In this section we consider the transformation
v(x) =rp⇤−12 u(r)|
r=ep⇤−12 x (3.5.1)
and then problem (3.1.6) turns out to be equivalent to 8<
We define ˆU as the transformation via (3.5.1) ofw, and for small " >0 we define
⇠ˆ1 = − 1
q−p⇤ log"−log ˆΛ1
⇠ˆi+1−⇠ˆi = −log"−log ˆΛi+1 i= 1, . . . , k−1 (3.5.4) where the points ˆΛi are positive parameters. We look for a solution of (3.5.2) of the form
v(x) =
CHAPTER 3. “BUBBLE-TOWER” SOLUTIONS 45
It follows that the only critical point of ˆΨk is nondegenerate and given by Λˆ⇤ =
The finite-dimensional reduction can be worked in a way similar to the section 3.3, except for (3.3.9), (3.3.10), that get replaced by
kN(φ)k⇤ C(kφkmin⇤ {p⇤,2}+kφkmin{
The finite-dimensional variational problem and the conclusion of the theorem can be derived in a way analogous to the one of section 3.4. ⇤
Bibliography
[1] T. Aubin, Probl`emes isop´erim´etriques et espaces de sobolev, J. Differential Geom., 11 (1976), pp. 573–598.
[2] R. Bam´on, M. Del Pino, and I. Flores, Ground states of semilinear elliptic equations: A geometric approach, Ann. I. H. Poincar´e, 17 (2000), pp. 551–581.
[3] L. A. Caffarelli, B. Gidas, and J. Spruck,Asymptotic symmetry and local behavior of semilin-ear elliptic equations involving critical sobolev growth, Comm. Pures Appl. Math, 42 (1989), pp. 271–
297.
[4] C. Chen and C.-S. Lin, Blowing up with infinite energy of conformal metrics on sn, Comm. in Partial Differetial Equations, 24 (1999), pp. 785–799.
[5] A. Contreras and M. del Pino, Nodal bubble-tower solutions to radial elliptic problems near criticality, Discrete and Continuous Dynamical Systems, 16 (2006), pp. 525–539.
[6] M. del Pino, J. Dolbeault, and M. Musso,“bubble-tower” radial solutions in the slightly super critical Brezis-Nirenberg problem, J. Differential Equations, 193 (2003), pp. 280–306.
[7] , The Brezis-Nirenberg problem near criticality in dimension 3, J. Math. Pures App, (9) 83 (2004), pp. 1405–1456.
[8] M. del Pino and I. Guerra,Ground states of a prescribed curvature equation, Journal of Differ-ential Equations, 241 (2007), pp. 112–129.
[9] M. del Pino, M. Musso, and A. Pistoia, Super critical boundary bubbling in a semilinear neumann problem, Ann. I. H. Poincar´e, 22 (2005), pp. 45–82.
[10] V. Felli and S. Terracini, Fountain-like solutions for nonlinear equations with critical hardy potential, Commun. Contemp. Math., 7 (2005), pp. 867–904.
[11] A. Floer and A. Weinstein,Nonspreading wave packets for the cubic Schr¨odinger equation with bounded potential, J. Funct. Anal., 69 (1986), pp. 397–408.
[12] I. Flores,A resonance phenomenon for ground states of an elliptic equation of emden-fowler type, J. Differential Equations, 198 (2004), pp. 1–15.
[13] R. Fowler,Further studies on emdens and similar differential equations, Quart. J. Math, 2 (1931), pp. 259–288.
[14] Y. Ge, R. Jing, and F. Pacard, Bubble towers for supercritical semilinear elliptic equations, Journal of functional Analysis, 221 (2005), pp. 251–302.
46
BIBLIOGRAPHY 47 [15] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,
Comm. in Partial Differetial Equations, 6 (1981), pp. 883–901.
[16] C.-S. Lin and W.-M. Ni, A counterexample to the nodal line conjecture and a related semilinear equation, Proc. Amer. Math., 102 (1988), pp. 271–277.
[17] G. Talenti,Best constant in Sobolev inequality, Ann Mat. Pura Appl., 110 (1976), pp. 353–372.
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Chapter 4
Relative equilibria in continuous stellar dynamics
This chapter is devoted to the study of a three dimensional continuous model of gravitating matter rotating at constant angular velocity. In the rotating reference frame, by a finite dimensional reduction, we prove the existence of non radial stationary solutions whose supports are made of an arbitrarily large number of disjoint 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 therelative equilibria inN-body dynamics.
This work is a joint work with M. del Pino and J. Dolbeault, which has already been published as Relative Equilibria in Continuous Stellar Dynamics, in the journal Communications in Mathematical Physics, December 2010, Volume 300, Issue 3, pp. 765-788.
4.1 Introduction and statement of the main results
We consider the Vlasov-Poisson system 8>
><
>>
:
@tf+v· rxf − rxφ· rvf = 0 φ=− 1
4⇡| · |⇤⇢ , ⇢:=
Z
R3
f dv
(4.1.1)
which models the dynamics of a cloud of particles moving under the action of a mean field gravitational potential φsolving the Poisson equation: ∆φ=⇢. Kinetic models like system (4.1.1) are typically used to describe gaseous stars or globular clusters. Here f = f(t, x, v) is the so-called distribution function, a nonnegative function in L1(R, L1(R3 ⇥R3)) depending on time t 2 R, position x 2 R3 and velocity v2R3, which represents a density of particles in thephase space, R3⇥R3. The function ⇢ is the spatial density function and depends only on tand x. The total mass is conserved and hence
ZZ
R3⇥R3
f(t, x, v)dx dv= Z
R3
⇢(t, x)dx=M does not depend on t.
The first equation in (4.1.1) is theVlasov equation,also known as thecollisionless Boltzmann equation in the astrophysical literature; see [22]. 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
49
dynamics of discrete particle systems can be formally recovered by considering empirical distributions, namely measure valued solutions made of a sum of Dirac masses, and neglecting the self-consistent gravitational terms associated to the interaction of each Dirac mass with itself.
It is also possible to relate (4.1.1) with discrete systems as follows. Consider the case of N gaseous spheres, far away one to each other, in such a way that they weakly interact through gravitation. In terms of system (4.1.1), such a solution should be represented by a distribution function f, whose space density ⇢ is compactly supported, with several nearly spherical components. At large scale, the location of these spheres is governed at leading order by theN-body gravitational problem.
The purpose of this study is to unveil this link by constructing a special class of solutions: we will build time-periodic, non radially symmetric solutions, 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 an axis which contains the center of gravity of the system, so that the centrifugal force counter-balances the attraction due to gravitation. Let us give some details.
Consider N point particles with masses mj, located at points xj(t) 2 R3 and assume that their dynamics is governed by Newton’s gravitational equations
mj d2xj x1 +i x2 and rewrite system (4.1.2) in coordinates relative to a reference frame rotating at a constant velocity ! >0 around thex3-axis. This amounts to carry out the change of variables
x= (ei ω tz0, z3), z0=z1+i z2. In terms of the coordinates (z0, z3), system (4.1.2) then reads
d2zj We consider solutions which are stationary in the rotating frame, namely constant solutions (z1, . . . zN) of system (4.1.3). Clearly allzj’s have their third component with the same value, which we assume zero.
Hence, we have that
zk= (⇠k,0), ⇠k2C, where the ⇠k’s are constants and satisfy the system of equations
XN In the original reference frame, the solution of (4.1.2) obeys to a rigid motion of rotation around the center of mass, with constant angular velocity !. This solution is known as a relative equilibrium, thus taking the form
xωj(t) = (ei ω t⇠j,0), ⇠j 2C, j= 1, . . . N .
System (4.1.4) has a variational formulation. In fact a vector (⇠1, . . . ⇠N) solves (4.1.4) if and only if it is a critical point of the function
Vmω(⇠1, . . . ⇠N) := 1
CHAPTER 4. RELATIVE EQUILIBRIA IN CONTINUOUS STELLAR DYNAMICS 51 Here m denotes (mj)Nj=1. A further simplification is achieved by considering the scaling
⇠j =!−2/3⇣j , Vmω(⇠1, . . .⇠N) =!2/3Vm(⇣1, . . .⇣N) (4.1.5)
This function has in general many critical points, which are allrelative equilibria. For instance,Vmclearly has a global minimum point.
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 mj, j = 1, . . . N, and any sufficiently small ! > 0, there exists a solution fω(t, x, v) of equation (4.1.1) which is 2πω-periodic in time and whose spatial density takes the form
⇢(t, x) :=
Here o(1) means that the remainder term uniformly converges to 0 as !!0+ and identically vanishes away from [Nj=1BR(xωj(t)), for some R > 0, independent of !. The functions ⇢j(y) are non-negative, radially symmetric, non-increasing, compactly supported functions, independent of!, withR
R3⇢j(y)dy =
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 around the x3-axis. The radii of these balls are very small compared with their distance to the axis. We shall call such a solution arelative equilibrium of (4.1.1), by extension of the discrete notion. The construction provides much more accurate informations on the solution. In particular, the building blocks ⇢j are obtained as minimizers of an explicit reduced free energy functional, under suitable mass constraints.
It is also natural to consider other discrete relative equilibria, namely critical points of the energyVm
that may or may not be globally minimizing, and ask whether associated relative equilibria of system (4.1.1) exist. There are plenty of relative equilibria of the N-body problem. For instance, if all masses mj are equal to some m⇤ > 0, a critical point is found by locating the ⇣j’s at the vertices of a regular polygon:
⇣j =r e2i π(j−1)/N , j = 1, . . . N , (4.1.6) wherer is such that
d
i.e. r= (aNm⇤/(4⇡))1/3. This configuration is called theLagrange solution,see [52]. The counterpart in terms of continuum mechanics goes as follows.
Theorem 2 Let (⇣1, . . .⇣N) be a regular polygon, namely with ⇣j given by (4.1.6), and assume that all masses are equal. Then there exists a solutionfω exactly as in Theorem 1, but withlimω!0+(⇣1ω, . . .⇣Nω) = (⇣1, . . .⇣N).
Further examples of relative equilibria in the N-body problem can be obtained for instance by setting N −1 point particles of the same mass at the vertices of a regular polygon centered at 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 functionalVm are always degenerate because of their invariance under rotations:
for any↵2Rwe have
Vm(⇣1, . . .⇣N) =Vm(ei α⇣1, . . . ei α⇣N).
Let ¯⇣ = (¯⇣1, . . .⇣¯N) be a critical point of Vm with ¯⇣` 6= 0. After a uniquely defined rotation, we may assume that ¯⇣`2 = 0. Moreover, we have a critical point of the function of 2N −1 real variables
V˜m(⇣1, . . .⇣`1, . . .⇣N) := Vm(⇣1, . . .(⇣`1,0), . . .⇣N).
We shall say that a critical point ofVmisnon-degenerate up to rotationsif the matrixD2V˜m(¯⇣1, . . .⇣¯`1, . . .⇣¯N) is non-singular. This property is clearly independent of the choice of `.
Palmore in [47, 48, 49, 50, 51] has obtained classification results for the relative equilibria. In particular, it turns out that for almost every choice of masses mj, all critical points of the functionalVm are non-degenerate up to rotations. 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 4.8 with a list of relevant references. These results have a counterpart in terms of relative equilibria of system (4.1.1).
Theorem 3 Let(⇣1, . . .⇣N) be a non-degenerate critical point of Vm up to rotations. Then there exists a solution f! as in Theorem 1, which satisfies, as in Theorem 2, lim!!0+(⇣1!, . . .⇣N!) = (⇣1, . . .⇣N).
This chapter 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 gravitational potential. The construction of these critical points is detailed in Section 4.3. Sections 4.4 and 4.5 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 (⇠j)Nj=1 related to the location of the N 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 (⇠j)Nj=1: this is the variational reduction described in Section 4.6. The proof of Theorems 1, 2 and 3 is given in Section 4.7 while known results on relative equilibria for theN-body, discrete problem are summarized in 4.8.