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Relative equilibria in continuous stellar dynamics Juan Campos, Manuel del Pino, and Jean Dolbeault 0 M.delPino:DepartamentodeIngenier´ıaMatema´ticaandCMM,UniversidaddeChile,Casilla 170Correo3,Santiago, Chile.E-mail:[email protected] 1 J. Campos, J. Dolbeault: Ceremade (UMR CNRS n◦ 7534), Université Paris-Dauphine,Place 0 de Lattre de Tassigny, 75775 Paris Cédex 16, France. E-mail: [email protected], 2 [email protected] n a January27,2010 J 7 Abstract: We study a three dimensionalcontinuous model of gravitatingmat- 2 ter rotating at constant angular velocity. In the rotating reference frame, by ] a finite dimensional reduction, we prove the existence of non radial stationary P solutions whose supports are made of an arbitrarily large number of disjoint A compact sets, in the low angular velocity and large scale limit. At first order, . the solutions behave like point particles, thus making the link with the relative h equilibria in N-body dynamics. t a m Keywords. Stellardynamics–gravitation–mass–rotation–angularvelocity [ – solutions with compact support – radial solutions – symmetry breaking – orbital stability – elliptic equations – variational methods – critical points – 1 finite dimensional reduction – N-body problems – relative equilibria v 6 Mathematical subject classification (2000). Primary: 35J20; Secondary: 6 35B20, 35B38,35J60, 35J70,35Q35,76P05, 76S05, 82B40 8 4 . 1 0 1. Introduction and statement of the main results 0 1 We consider the Vlasov-Poissonsystem : v ∂ f +v f φ f =0 i t x x v X ·∇ −∇ ·∇ (1) r  1 a φ=−4π ∗ρ, ρ:= R3f dv |·| Z which models the dynamics of a cloud of particles moving under the action of a mean field gravitational potential φ solving the Poisson equation: ∆φ=ρ. Kinetic models like system (1) are typically used to describe gaseous stars or globular clusters. Here f = f(t,x,v) is the so-called distribution function, a 2 JuanCampos,ManueldelPino,andJeanDolbeault nonnegative function in L (R,L1(R3 R3)) depending on time t R, position ∞ x R3 and velocity v R3, which rep×resents a density of particles∈in the phase sp∈ace, R3 R3. The fu∈nction ρ is the spatial density function and depends only × on t and x. The total mass is conserved and hence f(t,x,v)dxdv = ρ(t,x)dx=M R3 R3 R3 ZZ × Z does not depend on t. Thefirstequationin(1)istheVlasovequation,alsoknownasthecollisionless Boltzmann equation in the astrophysical literature; see [5]. It is obtained by writing that the mass is transported by the flow of Newton’s equations, when the gravitational field is computed as a mean field potential. Reciprocally, the dynamics of discrete particle systems can be formally recovered by considering empiricaldistributions,namelymeasurevaluedsolutionsmadeofasumofDirac masses, and neglecting the self-consistent gravitational terms associated to the interaction of each Dirac mass with itself. It is also possible to relate (1) with discrete systems as follows. Consider the case of N gaseous spheres, far away one to each other, in such a way that theyweaklyinteractthroughgravitation.Intermsofsystem(1),suchasolution should be represented by a distribution function f, whose space density ρ is compactly supported, with several nearly spherical components. At large scale, the location of these spheres is governed at leading order by the N-body gravitational problem. The purposeofthis paperis tounveilthis linkbyconstructingaspecialclass ofsolutions:wewillbuildtime-periodic,nonradiallysymmetricsolutions,which generalize to kinetic equations the notion of relative equilibria for the discrete N-body problem. Such solutions have a planar solid motion of rotation around anaxiswhichcontainsthecenterofgravityofthesystem,sothatthecentrifugal forcecounter-balancestheattractionduetogravitation.Letusgivesomedetails. Consider N point particles with massesm , locatedatpoints x (t) R3 and j j ∈ assume that their dynamics is governedby Newton’s gravitationalequations d2x N m m x x j j k k j m = − , j =1,...N . (2) j dt2 4π x x 3 k j j=k=1 | − | 6X Let us write x R3 as x = (x,x3) R2 R C R where, using complex ′ notations,x =∈(x1,x2) x1+ix2and∈rewr×itesy≈stem×(2)incoordinatesrelative ′ to a reference frame rot≈ating at a constant velocity ω > 0 around the x3-axis. This amounts to carry out the change of variables x=(eiωtz′, z3), z′ =z1+iz2. In terms of the coordinates (z ,z3), system (2) then reads ′ dd2tz2j = N m4πk zzk−zzj3 +ω2(zj′,0)+ 2ω iddztj′,0 , j =1,...N . (3) k j j6=Xk=1 | − | (cid:16) (cid:17) Multiplecomponents configurations incontinuous stellardynamics 3 We consider solutions which are stationary in the rotating frame, namely constant solutions (z ,...z ) of system (3). Clearly all z ’s have their third com- 1 N j ponent with the same value, which we assume zero. Hence, we have that z =(ξ ,0), ξ C, k k k ∈ where the ξ ’s are constants and satisfy the system of equations k N m ξ ξ k k− j + ω2ξ , j =1,...N . (4) 4π ξ ξ 3 j k j k=j=1 | − | 6X In the original reference frame, the solution of (2) obeys to a rigid motion of rotation around the center of mass, with constant angular velocity ω. This solution is known as a relative equilibrium, thus taking the form xω(t)=(eiωtξ ,0), ξ C, j =1,...N . j j j ∈ System (4) has a variational formulation. In fact a vector (ξ ,...ξ ) solves (4) 1 N if and only if it is a critical point of the function 1 N m m ω2 N ω(ξ ,...ξ ) := j k + m ξ 2. Vm 1 N 8π ξ ξ 2 j| j| k j j=k=1 | − | j=1 6X X Here m denotes (m )N . A further simplification is achievedby considering the j j=1 scaling ξ =ω 2/3ζ , ω(ξ ,...ξ )=ω2/3 (ζ ,...ζ ) (5) j − j Vm 1 N Vm 1 N where N N 1 m m 1 (ζ ,...ζ ) := j k + m ζ 2. m 1 N j j V 8π ζ ζ 2 | | k j j=k=1| − | j=1 6X X Thisfunctionhasingeneralmanycriticalpoints,whichareallrelativeequilibria. For instance, clearly has a global minimum point. m V Our aim is to construct solutions of gravitational models in continuum mechanics based on the theory of relative equilibria. We have the following result. Theorem 1. Given masses m , j = 1,...N, and any sufficiently small ω > 0, j thereexistsasolution f (t,x,v) of equation (1)which is 2π-periodic in timeand ω ω whose spatial density takes the form N ρ(t,x):= f dv = ρ (x xω(t)) + o(1). R3 ω j − j Z i=1 X Here o(1) means that the remainder term uniformly converges to 0 as ω 0 + andidenticallyvanishesawayfrom N B (xω(t)),forsomeR>0,indepen→dent ∪j=1 R j of ω. The functions ρ (y) are non-negative, radially symmetric, non-increasing, j compactly supported functions, independent of ω, with ρ (y) dy = m and R3 j j the points xω(t) are such that j R xω(t)=ω 2/3(eiωtζω,0), ζω C, j =1,...N j − j j ∈ 4 JuanCampos,ManueldelPino,andJeanDolbeault and lim (ζω,...ζω)=min , lim (ζω,...ζω)=0. ω→0+Vm 1 N CN Vm ω→0+∇Vm 1 N The solution of Theorem 1 has a spatial density which is nearly spherically symmetric on each component of its support and these ball-like components rotate at constant, very small, angular velocity aroundthe x3-axis. The radii of theseballsareverysmallcomparedwiththeirdistancetotheaxis.Weshallcall such a solution a relative equilibrium of (1), by extensionof the discrete notion. The construction provides much more accurate informations on the solution. In particular, the building blocks ρ are obtained as minimizers of an explicit j reduced free energy functional, under suitable mass constraints. It is alsonaturalto considerotherdiscrete relativeequilibria,namely critical points of the energy that may or may not be globally minimizing, and ask m V whether associated relative equilibria of system (1) exist. There are plenty of relative equilibria of the N-body problem. For instance, if all masses m are j equal to some m > 0, a critical point is found by locating the ζ ’s at the j vertices of a regul∗ar polygon: ζ =re2iπ(j 1)/N , j =1,...N , (6) j − where r is such that N 1 d aN m∗ + 1r2 =0 with aN := 1 − 1 , dr 4π r 2 √2 1 cos(2πj/N) h i Xj=1 − p i.e. r = (a m /(4π))1/3. This configuration is called the Lagrange solution, N see [35]. The cou∗nterpart in terms of continuum mechanics goes as follows. Theorem 2. Let (ζ ,...ζ ) be a regular polygon, namely with ζ given by (6), 1 N j and assume that all masses are equal. Then there exists a solution f exactly as ω in Theorem 1, but with lim (ζω,...ζω)=(ζ ,...ζ ). ω→0+ 1 N 1 N Further examples of relative equilibria in the N-body problem can be obtained for instance by setting N 1 points particles of the same mass at the vertices − of a regular polygoncenteredat the origin,then adding one more point particle at the center (not necessarily with the same mass), and finally adjusting the radius. Another family of solutions, known as the Euler–Moulton solutions is constituted by arrays of aligned points. Critical points of the functional are always degenerate because of their m invariance under rotations: for any αV R we have ∈ (ζ ,...ζ )= (eiαζ ,...eiαζ ). m 1 N m 1 N V V Letζ¯=(ζ¯ ,...ζ¯ )beacriticalpointof withζ¯ =0.Afterauniquelydefined 1 N m ℓ rotation, we may assume that ζ¯ =0. MVoreover,w6e have a critical point of the ℓ2 function of 2N 1 real variables − ˜ (ζ ,...ζ ,...ζ ) := (ζ ,...(ζ ,0),...ζ ). m 1 ℓ1 N m 1 ℓ1 N V V Multiplecomponents configurations incontinuous stellardynamics 5 We shall say that a critical point of is non-degenerate up to rotations if the m matrixD2˜ (ζ¯ ,...ζ¯ ,...ζ¯ )isnonV-singular.Thispropertyisclearlyindepen- m 1 ℓ1 N V dent of the choice of ℓ. Palmore in [30,31,32,33,34] has obtained classification results for the relative equilibria.In particular,it turns out that for almost every choice of masses m , all critical points of the functional are non-degenerate up to rotations. j m V Moreover, in such a case there exist at least [2N 1(N 2)+1](N 2)! such − − − distinct critical points. Many other results on relative equilibria are available in the literature.We have collected some of them in Appendix A with a list of rel- evant references.These results have a counterpartin terms of relative equilibria of system (1). Theorem 3. Let (ζ ,...ζ ) be a non-degenerate critical point of up to ro- 1 N m V tations. Then there exists a solution f as in Theorem 1, which satisfies, as in ω Theorem 2, lim (ζω,...ζω)=(ζ ,...ζ ). ω→0+ 1 N 1 N This paper is organized as follows. In the next section, we explain how the search for relative equilibria for the Vlasov-Poisson system can be reduced to the study of critical points of a functional acting on the gravitationalpotential. The construction of these critical points is detailed in Section 3. Sections 4 and 5 are respectively devoted to the linearization of the problem around a superposition of solutions of the problem with zero angular velocity, and to the existence of a solution of a nonlinear problem with appropriate orthogonality constraints depending on parameters (ξ )N related to the location of the N j j=1 components of the support of the spatial density. Solving the original problem amounts to make all corresponding Lagrange multipliers equal to zero, which is equivalent to find a critical point of a function depending on (ξ )N : this is the j j=1 variational reduction described in Section 6. The proof of Theorems 1, 2 and 3 is given in Section 7 while known results on relative equilibria for the N-body, discrete problem are summarized in Appendix A. 2. The setup Guided by the representation (3) of the N-body problem in a rotating frame, we change variables in equation (1), replacing x = (x,x3) and v = (v ,v3) ′ ′ respectively by (eiωtx,x3) and (iωx +eiωtv ,v3). ′ ′ ′ Written in these new coordinates, Problem (1) becomes ∂∂ft +v·∇xf −∇xU ·∇vf −ω2x′·∇v′f +2ωiv′·∇v′f =0,  (7) 1 U =−4π ∗ρ, ρ:= R3f dv . |·| Z ThelasttwotermsintheequationtakeintoaccountthecentrifugalandCoriolis forceeffects.System(7)canberegardedasthecontinuousversionofproblem(3). Accordingly, a relative equilibrium of System (1) will simply correspond to a stationary state of (7). 6 JuanCampos,ManueldelPino,andJeanDolbeault Such stationary solutions of (7) can be found by considering for instance critical points of the free energy functional 1 1 [f]:= β(f)dxdv+ v 2 ω2 x 2 f dxdv U 2 dx ′ F R3 R3 2 R3 R3 | | − | | − 2 R3|∇ | ZZ × ZZ × (cid:0) (cid:1) Z for some arbitrary convex function β, under the mass constraint f dxdv =M . R3 R3 ZZ × A typical example of such a function is 1 β(f)= κq 1fq (8) q q− for some q (1, ) and some positive constant κ , to be fixed later. The corre- q ∈ ∞ sponding solution is knownas the solution of the polytropic gas model, see [3,4, 5,40,44]. When dealing with stationary solutions, it is not very difficult to rewrite the problemintermsofthepotential.Acriticalpointof underthemassconstraint F f dxdv =M is indeed given in terms of U by R3 R3 × RR f(x,v)=γ λ+ 1 v 2+U(x) 1ω2 x 2 (9) 2| | − 2 | ′| where γ is, up to a sign, an a(cid:0)ppropriate generalized inverse(cid:1)of β . In case (8), ′ γ(s) = κ−q1(−s)1+/(q−1), where s+ = (s+|s|)/2 denotes the positive part of s. The parameterλstands for the Lagrangemultiplier associatedto the mass constraint,atleastiff hasasingleconnectedcomponent.Atthispoint,oneshould mention that the analysisis notexactly as simple as writtenabove.Identity (9) indeedholdsonlycomponentbycomponentofthesupportofthesolution,ifthis support has more than one connected component, and the Lagrangemultipliers have to be defined for each component. The fact that M U(x) x∼ −4π x | |→∞ | | is dominatedby 1ω2 x 2 as x is alsoa seriouscauseoftrouble,which −2 | ′| | ′|→∞ clearly discards the possibility that the free energy functional can be bounded frombelowifω =0.Thisissuehasbeenstudiedin[9],inthecaseoftheso-called 6 flat systems. Finding a stationary solution in the rotating frame amounts to solving a non-linear Poisson equation, namely ∆U =g λ+U(x) 1ω2 x 2 ifx supp(ρ) (10) − 2 | ′| ∈ and ∆U =0 otherwise, w(cid:0)here g is defined by (cid:1) g(µ):= γ µ+ 1 v 2 dv . R3 2| | Z (cid:0) (cid:1) Multiplecomponents configurations incontinuous stellardynamics 7 Hence, the problem can also be reduced to look for a critical point of the functional 1 [U]:= U 2 dx+ G λ+U(x) 1ω2 x 2 dx λρdx, J 2 R3|∇ | R3 − 2 | ′| − R3 Z Z Z (cid:0) (cid:1) where λ = λ[x,U] is now a functional which is constant with respect to x, with value λ , on each connected component K of the support of ρ(x) = i i g λ[x,U]+U(x) 1ω2 x 2 , and implicitly determined by the condition − 2 | ′| (cid:0) (cid:1) g λi+U(x)− 21ω2|x′|2 dx=mi . ZKi (cid:0) (cid:1) By G,wedenote a primitiveofg andthe totalmassis M = N m .Hence we i=1 i can rewrite as J P N 1 [U]= U 2 dx+ G λ +U(x) 1ω2 x 2 dx m λ . J 2ZR3|∇ | i=1(cid:20)ZKi i − 2 | ′| − i i(cid:21) X (cid:0) (cid:1) We may also observethat criticalpoints of correspondto criticalpoints of F the reduced free energy functional 1 1 [ρ]:= h(ρ) ω2 x′ 2ρ dx U 2 dx G R3 − 2 | | − 2 R3|∇ | Z (cid:16) (cid:17) Z acting on the spatial densities if h(ρ) = ρg 1( s)ds. Also notice that, using 0 − − the same function γ as in (9), to each distribution function f, we can associate R a local equilibrium, or local Gibbs state, G (x,v)=γ µ(ρ(x))+ 1 v 2 f 2| | where µ is such that g(µ) = ρ. This(cid:0)identity defines(cid:1)µ = µ(ρ) = g 1(ρ) as a − function of ρ. Furthermore, by convexity, it follows that [f] [G ]= [ρ] if f F ≥F G ρ(x)= f(x,v)dv, with equality if f is a local Gibbs state. See [11] for more R3 details. R Summarizing, the heuristics are now as follows. The various components K i of the support of the spatial density ρ of a critical point are assumed to be far away from each other so that the dynamics of their center of mass is described bytheN-bodypointparticlessystem,atfirstorder.OneachcomponentK ,the i solutionisaperturbationofanisolatedminimizerofthefreeenergyfunctional F (withoutangularrotation)under the constraintthatthe massis equaltom .In i the spatial density picture, on K , the solution is a perturbation of a minimizer i of the reduced free energy functional . G Tofurthersimplifythepresentationofourresults,weshallfocusonthemodel of polytropic gases corresponding to (8). In such a case, with p:= 1 + 3, g is q 1 2 given by − g(µ)=( µ)p − + if the constant κq is fixed so that κq =4π√2R0+∞√t(1+t)q−11 dt=(2π)23Γ(Γ32(+q−qq1−q)1). 8 JuanCampos,ManueldelPino,andJeanDolbeault Free energy functionals have been very much studied over the last years, not only to characterizespecial stationary states, but also because they provide a framework to deal with orbital stability, which is a fundamental issue in the mechanicsofgravitation.Theuseofafreeenergyfunctional,whoseentropypart, β(f)dxdv is sometimes also called the Casimir energy functional, goes R3 R3 back×to the work of V.I. Arnold (see [1,2,45]). The variational characterization RR of special stationary solutions and their orbital stability have been studied by Y.GuoandG.Reininaseriesofpapers[16,17,18,19,36,37,38,39]andbymany other authors, see for instance [9,10,22,23,24,25,40,41,44]. The main drawback of such approaches is that stationary solutions which are characterized by these techniques are in some sense trivial: radial, with a single simply connected component support. Here we use a different approach to constructthe solutions,whichgoes back to [13] inthe contextof Schrödinger equations. We are not aware of attempts to use dimensional reduction coupled to power-law non-linearities and Poisson force fields except in the similar case of a nonlinear Schrödinger equation with power law nonlinearity and repulsive Coulombforces(see [8]),orin the caseofanattractiveHartree-Fockmodel(see [21]). Technically, our results turn out to be closely related to the ones in [6,7]. Compared to previous results on gravitational systems, the main interest of our approach is to provide a much richer set of solutions, which is definitely of interestinastrophysicsfordescribingcomplexpatternslikebinarygaseousstars or even more complex objects. The need of such an improvement was pointed for instance in [20]. An earlier attempt in this direction has been done in the frameworkofWasserstein’sdistanceandmasstransporttheoryin[27].Thepoint ofthis paperisthatwecantakeadvantageofthe knowledgeofspecialsolutions of the N-body problem to produce solutions of the corresponding problem in continuum mechanics, which are still reminiscent of the discrete system. 3. Construction of relative equilibria 3.1. Some notations. We denote by x=(x,x3) R2 R a genericpoint in R3. ′ ∈ × We may reformulate Problem (10) in terms of the potential u = U as fol- − lows. Given N positive numbers λ ,...λ and a small positive parameter ω, 1 N we consider the problem of finding N non-empty, compact, disjoint, connected subsets K of R3, i=1, 2...N, and a positive solution u of the problem i N −∆u= ρi inR3, ρi := u−λi+ 21ω2|x′|2 p+ χi , (11) i=1 X (cid:0) (cid:1) lim u(x)=0, (12) x | |→∞ where χ denotes the characteristic function of K . We define the mass and the i i center of mass associated to each component by 1 mω := ρ dx and xω := xρ dx. i ZR3 i i mi ZR3 i Weshallfindasolutionof (11)asacriticalpointofJ[u]= [ u]incaseof (8), J − with the notations of Section 2, that is when G( s) = 1 sp+1 for any s R. − p+1 + ∈ Multiplecomponents configurations incontinuous stellardynamics 9 The functional J reads N J[u]= 21 R3|∇u|2 dx− p+1 1 R3 u−λi+ 21ω2|x′|2 p++1χi dx. (13) Z i=1Z X (cid:0) (cid:1) Heuristically,ourmethodgoesasfollows.Wefirstconsidertheso-calledbasic cell problem:wecharacterizethesolutionwithasinglecomponentsupport,when ω = 0 and then build an ansatz by considering approximate solutions made of the superposition of basic cell solutions located close to relative equilibrium points, when they are far apart from each other. This can be done using the scaling invariance, in the low angular velocity limit ω 0 . The proof of our + → main results will be given in Sections 4–7. It relies on a dimensional reduction of the variational problem: we shall prove that for a well chosen u, J[u] = whNii=c1hλa5ir−epuen∗if−ormVmωin(ξ1ω,.>..ξ0N, )sm+allo.(1H)enfocresfionmdeincgonasctarintticea∗l,puopinttofoor(1J)wteilrlmbse, rPeduced to look for a critical point of ω as a function of (ξ ,...ξ ). Vm 1 N 3.2. The basic cell problem. Let us consider the following problem ∆w=(w 1)p inR3. (14) − − + Lemma 1.Under the condition lim w(x)=0, Equation (14) has a unique x solution, up to translations, which is| |p→os∞itive and radially symmetric. Proof. Since p is subcritical, it is well known that the problem ∆Z =Zp inB (0) 1 − with homogeneous Dirichlet boundary conditions, Z = 0, on ∂B (0), has a 1 unique positive solution, which is also radially symmetric (see [15]). For any R > 0, the function Z (x) := R 2/(p 1)Z(x/R) is the unique radial, positive R − − solution of ∆Z =Zp inB (0) − R R R with homogeneous Dirichlet boundary conditions on ∂B (0). According to [14, R 15],anypositivesolutionof (14)isradiallysymmetric,upto translations.Find- ing such a solution w of (14) is equivalent to finding numbers R > 0 and m > 0 such that the function, defined by pieces as w = Z +1 in B and R R w(∗x) = m /(4π x) for any x R3 such that x > R, is of class C1. These numbers ar∗e ther|ef|ore uniquely∈determined by | | w(R−)=1= m∗ =w(R+), w′(R−)=R−p−21−1Z′(1)= m∗ =w′(R+), 4πR −4πR2 which uniquely determines the solution of (14). (cid:3) 10 JuanCampos,ManueldelPino,andJeanDolbeault Now let us consider the slightly more general problem ∆wλ =(wλ λ)p inR3 − − + with lim wλ(x) = 0. For any λ > 0, it is straightforward to check that it x has a un|iq|u→e∞radial solution given by wλ(x)=λw λ(p−1)/2x x R3. ∀ ∈ Let us observe, for later reference,(cid:0)that (cid:1) (wλ λ)p dx=λ(3 p)/2 (w 1)p dx=:λ(3 p)/2m . (15) ZR3 − + − ZR3 − + − ∗ Moreover, wλ is given by m wλ(x)= ∗ λ(3−p)/2 x R3 such that x >Rλ−(p−1)/2. 4π x ∀ ∈ | | | | 3.3. The ansatz. We consider now a first approximation of a solution of (11)- (12),built asa superpositionofthe radiallysymmetricfunctions wλi translated to points ξ , i=1,...N in R2 0 , far away from each other: i ×{ } N w (x):=wλi(x ξ ), W := w . i i ξ i − i=1 X Recall that we are given the masses m ,...m . We choose, according to for- 1 N mula (15), the positive numbers λ so that i (w λ )p dx = m for all i=1,...N . R3 i− i + i Z By ξ we denote the array (ξ , ξ ,...ξ ). 1 2 N We shall assume in what follows that the points ξ are such that for a large, i fixed µ>0, and all small ω >0 we have ξ <µω 2/3, ξ ξ >µ 1ω 2/3 . (16) i − i j − − | | | − | Equivalently, ζ <µ , ζ ζ >µ 1 where ζ :=ω 2/3ξ . (17) i i j − i − i | | | − | We look for a solution of (11) of the form u=W +φ ξ foraconvenientchoiceofthepointsξ ,whereφisgloballyuniformlysmallwhen i compared with W . For this purpose, we consider a fixed number R > 1 such ξ that supp(wλi λ ) B (0) i=1, 2...N i + R 1 − ⊂ − ∀