement of the results We fix a Casimir functional C, i.e., a function Φ such as Φ(f)=f1+1/k,f≥0, with 0<k<3/2, (2.1) more generally: Φ∈C1([0,∞[) with Φ(0)=0=Φ′(0), and 4 (Φ1) Φ is strictly convex, (Φ2) Φ(f)≥Cf1+1/k for f≥0 large, with 0<k<3/2, (Φ3) Φ(f)≤Cf1+1/k′ for f≥0 small, with 0<k′<3/2. For a given constantM>0 we wantto minimize the energy-Casimirfunctional H over the constraint set C F := f∈L1(R6)| fdvdx=M, E (f)+C(f)<∞ . M + kin (cid:26) ZZ (cid:27) Here L1(R6) denotes the set of non-negative, integrable functions on R6. One + can show that the potential energy is defined on this set and the two forms of E givenin(1.2)areequal,cf. [6, Lemma 1]. Due to conservationofmass the pot constraint set F is invariant under solutions of the Vlasov-Poissonsystem. M The mainstepinthe stability analysisis to establishthe following theorem: Theorem 1 The energy-Casimir functional H is bounded from below on F C M with h :=inf H <0. Let (f )⊂F be a minimizing sequence of H , i.e., M FM C j M C H (f )→h . Then there exists a function f ∈F , a subsequence, again de- C j M 0 M noted by (f ) and a sequence (a )⊂R3 of shift vectors such that for the induced j j gravitational fields Taj∇U =∇U (·+a )→∇U in L2(R3), j→∞. fj fj j f0 The state f minimizes the energy-Casimir functional: H (f )=h . 0 C 0 M We shall see shortly that to conclude stability of the state f the theorem 0 is needed in the above form; mere existence of a minimizer is not sufficient. A proof via a reduced functional is given below. The main difficulty is seen from thefollowingsketchoftheargument: Toobtainalowerboundforthefunctional ontheconstraintsetiseasy,andbyAssumption(Φ2)minimizingsequencescan be seen to be bounded in L1+1/k. A standard analysis result then implies that suchasequencehasaweaklyconvergentsubsequence,whichmeansthatforany test function g from the dual space L1+k the convergence f g→ f g holds, j 0 cf. [7, Sec. 2.18]. The weak limit f is the candidate for the minimizer, and one 0 R R has to pass the limit into the energy-Casimir functional. This is easy for the kinetic energy, the latter being linear, and for the Casimir functional which is convex due to Assumption (Φ1) it relies on Mazur’s lemma, cf. [7, Sec. 2.13]. The difficult part is the potential energy, for which one has to prove that the inducedgravitationalfields convergestronglyinL2. Thisproblemwillwedealt with on the level of the reduced functional in Section 5. Since our minimization problem is invariant under spatial translations we obtain a trivial minimizing sequence by shifting a given minimizer in space. If for example we shift it off to spatial infinity we cannot obtain a subsequence whichtendsweaklytoaminimizer,unlesswemovewiththesequence. Hencethe spatial shifts in the theorem arise from the physical properties of our problem. 5 The Euler-Lagrange identity corresponding to our constrained variational problem implies that any minimizer is a steady state of the Vlasov-Poisson system. For the proof we refer to [8] or [9]: Theorem 2 Let f ∈F be a minimizer with potential U . Then 0 M 0 (Φ′)−1(E −E),E<E , 1 f (x,v)= 0 0 where E= |v|2+U (x) 0 0 ,else 2 0 (cid:26) with Lagrange multiplier E . In particular, f is a steady state of the Vlasov- 0 0 Poisson system. For example, the choice (2.1) yields the polytropic steady state k (E −E)k ,E<E , f (x,v)= 0 0 0 k+1 0 ,else. (cid:26) It should be noted that the assumptions on Φ easily translateinto assumptions on the steady state f as a function of the particle energy, the main one being 0 thatthis functionis strictlydecreasingonits support. Variousadditionalprop- erties can be derived for these minimizers/steady states, in particular they are necessarilysphericallysymmetric,cf.[6,Thm.3]or[9,Thm.2]. Non-symmetric steady states will be briefly considered in the last section. To deduce our stability result we expand H about the minimizer f : C 0 1 H (f)−H (f )=d(f,f )− |∇U −∇U |2dx (2.2) C C 0 0 f 0 8π Z where for f∈F , M d(f,f ):= [Φ(f)−Φ(f )+E(f−f )]dvdx 0 0 0 ZZ ≥ [Φ′(f )+(E−E )](f−f )dvdx≥0 0 0 0 ZZ with d(f,f )=0iff f=f . This is due to the strict convexityof Φ, and the fact 0 0 that on the support of f the bracket vanishes by Theorem 2; note also that 0 (f−f )=0forf∈F . ForsuitableΦ’s,weevenhaved(f,f )≥c |f−f |2. 0 M 0 0 ThepointnowisthataccordingtoTheorem1thetermwiththenegativesignin RR RR (2.2)tendstozeroalonganyminimizingsequence. Thisallowsustousethesum ofthetwopositivedefinite termsintheexpansionasourmeasureofdistancein thestabilityresult. Asmentionedintheintroduction,initialdatafromthespace C1(R6) of continuously differentiable, compactly supported functions launch c smoothglobal-in-timesolutionsoftheVlasov-Poissonsystemwhichpreserveall the physically conserved quantities. As above, Taf(x,v):=f(x+a,v). Theorem 3 Foranyǫ>0thereexistsaδ>0suchthatforanysolutiont7→f(t) of the Vlasov-Poisson system with f(0)∈C1(R6)∩F the initial estimate c M 1 d(f(0),f )+ |∇U −∇U |2dx<δ 0 8π f(0) 0 Z 6 implies that for any t≥0 there is a shift vector a∈R3 such that 1 d(Taf(t),f )+ |Ta∇U −∇U |2dx<ǫ, t≥0, 0 8π f(t) 0 Z (provided the minimizer f is unique up to shifts). 0 We will comment on the uniqueness assumption (and why it comes in paren- thesis) shortly, but first we give the proof of this result, which is surprisingly simple—the difficulty resides in the proof of Theorem 1. Proof: Assume the assertion is false. Then there exist ǫ>0, t >0, f (0)∈ j j C1(R6)∩F such that for j∈N, c M 1 1 d(f (0),f )+ |∇U −∇U |2dx< , j 0 8π fj(0) 0 j Z but for any shift vector a∈R3, 1 d(Taf (t ),f )+ |Ta∇U −∇U |2dx≥ǫ. j j 0 8π fj(tj) 0 Z SinceH isconserved,(2.2)andtheassumptionontheinitialdataimpliesthat C H (f (t ))=H (f (0))→H (f ), i.e., (f (t ))⊂F is a minimizing sequence. C j j C j C 0 j j M HencebyTheorem1, |∇U −∇U |2→0uptosubsequencesandshiftsinx, fj(tj) 0 provided that there is no other minimizer to which this sequence can converge. By (2.2), d(f (t ),f )R→0 as well, which is the desired contradiction. 2 j j 0 Some comments are in order: For the polytropic steady states one canshow thatforagivenmassM theminimizerisindeeduniqueuptoshifts,asassumed above, cf. [6, Thm. 3]. In general, minimizers do not seem to be unique; for a numerically verified example of non-uniqueness we refer to [9, Rem. 3 (b)]. However, minimizers always seem to be isolated up to shifts which is sufficient for the above statement to still hold true, cf. [9, Thm. 3]. If there were a continuumofminimizersthenthissetofminimizersasawholewouldbestable, cf. [10, p. 242]. Finally, for a closely related approach to which we will come backinthe lastsectionitisshownin[11]thatthe assertionofTheorem3holds without f being unique or isolated. We kept the former assumption to make 0 the proof simple. The spatial shifts appearing in the stability statement are again due to the spatial invariance of the system: If we perturb f by giving all the particles an 0 additional, fixed velocity, then in space the corresponding solution travels off from f at a linear rate in t, no matter how small the perturbation was. 0 A nice feature of the result is that the same quantity is used to measure the deviation initially and at later times t. In infinite dimensional dynamical systems control in a strong norm initially can be necessary to gain control in a weaker norm at later times. Whether our concept of distance is appropriate from a physics point of view is open to debate—it is simply what comes out of the energy-Casimir method. For the polytropic steady states our approach has been extended to yield stability with respect to the L1-norm of f, cf. [12]. Definitely a weak point is the fact that the proof is not constructive: Given an ǫ we do not know how small the corresponding δ must be. 7 3 The reduced variational problem We wish to factor out the dependence on the velocity variablein our minimiza- tion problem. Starting from a given function f=f(x,v) with induced spatial density ρ =ρ (x) we clearly decrease H (f) by minimizing for each point x f f C overallfunctionsg=g(v)whichhaveasintegralthevalueρ (x). Thisprocedure f does not affect the potential energy and reduces the sum of the kinetic energy andthe Casimir functional into a new functional whichno longerdepends onf directly but only on ρ . More precisely, with f 1 Ψ(s):=inf |v|2g+Φ(g) dv |g∈L1(R3), g(v)dv=s (3.1) 2 + (cid:26)Z (cid:18) (cid:19) Z (cid:27) we have the estimate H (f)≥H (ρ ) where C r f H (ρ):= Ψ(ρ(x))dx+E (ρ). r pot Z We now wish to minimize H over the constraint set r R := ρ∈L1(R3)| Ψ(ρ)<∞, ρ=M . M + (cid:26) Z Z (cid:27) These constructions owe much to [13]. Before we analyze the reduced problem we make sure that we can lift any information gained for the latter back to the level of the original problem: Theorem 4 For all f∈F the estimate H (f)≥H (ρ ) holds, with equality M C r f if f=f is a minimizer. Let ρ ∈R minimize H and U =U . Then 0 0 M r 0 ρ0 (Ψ′)−1(E −U ) ,U <E , ρ = 0 0 0 0 (3.2) 0 0 ,U ≥E 0 0 (cid:26) with Lagrange multiplier E , and 0 (Φ′)−1(E −E),E<E , 1 f := 0 0 E=E(x,v):= |v|2+U (x), 0 0 ,E≥E0. 2 0 (cid:26) lies in F and minimizes H . If on the other hand f ∈F minimizes H M C 0 M C then ρ ∈R minimizes H . f0 M r Equation(3.2)isnothingbuttheEuler-Lagrangeidentity forthereducedprob- lem, cf. [9]; the theorem is proven in detail in [14, Sec. 2]. Let us now consider the reduced variational problem in its own right. The function Ψ defining the reduced functional is taken from the following class: Ψ∈C1([0,∞[), Ψ(0)=0=Ψ′(0), and (Ψ1) Ψ is strictly convex. (Ψ2) Ψ(ρ)≥Cρ1+1/n for ρ≥0 large, with 0<n<3, (Ψ3) Ψ(ρ)≤Cρ1+1/n′ for ρ≥0 small, with 0<n′<3. In Section 5 we shall prove the following central result: 8 Theorem 5 The functional H is bounded from below on R . Let (ρ )⊂R r M j M be a minimizing sequence of H . Then there exists a sequence of shift vectors r (a )⊂R3 and a subsequence, again denoted by (ρ ), such that j j Tajρ ⇀ρ weakly in L1+1/n(R3), j→∞, j 0 Taj∇U →∇U strongly in L2(R3), j→∞, ρj 0 and ρ ∈R is a minimizer of H . 0 M r We need to translate the conditions on Ψ back into conditions on Φ. To do so we denote the Legendre transform of a function h:R→]−∞,∞] by h(λ):=supr∈R(λr−h(r)). If Ψ arises from Φ by reduction, i.e., by formula (3.1) then 1 Ψ(λ)= Φ λ− |v|2 dv. 2 Z (cid:18) (cid:19) This more explicit relation between Φ and Ψ is established in [14, Sec. 2], and it allows us to show that the assumptions on Φ imply the ones on Ψ if the parameters k and n are related by n=k+3/2, with the same relation holding for the primed parameters. Theorem 4 connects our two variational problems in the appropriate way to derive Theorem 1 from Theorem 5: Firstly, H is bounded from below on C F since this is true for H on R . Let (f )⊂F be a minimizing sequence M r M j M for H . By Theorem 4, (ρ )⊂R is a minimizing sequence for H . Again C fj M r by Theorem 4 we can lift the minimizer ρ of H obtained in Theorem 5 to a 0 r minimizer f of H , and the proof of Theorem 1 is complete. 0 C Before we consider some of the ideas involved in the proof of Theorem 5 we reinterpret it in terms of the Euler-Poissonsystem. 4 Pay-off of reduction—The Euler-Poisson sys- tem If ρ ∈R minimizes the reduced functional H , then ρ supplemented with 0 M r 0 the velocity field u =0 is a steady state of the Euler-Poissonsystem 0 ∂ ρ+∇·(ρu)=0, t ρ∂ u+ρ(u·∇)u=−∇p−ρ∇U, t △U=4πρ, lim U(t,x)=0, |x|→∞ with equation of state p=P(ρ):=ρΨ′(ρ)−Ψ(ρ). 9 This follows from the Euler-Lagrange identity (3.2). Here u and p denote the velocity field and the pressure of an ideal, compressible fluid with mass density ρ, and the fluid self-interacts via its induced gravitational potential U. This systemis sometimes used asa simple modelfor a gaseous,barotropicstar. The beautiful thing now is that obviously (ρ ,u =0) minimizes the energy 0 0 1 H(ρ,u):= |u|2ρdx+ Ψ(ρ)dx+E (ρ) pot 2 Z Z of the system, which is a conservedquantity. Expanding as before we find that 1 1 H(ρ,u)−H(ρ ,0)= |u|2ρdx+d(ρ,ρ )− |∇U −∇U |2dx 0 0 ρ 0 2 8π Z Z where for ρ∈R , M d(ρ,ρ ):= [Ψ(ρ)−Ψ(ρ )+(U −E )(ρ−ρ )]dx≥0, 0 0 0 0 0 Z withequalityiffρ=ρ . ThesameproofasfortheVlasov-Poissonsystemimplies 0 a stability result for the Euler-Poisson system—the term with the unfavorable sign in the expansion again tends to zero along minimizing sequences, cf. The- orem 5. However, there is an important caveat: While for the Vlasov-Poisson system we have global-in-time solutions for sufficiently nice data, and these so- lutions really preserve all the conserved quantities, no such result is available for the Euler-Poissonsystem, and we only obtain a Conditionalstabilityresult: Foreveryǫ>0thereexistsaδ>0suchthat for every solution t7→(ρ(t),u(t)) with ρ(0)∈R which preserves energy and mass M the initial estimate 1 1 |u(0)|2ρ(0)dx+d(ρ(0),ρ )+ |∇U −∇U |2dx<δ 2 0 8π ρ(0) 0 Z Z implies that as long as the solution exists, 1 1 |u(t)|2ρ(t)dx+d(ρ(t),ρ )+ |∇U −∇U |2dx<ǫ 2 0 8π ρ(t) 0 Z Z up to shifts in x (provided the minimizer is unique up to such shifts). The same comments as on Theorem 3 apply also in this case. Because of the above caveat we prefer not to call this a theorem, although as far as the stability analysis itself is concerned it is perfectly rigorous. The open problem is whether a suitable concept of solution to the initial value problem exists. Nowthatminimizersofthereducedfunctionalareidentifiedassteadystates of the Euler-Poisson system it is instructive to reconsider the reduction proce- dureleading fromthe kinetic tothe fluiddynamics picture. Firstwe recallthat for the Legendre transform h′(ξ)=η⇐⇒h(ξ)+h(η)=ξη⇐⇒ h ′(η)=ξ. (cid:0) (cid:1) 10