CCA mini-project:“Concentration-compactness in PDEs and orbital stability of

galaxy configurations”

Milana Gatarić

Supervisor: Clément Mouhot

University of CambridgeCambridge Centre for Analysis

June 9, 2012

Abstract

In this mini-project, the Vlasov–Poisson system is studied in its form which is used todescribe time evolution of a galaxy. The main properties are carefully revisited and a fewresults on nonlinear stability are given. The major part of the report is dedicated to theconcentration-compactness principle, which can be used in more general context in order tostudy stability of ground states in partial differential equations. Also, the concentration-compactness principle applied to the nonlinear Schrödinger equation is given. Finally, afew important recent results on the nonlinear stability of the Vlasov–Poisson system arerevisited, which use the notion of rearrangements.

Contents

1 The gravitational Vlasov–Poisson system 31.1 From the Newtonian N-body problem to the Vlasov–Poisson system . . . . . . . 31.2 Properties of the gravitational Vlasov–Poisson system . . . . . . . . . . . . . . . 51.3 Steady galaxy configurations and orbital stability . . . . . . . . . . . . . . . . . . 9

2 Nonlinear stability by concentration-compactness 112.1 The concentration-compactness principle in a simple setting . . . . . . . . . . . . 112.2 The general concentration-compactness principle . . . . . . . . . . . . . . . . . . 162.3 The orbital stability and the nonlinear Schrödinger equation . . . . . . . . . . . . 192.4 The orbital stability and the Vlasov–Poisson system . . . . . . . . . . . . . . . . 21

3 Nonlinear stability by rearrangements 223.1 Rearrangement according to the microscopic energy . . . . . . . . . . . . . . . . . 233.2 Monotonicity of the Hamiltonian and the reduced Hamiltonian . . . . . . . . . . 253.3 Coercivity inequality for the reduced Hamiltonian . . . . . . . . . . . . . . . . . . 253.4 Compactness of the minimizing sequence . . . . . . . . . . . . . . . . . . . . . . . 273.5 Completion of the proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . 28

1

Introduction

Describing galaxy configurations which are stable over time is an interesting and mathematicallychallenging problem from astrophysics. There are around 1011 gravitationally interacting starsin a typical galaxy and keeping track of the dynamics of each of them is not feasible. Instead,a statistical description of a galaxy is employed so that one can equivalently study an evolutionof the statistical distribution of a N -body system for large N , i.e. an evolution of a galaxy forlarge number of stars.

This statistical approach is characteristic for the kinetic theory and it was first applied tothe problem of describing a collisional gas, which gave rise to the Boltzmann equation in 1860’s.However, as opposed to the particles in a gas, collisions among stars in a galaxy are sufficientlyrare so that they can be neglected. Here, bodies of a system, i.e. stars, interact at the distanceand only by the gravitational field which they create collectively. Therefore, to such a system,the non-collisional or so-called the mean-field limit can be applied, discovered by Vlasov in1930’s. As a result, the Vlasov–Poisson equation is obtained, which is a nonlinear reversiblepartial-differential equation typically used to describe time evolution of a plasma or galaxy.

In this mini-project, we study the Vlasov–Poisson system in its gravitational form. We areinterested in the stability of its stationary solutions, which model the steady galaxy states. Itwas shown in [LMR08] that some of the galaxy models can be characterized as the ground statesof a certain energy functional and therefore, we are interested in the stability of the minimizersof such a functional. As we explain later, this can be derived as a direct consequence of a resultthat all such minimizing sequences are compact. However, first, one had to find a tool to provethe desired compactness.

Such question of the orbital stability is typical for any PDE system. In particular, a verysimilar problem of analyzing the energy functional in order to derive the stability of its minimizersis recognized in some nonlinear Schrödinger equations, [CL82]. The compactness of minimizingsequences in such circumstances was successfully proved in the Schrödinger and then in theVlasov–Poisson equations by using the concentration-compactness principle introduced by Lionsin [Lio84a, Lio84b]. More generally, one can use this principle to minimize a functional on abounded and closed subset of an infinite-dimensional function space defined on a unboundeddomain such as RN , and therefore, this is a very useful tool in variety of problems.

Let us mention here that the concentration-compactness was successfully used to prove onlythe orbital stability of certain group of steady galaxy models, so-called polytopic models, whilefor more general models different approach had to be applied. In 1960’s [Ant61, Ant65], Antonovresolved the problem of stability for more general models, but only in a linearized framework,and posed the conjecture of nonlinear stability of spherically-symmetric, isotropic and decreas-ing models. This conjecture of nonlinear stability was finally resolved by Lemou, Méhats andRaphaël in 2011 [LMR11b]. Still, in the absence of a spherical symmetry, the question of stabilityis open even for the linearized problem.

In Section 1, we revised the main features of the gravitational Vlasov–Poisson system, mainlyby following the recent review paper [Mou12]. Section 2 is devoted to the concept of theconcentration-compactness principle, which is firstly described in a simple setting and thena general framework is given. Also, we summarize how this method can be used to show thestability of solitary waves of some Schrödinger equation [CL82] and the stability of polytropicstates of the Vlasov–Poisson system [LMR08]. Finally, in the Section 3, we sketch the proof ofthe resolution of the conjecture given in [LMR11b], also by following the review paper [Mou12].

2

1 The gravitational Vlasov–Poisson system

1.1 From the Newtonian N-body problem to the Vlasov–Poisson system

Consider a system of N bodies of the same mass normalized to 1. Each body is associatedwith an index 1 ≤ i ≤ N , a position xi and a velocity vi. Binary interactions are assumedtrough an interaction potential φ, without an external field. For such a system, the Hamiltonianformulation of N -body problem is

x′i(t) = vi(t) =∂H

∂vi(X(t), V (t)) ,

v′i(t) = −∑i 6=j∇xψ (xi − xj) = −∂H

∂xi(X(t), V (t)) ,

(1.1)

for 1 ≤ i ≤ N and for the Hamiltonian

H (X,V ) =N∑i=1

|vi|2

2+∑i<j

ψ (xi − xj) ,

where X = (x1, . . . , xN ), V = (v1, . . . , vN ) for each xi, vi ∈ R3. This Hamiltonian H is calledthe microscopic Hamiltonian. It is preserved along the evolution of the Newton equations (1.1),which we denote with St. Hence, one has

∀t ∈ R, H (St (X,V )) = H (X,V ) .

Instead of this microscopic viewpoint, one can shift to a more statistical point of view andconsider a joint distribution function for all particles f (N) = f (N) (t,X, V ). The claim is thatf (N) evolves according to the following N -body Liouville equation

∂tf(N) +

N∑i=1

(∂H

∂vi· ∂f

(N)

∂xi− ∂H

∂xi· ∂f

(N)

∂vi

)= ∂tf

(N) +{f (N), H

}= 0, (1.2)

where {·, ·} denotes Poisson’s brackets on(R3)N × (R3

)N defined as{f (N), H

}= ∇Xf (N) · ∇V g(N) −∇Xg(N) · ∇V f (N).

The equation (1.2) is derived from the ODE system (1.1) by imposing that the distribution f (N)

is preserved when following the trajectories of each particle, i.e.

∀t ≥ 0, f (N) (t, St (X,V )) = f (N) (0, X, V ) .

The solution of (1.2) satisfies the following conservation laws:

- the conservation of the sign

f (N) (0, X, V ) ≥ 0 =⇒ f (N) (t,X, V ) ≥ 0, ∀t ≥ 0;

- the conservation of the mass∫(R3)N×(R3)N

f (N) (t,X, V ) dxdv =

∫(R3)N×(R3)N

f (N) (0, X, V ) dxdv, ∀t ≥ 0;

3

- the conservation of any Casimir functional∫(R3)N×(R3)N

C(f (N) (t,X, V )

)dxdv =

∫(R3)N×(R3)N

C(f (N) (0, X, V )

)dxdv, ∀t ≥ 0,

where C ∈ C1 (R+,R+), C (0) = 0.

To simplify the description of the system, one can introduce a distribution of one bodyf = f (1)

f (t, x, v) :=

∫(R3)N−1×(R3)N

f (N) (t,X, V ) dx2 . . . dxNdv2 . . . dvN ,

which is determined as a marginal distribution of f (N). To obtain the equation which de-scribes how the one-body distribution evolves, one integrates the Liouville equation (1.2) inx2, v2, . . . , xN , vN variables and gets

∂f

∂t+ v · ∇xf − (N − 1)

∫R3×R3

∇xψ (x− y) · ∇vf (2) (x, y, v, v∗) dydv∗ = 0. (1.3)

This equation depends on the two-particle distribution f (2), i.e the second marginal

f (2) (t, x1, x2, v1, v2) :=

∫R3×R3

f (N)dx3 . . . dxNdv3 . . . dvN ,

which contains the information on the correlations in the system.Now, the goal is to take the limit as N → ∞ and obtain an equation on the first marginal

only. The two following assumptions are required when taking the mean field limit, also calledthe Vlasov limit:

- each binary interaction is of the order O (1/N) and therefore the appropriate scaling ofthe interaction potential is

ψ =ψ̄

N,

which means that the influence of one star to another star is negligible across the entiregalaxy, while the collective action of the entire galaxy on a given star is important and itis of the order O ((N − 1) /N) = O (1);

- the “chaos property”

f (2) ∼ f × f, as N →∞,

which indicates weak correlations in the system.

Under these assumptions, the following equation is obtained from (1.3) when N →∞

∂f

∂t+ v · ∇xf −

∫R3×R3

∇xψ̄ (x− y) · ∇vf (x, v) f (y, v∗) dydv∗ = 0,

which is closed on the distribution f and can be rewritten as

∂f

∂t+ v · ∇xf −

(∫R3

∇xψ̄ (x− y)

(∫R3

f (y, v∗) dv∗)dy)· ∇vf (x, v) = 0. (1.4)

4

In (1.4), if the microscopic potential is chosen such that it corresponds to the gravitationalinteractions

ψ̄ = − 1

4π|x|,

one obtains the gravitational Vlasov–Poisson system

∂tf + v · ∇xf −∇xφf · ∇vf = 0, t ∈ R+, x ∈ R3, v ∈ R3,

f (t = 0, x, v) = fin (x, v) ,(1.5)

with the gravitational mean field φf defined by

φf := − 1

4π|x|∗ ρf , ρf :=

∫R3

f (t, x, v) dv. (1.6)

where ρf (t, x) gives the spatial density. Observe that the field φf solves the Poisson ellipticequation

∆φf = ρf , (1.7)

which explains the name of the Vlasov–Poisson system. We note here that the minus sign inthe definition of the potential in (1.6) is due to the attractive interactions in the gravitationalmodels. In the case when the Vlasov–Poisson system is used to describe the plasma, here wewould have the plus sign, which corresponds to the repulsive interactions.

1.2 Properties of the gravitational Vlasov–Poisson system

The Vlasov–Poisson system (1.5)–(1.6) is nonlinear due to the mean-field term, i.e. due to ∇xφf .Nevertheless, it shares a similar structure with the Liouville equation (1.2), which is a linearPDE, in the following way. The Vlasov–Poisson equation can be written in the form

∂tf + {f,Ef} = 0,

where {·, ·} denotes the Poisson brackets on R3 × R3 and

Ef = Ef (t, x, v) =|v|2

2+ φf (t, x)

defines the microscopic mean-field Hamiltonian. Also, note here, that the system (1.5)–(1.6) isassociated with the solutions to the differential system

x′(t) = v(t) =∂Ef∂v

, (1.8)

v′(t) = −∇xφf (x(t)) = −∂Ef∂x

. (1.9)

Next, it is shown that the quantity called macroscopic Hamiltonian and given by

H (f) =

∫R3×R3

|v|2

2f (t, x, v) dxdv +

∫R3×R3

φf (t, x)

2f (t, x, v) dxdv

is preserved along the evolution of the Vlasov–Poisson system (1.5)–(1.6). First, observe that itcan be rewritten as follows

H (f) =

∫R3×R3

|v|2

2f (t, x, v) dxdv −

∫R3

|∇xφf (t, x) |2

2dx, (1.10)

5

by using the definition of ρf , the Poisson equation (1.7) and by applying the partial integrationon 1

2

∫R3 φf (t, x) ∆φf (t, x) dx. Now, by differentiating the macroscopic Hamiltonian in time,

one gets the following

d

dt

(∫R3×R3

|v|2

2fdxdv −

∫R3

|∇xφf |2

2dx)

=

∫R3×R3

|v|2

2

∂f

∂tdxdv −

∫R3

∇xφf · ∇x∂φf∂t

dx

=

∫R3×R3

|v|2

2

∂f

∂tdxdv +

∫R3

φf∆x∂φf∂t

dx

=

∫R3×R3

|v|2

2

∂f

∂tdxdv +

∫R3

φf∂ρf∂t

dx

=

∫R3×R3

(|v|2

2+ φf

)∂f

∂tdxdv

=

∫R3×R3

Ef (−v · ∇xf +∇xφf · ∇vf) dxdv

=

∫R3×R3

(∇xφfvf − v · ∇xφff) dxdv

= 0.

Note that the Hamiltonian H can be written as the difference of two positive terms, which wewill denote as follows

H (f) = Hkin (f)−Hpot (f) .

Therefore, it is possible to have the situation when the total energy of the system H staysconstant over time, but the kinetic energy Hkin and potential energy Hpot diverge. That means,it is possible to have a transfer between kinetic and potential energy, which may diverge whilebalancing. This creates the mathematical difficulties when analyzing the stability of the system(1.5)–(1.6). It is not enough to have a control over the total energy, but the kinetic and thepotential energy have to be controlled separately as well. Luckily, it is possible to prove a keyinterpolation inequality, which gives the control of the potential energy by the kinetic energy.Thus, it is sufficient to bound the kinetic energy. Let us mention here that by the analogy withthe dispersive partial differential equations, in particular with the focusing nonlinear Schrödingerequation, where the Hamiltonian is also composed of two terms of opposite sign, the Vlasov-Poisson equation is called a focusing equation.

Before commenting more on these energy terms and proving the key interpolation inequality,let us mention another very important conservation law for the Vlasov–Poisson system (1.5)–(1.6). Namely, that for all Casimir functionals it is true that∫

R3×R3

C (f (t, x, v)) dxdv =

∫R3×R3

C (f (0, x, v)) dxdv ∀t ≥ 0.

This is verified by the following computation

∂

∂t

(∫R3×R3

C (f (t, x, v)) dxdv)

=

∫R3×R3

C′ (f)∂f

∂tdxdv

=

∫R3×R3

C′ (f) (−v · ∇xf +∇xφf · ∇vf) dxdv

= −∫R3

v

(∫R3

∇xC(f)dx)dv +

∫R3

∇xφf(∫

R3

∇vC(f)dv)dx

= 0,

6

where one uses that the function C vanishes when f is equal to zero, i.e. when x and v go toinfinity. This property of the conservation of a Casimir functional leads to the preservations ofall Lp norms for the solution f , i.e.

‖f (t, x, v) ‖Lp = ‖fin (x, v) ‖Lp∀t ≥ 0.

The following classical result, gives is the key interpolation inequality, which says that thepotential energy can be controlled by the kinetic energy and the invariant of the system, i.e. theLp norm of a solution f .

Theorem 1.1. For every p ∈ [1,∞], the integrability of the spatial density is estimated as follows

‖ρf‖L

5p−33p−1 (R3)

≤ C‖f‖2p

5p−3

Lp(R6)Hkin (f)

3p−35p−3 ,

and in the limiting case when p =∞

‖ρf‖L

53 (R3)

≤ C‖f‖25

L∞(R6)Hkin (f)

35 ,

for some constant C > 0.The following control of the potential energy for p > pc = 9/7 is deduced

Hpot (f) ≤ C‖f‖7p−96(p−1)

L1(R6)‖f‖

p3(p−1)

Lp(R6)Hkin (f)

12 ,

for some constant C > 0.

Proof. For any p > 1 and R > 0, it is possible to decompose the spatial density in the followingway ∫

R3

fdv =

∫|v|≤R

fdv +

∫|v|>R

fdv

≤ ‖f‖Lpv

(∫|v|≤R

dv

) (p−1)p

+R−2

∫R3

f |v|2dv

≤ ‖f‖Lpv

(4πR3

3

) (p−1)p

+R−2

∫R3

f |v|2dv.

By the optimization over the parameter R, one obtains∫R3

fdv ≤ C‖f (x, ·) ‖2p

(5p−3)

Lp(R3)

(∫R3

f |v|2dv) (3p−3)

(5p−3)

,

and then by the Cauchy–Schwarz inequality

‖ρf‖L

(5p−3)(3p−1) (R3)

≤ C

∫R3

‖f (x, ·) ‖2p

(3p−1)

Lp(R3)

(∫R3

f |v|2dv) (3p−3)

(3p−1)

dx

(3p−1)(5p−3)

which gives the first inequality from the statement of the theorem.To obtain the control on the potential energy, we need the inequality

‖∇xφf‖L2(R3) ≤ C‖ρf‖L6/5(R3),

7

which can be justified from he Poisson equation (1.7), by using the formula ∇xφf = x4π|x|3 ∗ ρf

and the Hardy–Littlewood–Sobolev inequality. Therefore, one has

Hpot (f) = ‖∇xφf‖2L2(R3) ≤ C‖ρf‖2L6/5(R3)

.

For any p > pc = 9/7, one has (5p− 3) / (3p− 1) > 6/5 and then it is possible to write thefollowing

‖ρf‖L6/5(R3) ≤ ‖ρf‖7p−9

12p−12

L1(R3)‖ρf‖

5p−312p−12

L5p−33p−1 (R3)

≤ ‖f‖7p−9

12p−12

L1(R3)‖ρf‖

5p−312p−12

L5p−33p−1 (R3)

.

By combining all this, together with the first inequality from the statement, one gets

Hpot (f) ≤ C‖f‖7p−96(p−1)

L1(R6)‖f‖

p3(p−1)

Lp(R6)Hkin (f)

12 ,

for any p > pc = 9/7, which finishes the proof.

Note that by this inequality, the potential energy is controlled by an interpolation betweenthe invariants of the system and the kinetic energy, with a power of the kinetic energy strictly lessthan 1. When it is possible to establish such control of the potential energy, the correspondingequation is called sub-critical. This is also the case with certain Schrödinger equation, which wewill discuss later. Let us just mention that when such control is possible but with a power of thekinetic energy equal to 1, one has a critical equation. When such control is only possible witha power strictly greater than 1, it is an over-critical equation, for which a general phenomenaof explosion in finite time is expected. In the case of Vlasov–Poisson system in dimension 3,which is sub-critical and which we consider in this presentation, as suggested by the results inthe theorem above, the main difficulty is to establish the control of the kinetic energy.

Concerning the existence of a solution of the Vlasov–Poisson system (1.5)-(1.6), firstly, it wasshown how to construct C1 solutions with the compact support, in [Pfa92], about twenty yearsago. Secondly, at the same time, in [LP91] it was discovered how to build a unique global solutionin L1

((1 + |v|k

)dxdv

)∩ L∞ for k sufficiently large, with a rather weak regularity assumption

on the initial spatial density ρin. Moreover, the global weak solutions in the space

E ={f : f ≥ 0, f ∈ L1 ∩ L∞

(R6), |v|2f ∈ L1

(R6)}, (1.11)

under weaker assumptions on the initial data fin ∈ E , were constructed by Arsenev [Ars75],however, their uniqueness remains open. Solutions in E are continuous in time, with values inL1(R6)and they satisfy the following inequality

∀t ≥ 0, H (ft) ≤ H (fin) . (1.12)

In the later sections on nonlinear stability, we deal with these solutions.Let us mention here, one more property of the Vlasov–Poisson system. It allows for a large

group of symmetries and it is known that if f (t, x, v) is a solution, then for all t0 ∈ R, x0 ∈R3, λ0 ∈ R∗+, µ0 ∈ R∗+

g (t, x, v) :=µ0

λ20

f

(t+ t0λ0µ0

,x+ x0

λ0, µ0v

)is also a solution. Moreover, the translational invariance of Galilean holds, i.e. if f (t, x, v) is asolution, then for all v0 ∈ R3

g (t, x, v) := f (t, x+ tv0, v + v0)

is also a solution.

8

1.3 Steady galaxy configurations and orbital stability

A stationary solution f0 of the Vlasov–Poisson system (1.5)-(1.6) satisfies

∀x, v ∈ R3, v · ∇xf0 −∇xU (x) · ∇vf0 = 0.

∀x ∈ R3, ∆xU = ρf0 .(1.13)

Additionally, if a stationary solution f0 is spherical symmetric, it is invariant with respect tothe referential rotations, i.e.

∀ (x, v) ∈ R6, f0 (Ax,Av) = f0 (x, v)

for all orthogonal matrices A ∈ R3×3. Equivalently, one has that

f0 = f0 (|x|, |v|, x · v)

must hold.In the Vlasov–Poisson system with the spherical symmetry, the Poisson equation (1.7) be-

comes

1

r2

(r2U ′

)′=

∫R3

f0 (|x|, |v|, x · v) dv,

where the potential U depends only on r = |x|. In this case, the equations of the motion of thesystem (1.13) are

x′(t) = v,

v′(t) = −∇xU = − x

|x|U ′ (|x|) ,

and therefore, it follows that

ddt

(x(t) ∧ v(t)) =(x′(t) ∧ v(t)

)+(v′(t) ∧ x(t)

)= (v(t) ∧ v(t))−

(U ′ (|x|)|x|

x(t) ∧ x(t)

)= 0.

Beside the macroscopic Hamiltonian, this is another invariant of the system. Since the phasespace of the spherical system has dimension three, i.e. (|x|, |v|, x · v) ∈ R3, and since the systemnow possesses two independent invariants, it is left with one degrees of freedom and hence it iscompletely integrable. Having the completely integrable system, the Jeans theorem, see [BT87,Section 4], shows that f0 is a function depending only on the invariants of that system. Thus,the stationary solution can be written as

f0 (x, v) = F (E,L) ,

where

E (x, v) := Ef0 (x, v) =|v|2

2+ φf0 (t, x) ,

L (x, v) := |x ∧ v|2.

9

Therefore, here, one has

∆xφ =

∫R3

F (E,L) dv.

If F = F (E) depends only on E, it represents a isotropic spherical model, and if F = F (E,L)it is a anisotropic spherical model.

Among all spherical steady galaxy models, we mention two isotropic ones, which are themost common in the literature. The first one is the (normalized) polytrope

f0 (x, v) = F (E) = (E0 − E)n+ , (1.14)

where (·)+ := max{·, 0}, E0 is a non-positive constant which represents some energy thresholdand 0 < n ≤ 7

2 . For n > 72 , it leads to the solutions of an infinite mass and a noncompact

support by adjusting the parameter E0. For the limiting case when n = 72 , it has a finite mass

but non-compact support and it is called the Plummer model. The other model that we mentionis the King model, which is given by the following function

f0 (x, v) = F (E) =(eE0−E − 1

)+,

for a given constant E0 < 0. The King model describes isothermal galaxies, formally correspond-ing to n =∞ in the polytropes, which is widely used in astronomy. For more galaxy models see[BT87].

One of the central question when describing the Vlasov–Poisson system, or any PDE systemin general, is the stability of its stationary solutions, or in our case, the stability of steady galaxymodels f0. Here, we introduce the notion of the orbital stability, which will be discussed in moredetails in the following sections. A stationary solution f0 is nonlinearly stable if the solutions ftof the nonlinear Vlasov–Poisson system remain arbitrarily close to the stationary solution for alltimes t ≥ 0, provided that they start sufficiently close to the stationary solution, i.e. providedthat fin is close to f0. The distance is measured by the system invariants. This is usuallyexpressed as follows: for all ε > 0, there is ν > 0 such that

D(fin, f

0)

:=∥∥fin − f0

∥∥L1(R6)

+∣∣H (fin)−H

(f0)∣∣ ≤ ν =⇒ ∀t ≥ 0,D

(ft, f

0)≤ ε.

However, one must include the transformations under which the Vlasov–Poisson system is in-variant and then the notion of the orbital stability is reformulated in the following manner: thereis a translation function z = z(t) ∈ R3 such as

D(fin, f

0)≤ ν =⇒ ∀t ≥ 0,D

(ft, f

0 (· − z(t), ·))≤ ε. (1.15)

In the case when spherically symmetric perturbations are considered, the parameter z(t) isnecessarily always zero.

On the other hand, when one talks about linear stability, the solutions ft are replaced by thesolutions of the corresponding linearized problem. The question of the linear stability for thespherically-symmetric decreasing models, F ′ < 0, is resolved in the 1960s and 1970s as a resultof the pioneering work by Antonov [Ant61, Ant65]. For the systematical review on the resultsin linear stability see [Mou12] and [Guo08].

Concerning nonlinear stability, there are different variational and non-variational (direct)approaches. Also, for the detail review see [Mou12] and [Guo08]. Here, we will discuss thevariational approach by concentration-compactness technique, which is successfully applied tothe polytropic models. After that, we discuss the method by rearrangement used to resolve thequestion of orbital stability of more general models, precisely the spherically-symmetric, isotropic

10

and decreasing models under general perturbations. Also, this method is successfully appliedto prove orbital stability of spherically-symmetric, anisotropic and decreasing models under thespherically symmetric perturbations. Let us add that in the absence of spherical symmetry inthe stationary solution, the question of stability is open even for the linearized problem.

2 Nonlinear stability by concentration-compactness

It can be shown, that the minimizers of the microscopic Hamiltonian (1.10) are described bythe polytropic models (1.14). Later, we will state this result formally in the Theorem 2.9.Therefore, considering the stability of the gravitational Vlasov–Poisson system (1.5)-(1.6), oneis interested in the orbital stability of the states which are characterized as the minimizers ofthe functional H. Since those states are of the lowest energy, they are called the ground states.Thus, essentially, one deals with a problem of minimizing a functional F on the bounded andclosed subset E of an infinite-dimensional function space H, defined on a unbounded domain,such as RN , a typical problem from calculus of variations. In what follows we deal with a moregeneral framework which motivates the writing F instead of H.

In the infinite-dimensional function space H, we are faced with the difficulty that Bolzano–Weierstrass theorem does not hold, i.e. a bounded sequence in such space does not need tohave a convergent subsequence, in the sense of the strong topology. Equivalently, a closed andbounded subset in such space does not have to be compact. Hence, it is not possible easily toshow the existence of the solution of such minimization problem.

There are different ways to get around this difficulty of the loss of compactness, dependingon a certain variational problem and additional properties of the functional F . Often, one wantsto weaken the topology in order to recover the compactness of E, but this makes the continuity,or semicontinuity, of F harder to establish. The technique applicable in this situation is theconcentration-compactness principle, [Lio84a, Lio84b]. The idea here is to make the topologyweaker so that the functional F is continuous, but for which E is still not quite compact.It is not quite compact in the sense that a sequence x(n) in E still does not have to have asubsequence converging to a constant element x ∈ E in this weaker topology. However, theconcentration-compactness principle fully describes the failure of the compactness, which thencan be eliminated with the additional assumption on the functional F .

Basically, when a sequence x(n) does not have a subsequence converging to a constant elementx ∈ E, it may have a subsequence converging to another non-constant sequence y(n) in thisweaker topology. This sequence y(n) is some kind of an approximating sequence and it is fullydescribed by the concentration-compactness principle. It can be converging to a point or goingoff to infinity, or it can be a superposition of such sequences.

This principle is carefully described in simple settings in the next subsection, following whatis written in Terence Tao’s book [Tao09, Section 1.16]. Afterwards, the general frameworkis given. As this method was successfully applied to many problems, we chose one which isstructurally close to the problem of orbital stability of the polytropes in the gravitational Vlasov–Poisson system. In particularly, the orbital stability of ground solitary states of some nonlinearSchrödinger equation is shown first. Then, the similar procedure is applied in the Vlasov–Poissoncontext.

2.1 The concentration-compactness principle in a simple setting

Instead of working in a function space such as Sobolev space, first we demonstrate the concentra-tion-compactness principle in simple settings. The function space H is taken to be the space of

11

absolutely summable two-sided infinite sequences

l1 (Z) =

{(xm)m∈Z :

∑m∈Z|xm| <∞

}with the translation group Z. It is a well-known result, which will be used a lot here, thatfor every two-sided sequence x = (xm)m∈Z ∈ l1 (Z), one has xm → 0, as m → ±∞. There areseveral topologies defined on this space, and we will use the ones given in the following definition.

Definition 2.1. Let x(n), n = 1, 2, . . . be a sequence in l1 (Z), such that elements of this sequenceare two-sided sequences denoted as (x

(n)m )m∈Z, and let x = (xm)m∈Z be another element in l1 (Z).

(i) (Strong topology) The sequence x(n) converges to x, as n→∞, in the strong topology (orl1 topology) if the l1 distance ‖x(n) − x‖l1(Z) :=

∑m∈Z |x

(n)m − xm| converges to zero.

(ii) (Intermediate topology) The sequence x(n) converges to x, as n→∞, in the intermediatetopology (or l∞ topology) if the l∞ distance ‖x(n)−x‖l∞(Z) := supm∈Z |x

(n)m −xm| converges

to zero.

(iii) (Weak topology) The sequence x(n) converges to x, as n → ∞, in the weak topology (orpointwise topology) if x(n)

m → xm, as n→∞, for each m.

Here, the translation group is given by Z and the translation action is defined by the shiftoperator T h : l1 (Z)→ l1 (Z), for h ∈ Z, such that

T h (xm)m∈Z = (xm−h)m∈Z .

For h > 0, this is the right shift operator, which moves the two-sided infinite sequence as follows:

. . . x−1 x0 x1 x2 x3 . . .h = 1 . . . x−2 x−1 x0 x1 x2 . . .h = 2 . . . x−3 x−2 x−1 x0 x1 . . .

In the infinite-dimensional space l1 (Z), we take a closed and bounded set E to be

E =

{(xm)m∈Z ∈ l

1 (Z) :∑m∈Z|xm| = 1

}. (2.1)

This set is invariant under the above defined translation action. Also, note that this set is closedand bounded in strong topology, but it is not closed in weak or intermediate topology. This isbecause we can find a sequence in E converging to zero in weak and intermediate topology, butzero is not in E. For example, the sequence of the basis vectors e(n) = 1

2n+1

∑nk=−n δ

(n)k , with

the Kronecker symbol δ, which is a sequence of two-sided sequences from E, converges to zeroin the weak and intermediate topology.

The failure of the closure in the weak and intermediate topology causes the loss of compact-ness of E in the intermediate and strong topology. To demonstrate this, let us suppose that Eis compact in the strong or intermediate topologies. That would mean that every sequence inE has a convergent subsequence, in that particular topology. Also, the limit of the convergentsubsequence in these topologies has to be equal to its weak limit. On the other hand, there is,for example, the sequence e(n), which has the weak limit zero, but it does not converge to zeroin other two topologies, what gives the contradiction.

The failure of compactness of E in the intermediate topology is described by the followingconcentration-compactness lemma.

12

Lemma 2.2. Let x(n) be a sequence in E. Then, it has a subsequence, which is still denotedx(n), such that there exist xj ∈ l1 (Z), satisfying∑

j

‖xj‖l1(Z) ≤ 1, (2.2)

and there exist sequences h(n)j of integers obeying the asymptotic orthogonality condition

|h(n)k − h

(n)j | → ∞, as n→∞, (2.3)

for all k > j, so that the following decomposition of x(n) holds

x(n) =∑j

T h(n)j xj + w(n), (2.4)

where the remainder w(n) converges to zero in the intermediate topology.

Remark 2.3. If there is the equality in (2.2), it is possible to make w(n) converging in thestrong topology as well.

Remark 2.4. This lemma says that every sequence x(n) in E, i.e. every bounded sequence oftwo-sided sequences from l1 (Z), has a subsequence converging to the sum of sequences of verystructured form, in the intermediate topology. Analyzing the decomposition (2.4), we see thatthe mentioned subsequence can converge to zero in the intermediate topology, in which case wetake j to go over the empty set. Further, it can converge to any other element in l1 (Z) with the l1

norm less or equal to 1, in which case we have exactly one xj 6= 0 and the corresponding sequenceh

(n)j does not go to infinity. Otherwise, in the case when xj 6= 0 and h(n)

j goes off to infinity forat least one j, the sequence can not be compact, i.e. it can not have a convergent subsequence.We know that every subsequence of a compact sequence in E has to have the intermediate limitequal to its weak limit. However, the weak limit of such decomposed sequence x(n) is equal tozero, but the intermediate and strong limit are not. Therefore, it has no convergent subsequencein the intermediate (or strong) topology.

Proof. If the sequence x(n) converges to 0 in the intermediate topology, then it is enough totake j to go over the empty set, as mentioned in the Remark 2.4. So, let us suppose that x(n)

does not converge to 0 in this l∞ topology. This means that there exist a subsequence, whichwe denote the same, and ε1 > 0 such that ‖x(n)‖l∞ > ε1 for all n. Therefore, for each two-sidedsequence in the sequence x(n), it is possible to find an integer h(n)

1 such that |x(n)

h(n)1

| > ε1. Now

we want to move those "large" elements at the origin position, in each two-sided sequence. Ifwe pick all those integers h(n)

1 and shift the corresponding elements of x(n) to the left, we getthe rearranged sequence T−h

(n)1 x(n) such that all its two-sided sequences, at the origin position

m = 0, have a real number bounded below in magnitude by ε1.The sequence x(n) is a sequence of elements in E and that still holds for the shifted sequence

T−h(n)1 x(n). The set E is bounded and closed in l1, and therefore, after passing to another

subsequence which we still denote the same, we can conclude that shifted sequence convergeweakly to some limit x1 ∈ l1(Z). Since this shifted sequence has those "large" elements at theorigin, for the weak limit the inequality ‖x1‖l1(Z) ≥ ε1 has to hold. Now, by taking the inverseshift, it is obvious that we have obtained the following decomposition

x(n) = T h(n)1 x1 + w

(n)1 , (2.5)

13

where the remainder w(n)1 is such that T−h

(n)1 w

(n)1 converges weakly to zero. Both of these

sequences on the right hand side in (2.5) can be understood as asymptotically vanishing. Hence,it is possible to show

‖x(n)‖l1(Z) = ‖T h(n)1 x1‖l1(Z) + ‖w(n)

1 ‖l1(Z) + o (1) ,

where o (1) goes to zero as n→∞, and since l1 norm is invariant on the shift operator, one has

1 = ‖x1‖l1(Z) + limn→∞

‖w(n)1 ‖l1(Z).

Also, note here that "mass" of the remainder w(n)1 is eventually strictly less then that of the

original sequence, since

‖w(n)1 ‖l1(Z) ≤ 1− ε1 + o (1) .

Then, the same procedure is applied to the remainder w(n)1 . If it converges to zero in the

intermediate topology, the proof is done. Otherwise, as before, it is possible to find ε2 > 0 andsequence of integers h(n)

2 such that the shifted sequence T−h(n)2 w

(n)1 is bounded from below by ε2

at the origin positions. Therefore, after passing to another subsequence, one has that elementsof T−h

(n)2 w

(n)1 converge weakly to some non-trivial limit x2 ∈ l1 (Z), such that ‖x2‖l1(Z) ≥ ε2.

Now, it is possible to conclude that the orthogonality condition (2.3) must hold. Suppose itdoes not hold, i.e. |h(n)

1 − h(n)2 | does not go to infinity. Then, there is a subsequence for which

h(n)1 − h(n)

2 is equal to some constant c. In this case, one has

T−h(n)1 w

(n)1 = T−c

(T−h

(n)2 w

(n)1

),

which gives a contradiction because T−h(n)1 w

(n)1 converges weakly to zero and T−h

(n)2 w

(n)1 does

not.By undoing the shift, the following decompositions is obtained

w(n)1 = T h

(n)2 x2 + w

(n)2 ,

where the remainder w(n)2 is such that T−h

(n)2 w

(n)2 weakly converges to zero. Combining this

with (2.5), one gets

x(n) = T h(n)1 x1 + T h

(n)2 x2 + w

(n)2 .

Therefore,T−h

(n)1 x(n) = x1 + T h

(n)2 −h

(n)1 x2 + T−h

(n)1 w

(n)2

and one can see that T−h(n)1 w

(n)2 also converges weakly to zero. Moreover, by the asymptotic

orthogonality, the following is concluded

1 = ‖x1‖l1(Z) + ‖x2‖l1(Z) + limn→∞

‖w(n)2 ‖l1(Z).

Again, note that the norm of the remainder w(n)2 has decreased as follows

‖w(n)2 ‖l1(Z) ≤ 1− ε1 − ε2 + o (1) .

14

By proceeding in this manner, one gets finer and finer decomposition on further subsequences.The subsequences depend on j, but one can work with a single sequence throughout, by Cantordiagonalization argument. One stops if the remainder w(n)

j converges to zero in the intermediatetopology. However, the remainders are getting smaller and smaller, and one can observe thattheir asymptotic l∞ norm must decay to zero as j →∞, in the following way.

Notice that, at each step, we identified the large piece of x(n) and those pieces were exactlythe obstructions for the sequence to converge in the intermediate topology. Moreover, at eachstep, we decreased the total mass in play by "extracting the mass" of εj . One observes herethat, since the starting sequence is in E, the extracted mass can not exceed 1, i.e.

∑j εj ≤ 1

and (2.2) holds. In particular, the extracted mass εj must go to zero as j →∞. It is importantto chose εj in a “greedy“ manner and then the asymptotic l∞ norm of the remainders w(n)

j for

n → ∞ must decay to zero as j → ∞. For example, one can choose εj > 12 limn→∞ ‖w(n)

j−1‖l∞ ,where w(n)

0 is just the starting sequence x(n) and where one passes to another subsequence ofw

(n)j−1 to provide the existence of limn→∞ ‖w(n)

j−1‖l∞ . Since εj depends on this limit, one alsopasses to another subsequence, so that the arguments ‖wnj−1‖l∞ > εj for all n hold. Hence, ateach step j, the asymptotic l∞ norm of the remainder is reduced by 1

2 , so it has to converge tozero as j →∞. This gives the wanted decomposition (2.4) and completes the proof.

The result of this lemma is then used to establish the existence of the minimum of thefunctional F defined on E. This is essentially the concentration-compactness principle. Thefailure of the compactness has to be compensate with additional assumptions on F . Usually,it is the case that the assumptions are adapted to the particular problem. In the followingtheorem, which can be found in [Tao09], the typical hypotheses on F are taken.

Theorem 2.5. Let the set E be the subset in l1 (Z) given by (2.1), and let F : l1(Z)→ R be afunctional with the negative infimum I := infx∈E F (x) < 0 and with the following properties:

(a) (Continuity) F is continuous in the intermediate topology on E.

(b) (Homogeneity) F is homogeneous of some degree 1 < p <∞, i.e.

F (αx) = αpF (x)

for all α > 0 and x ∈ l1 (Z).

(c) (Invariance) F is invariant to the translation action defined by the shift operator T h, i.e.F(T hx

)= F (x) for all h ∈ Z and x ∈ l1 (Z).

(d) (Asymptotic additivity) If h(n)j , is a collection of sequences obeying the asymptotic orthog-

onality condition (2.3), and if xj ∈ l1 (Z) are such that∑

j ‖xj‖l1(Z) <∞, then

∑j

F (xj) <∞ and F

∑j

T h(n)j xj

=∑j

F (xj) + o (1) ,

where the quantity o (1) goes to zero as n → ∞. More generally, if w(n) is bounded in l1

and converges to zero in the intermediate topology, then

F

∑j

T h(n)j xj + w(n)

=∑j

F (xj) + o (1) .

15

Then F attains its minimum on E.

Proof. By the definition of the infimum, one can find the minimizing sequence x(n) ∈ E, i.e.the sequence in E such that F

(x(n)

)→ I, as n → ∞. After applying the Lemma 2.2 to this

minimizing sequence, one passes to the subsequence and obtains the decomposition (2.4). Sincethe asymptotic additivity assumption (d) holds, from the decomposition (2.4) one gets

F(x(n)) =∑j

F (xj) + o (1) .

Letting n to go to infinity, one obtains

I =∑j

F (xj) . (2.6)

From this, we have that I is finite, and now we want to show that it is equal to F (x0), for somex0 ∈ E. Since I is the infimum of F on the set E, one has I ≤ F (x) when ‖x‖l1(Z) = 1. Andsince l1 norm of xj

‖xj‖ equals to 1, by using the homogeneity assumption (b), for those xj 6= 0

one obtains

I ≤

(1

‖xj‖l1(Z)

)pF (xj) .

From here, by summing over all j and by using (2.6), one gets the inequality∑j

‖xj‖pl1(Z)

I ≤ I.

Since I < 0, one gets the following

1 ≤∑j

‖xj‖pl1(Z)≤

∑j

‖xj‖l1(Z)

p

≤ 1,

where the last inequality is due to (2.2). Hence, all these inequalities must be the equality and,since 1 < p < ∞, we conclude that all xj must vanish except one, which we denote x0. The l1

norm of x0 must be equal to 1 and, therefore, we have x0 ∈ E. Also, from the decomposition(2.4), we see that w(n) converges to zero in the strong topology and therefore x(n) → x0 strongly.Finally, from (2.6), it follows that F (x0) = I, which finishes the proof.

2.2 The general concentration-compactness principle

In this subsection, the concentration-compactness principle is given in more general settings,[Lio84a, Lio84b]. An analogous lemma to the one in the previous subsection is given below. Thecorrespondence of possible cases given in the following lemma and the decomposition (2.4) isclear from the Remark 2.4.

Lemma 2.6. Let {ρn}n≥1 be a sequence in L1(RN), satisfying

ρn ≥ 0,

∫RN

ρndx = 1.

Then there exists a subsequence, still denoted by {ρn}n≥1, satisfying one of the three followingpossibilities:

16

(a) (Compactness) For each n, there exists yn ∈ RN such that ρn (·+ yn) is tight, i.e.

∀ε > 0, ∃R <∞,∫yn+BR

ρn (x) dx ≥ 1− ε;

(b) (Vanishing) For all R <∞

limn→∞

supy∈RN

∫y+BR

ρn (x) dx = 0;

(c) (Dichotomy) There exists α ∈ (0, 1), such that for all ε > 0, there exists n0 ≥ 1, such thatfor all n ≥ n0, there exist non-negative functions ρ1

n, ρ2n ∈ L1

(RN)satisfying∣∣∣∣∫

RNρ1ndx− α

∣∣∣∣ ≤ ε, ∣∣∣∣∫RN

ρ2ndx− (1− α)

∣∣∣∣ ≤ ε anddist

(suppρ1

n, suppρ2n

)→∞ as n→∞,

such that the following holds

‖ρn −(ρ1n + ρ2

n

)‖L1(RN ) ≤ ε.

Proof. Consider the functions

Qn (r) = supy∈Rn

∫y+Br

ρn (x) dx.

(Qn)n is a sequence of nondecreasing, nonnegative, uniformly bounded functions on R+. Since∫RN ρndx = 1, one has that limr→∞Qn (r) = 1. This sequence has a pointwise convergentsubsequence, so there exists a nondecreasing, nonnegative function Q such that for all r ≥ 0

Qnk (r)→ Q (r) , k →∞.

It is clear that

limr→∞

Q (r) = α ∈ [0, 1].

If α = 0, then obviously the case (b) happens. If α = 1 it is possible to show that the case (a)happens and here we give a sketch of that argument. If one takes µ > 1

2 , then there exists R (µ)such that for all k ≥ 1 one has

Qnk (R) = supy∈RN

∫y+BR(µ)

ρnk (x) dx > µ.

Therefore, there exists asequence yk (µ) satisfying∫yk(µ)+BR(µ)

ρnk (x) dx > µ.

Also, there exist R(

12

)and a sequence yk = yk

(12

)for which∫

yk+BR( 12)ρnk (x) dx >

1

2.

17

Since µ > 12 and since one has that total mass of ρk for each k is equal to 1, i.e.

∫RN ρkdx = 1,

one concludes that balls B(yk(µ), R(µ)) and B(yk, R(

12

)) must intersect, i.e. |yk (µ) − yk| ≤

R(

12

)+ R (µ), for all µ ≥ 1

2 . If one takes R′ (µ) = R(

12

)+ 2R (µ), then one gets that the

sequence yk ∈ RN is such that for all k ≥ 1 and for all µ ≥ 12 the following holds∫

yk+BR′

ρnk (x) dx > µ,

which gives the case (a). Finally, if α ∈ (0, 1), the case (c) occurs. Let ε > 0 and choose R suchthat Q (R) > α− ε. Then for k large enough, one has α− ε < Qnk (R) < α+ ε. Moreover, onecan find Rk →∞ for k →∞ such that Qnk (Rk) ≤ α+ ε. So, there exists yk ∈ RN such that∫

yk+BR

ρnk (x) dx ∈ (α− ε, α+ ε) .

Then, if one takes ρ1k = ρnk1yk+BR and ρ2

k = ρnk1RN−(yk+BRk), it is clear that one gets the case(c), since ∫

RN

(ρnk − ρ

1k − ρ2

k

)dx =

∫R≤|x−yk|≤Rk

ρnk (x) dx

≤ Qnk (Rk)−Qnk (R) + 2ε

≤ (α+ ε)− (α− ε) + 2ε = 4ε.

By using this lemma, the goal is to minimize C1 real-valued functional F of the followingtype

F (u) =

∫RN

f (Au (x)) dx,

on some Hilbert function space H defined on RN . The constraint is defined with the help of theanother functional G, which is of similar type

G (u) =

∫RN

g (Bu (x)) dx,

such that G (0) = 0. Here, f and g are some real-valued functions on Rn and Rm, while A andB are some operators from H to another function space with values in Rn and Rm, invariant bythe translations of RN . Therefore, the functionals F and G are also invariant by the translationsof RN . Having this, the following minimization problem is considered

Iµ = inf {F (u) : u ∈ H,G (u) = µ} , (2.7)

for fixed µ > 0, so that I0 = 0.It can be checked that the sub-additivity condition

Iµ ≤ Iα + Iµ−α, ∀α ∈ [0, µ) (2.8)

always holds, see the argument [Lio84a, p. 113-114]. However, the concentration-compactnessprinciple states that all minimizing sequences of the problem (2.7) are relatively compact up toa translation if and only if the following strict sub-additivity condition holds

Iµ < Iα + Iµ−α, ∀α ∈ (0, µ) . (2.9)

18

Due to this relative compactness of all minimizing sequences, it is concluded that the solutionof (2.7) is attained.

That this condition (2.9) is necessary for the relative compactness of all minimizing sequencesis proved by the contraposition. If opposite of (2.9) is assumed, by (2.8) that means if Iµ =Iα + Iµ−α, ∀α ∈ (0, µ), holds, then it is possible to construct a minimizing sequence of (2.7)which is not relatively compact.

The proof of the other direction, i.e. the sufficiency of the condition (2.9) for the compactness,is based on the concentration-compactness lemma given above. It is used in order to show thatall minimizing sequences of (2.7) are relatively compact when the sub-additivity condition (2.9)holds. The lemma says that the only possible loss of compactness is due to the decompositionof the functions in H into at least two parts which are going infinitely away from each other.Since it can be shown that this possibility of failure of compactness is ruled out by the givenstrict inequality (2.9), see [Lio84a], the stated form of compactness is obtained.

We note here that for some particular problems it can be shown that sub-additivity condition(2.9) always holds. Also, for some other problems, this condition holds whenever the functionalF is homogeneous of some degree p ∈ (1,∞), the condition that we had in the Theorem 2.5from the previous subsection.

2.3 The orbital stability and the nonlinear Schrödinger equation

Here, we will see how the concentration-compactness method is used to prove the orbital stabilityof solitary waves for the nonlinear Schrödinger equation

i∂Φ

∂t(t, x) + ∆Φ (t, x) + |Φ (t, x) |p−1Φ (t, x) = 0, (t, x) ∈ R+ × RN ,

Φ (0, x) = Φ0 (x) , x ∈ RN ,(2.10)

where Φ0 is a complex-valued function in H1(RN). If p < 1 + 4

N , which is sub-critical case, it isknown result [GV75] that all solutions to (2.10) are global and bounded in H1. These solutionsverify the following two conservation laws

‖Φ (t, ·) ‖L2(RN ) = ‖Φ0‖L2(RN ), (2.11)

F (Φ (t, ·)) = F (Φ0) , (2.12)

for all t ≥ 0, where

F (Φ) =1

2

∫RN|∇Φ (t, x) |2dx− 1

p+ 1

∫RN|Φ (t, x) |p+1dx.

We would like to check stability of some special solutions of the equation (2.10). Thosesolutions are so-called solitary waves and they take the form

Φ (t, x) = eiλtu (x) , λ ∈ R,

where u (x) solves the equation

−∆u+ λu = |u|p−1u in RN .

Among these, we are interested in those of the lowest energy F , so-called ground states, forwhich it is assumed that

‖u‖L2(RN ) = µ,

19

for a given µ > 0. Therefore, we are interested in the solutions of the following minimizationproblem

Iµ = inf{F (u) : u ∈ H1

(RN), ‖u‖L2(RN ) = µ

}. (2.13)

Theorem 2.7. For given p < 1 + 4N and µ > 0, every minimization sequence of (2.13) is

relatively compact in H1 up to a translation in RN .Moreover, every solution u of the minimization problem (2.13) is of the form u (·) = eiθu0 (·+ y)

for some θ ∈ R, y ∈ RN , where u0 is the radially symmetric and positive solution of (2.13).

The proof of this theorem can be found in [CL82]. The first part says that Iµ from (2.13)is attained. It is just a consequence of the concentration-compactness principle described in theprevious subsection, since it can be proved that sub-additivity condition (2.9) for the problem(2.13) is always true. That is because for this problem it can be shown that 0 < Iµ = µ2/pI1, formore details see [Lio84b]. The second part describes the set of the ground solitary states, i.e.the set of the solutions of (2.13), which we denote by Sµ. The orbital stability of these solutionsin the set Sµ is a direct consequence of this result of relative compactness, and it is given in thefollowing theorem, which can be also found in [CL82].

Theorem 2.8. Let p < 1 + 4N and µ > 0. Then a ground state solitary u ∈ Sµ is orbitally

stable, i.e. for all ε > 0 there exists δ > 0 such that for any initial data Φ0 ∈ H1(RN)with

infu∈Sµ

‖u− Φ0‖H1(RN ) < δ,

there exists x(t) ∈ RN , such that the corresponding solution Φ (t, x) of (2.10) satisfies

infu∈Sµ

‖u− Φ (t, ·+ x(t)) ‖H1(RN ) < ε,

for all t ≥ 0.

Proof. This can be proved by the contradiction. If the claim were not true, then there wouldexist ε > 0, Φn

0 ∈ H1(RN)and tn ≥ 0 such that for all δ > 0 and all x(tn) ∈ RN

infu∈Sµ

‖u− Φn0‖H1(RN ) < δ and inf

u∈Sµ‖u− Φn (tn, ·+ x(tn)) ‖H1(RN ) > ε0 (2.14)

hold. Since u ∈ Sµ, we have that µ− δ < ‖Φn0‖L2(RN ) < µ+ δ, for all δ > 0, and moreover that

F (Φn0 )→ Iµ = F (u) and ‖Φn

0‖L2(RN ) → µ, as n→∞.

Therefore, by using the conservation laws (2.11) and (2.12), we see that we have constructedthe minimizing sequence {Φn (tn, ·)} of the problem (2.13). This sequence is relatively compactin H1 up to a translation, by the previous theorem, and hence

infu∈Sµ

‖u− Φn (tn, ·+ x(tn)) ‖H1(RN ) → 0, as n→∞, (2.15)

which contradicts (2.14).

20

2.4 The orbital stability and the Vlasov–Poisson system

Here, we will see how the concentration-compactness principle can be used to prove the orbitalstability of some steady states of the gravitational Vlasov–Poisson system (1.5)–(1.6) in thespace E given by (1.11). Mainly, we follow the results of the work [LMR08], where they make aparallel with the nonlinear Schrödinger equation.

The orbital stability is shown for the isotropic polytropes defined as

f0 (x, v) = F (E) = (−1− E)1p−1

+ =

(1− |v|

2

2− φf0

) 1p−1

+

(2.16)

for 9/7 = pcrit < p ≤ +∞. First, they are characterized as the ground states of the minimizationproblem given in the following theorem.

Theorem 2.9. Let f0 be defined with (2.16). Then the minimization problem

minf∈E,f 6=0

‖|v|2f‖12

Lp(R6)‖f‖

p3(p−1)

L1(R6)‖f‖

7p−96(p−1)

L1(R6)

‖∇xφf‖2L2(R3)

is attained on the family with four parameters

γf0

(x− x0

λ, µv

), γ ∈ R∗+, λ ∈ R∗+, µ ∈ R∗+, x0 ∈ R3.

By this theorem, the polytropic models (2.16) are characterized in terms of a best constantin the interpolation inequality given in the Theorem 1.1. Therefore, it is straightforward to seethat the polytropic models are actually the ground states of the minimization problem of theform

inf{H (f) : f ∈ E , ‖f‖L1(R6) = M1, ‖f‖Lp(R6) = Mp

}, (2.17)

for given M1,Mp > 0.Next, by using the concentration-compactness principle, it is possible to show the compact-

ness of the minimizing sequence of the problem (2.17), in even more general form. Namely, forpcrit < p < +∞ considered is the minimization sequence of the the problem

I (M1,Mj) = inf

{H (f) : f ∈ E ,

∫R6

f = M1,

∫R6

j (f) = Mj

}, (2.18)

where j is a strictly convex continuous non-negative function on R+ with

∀t ≥ 0, j(t) ≥ Ctp and limt→0

j(t)

t= 0.

The claim is that if the following sub-additivity condition holds

I (M1,Mj) < I (αM1, βMj) + I ((1− α)M1, (1− β)Mj) , (2.19)

for all α ∈ (0, 1) and β ∈ (0, 1), then every minimizing sequence of (2.18) is relatively compactup to a translation in the energy space E . The result is a bit different for the case p = ∞, see[LMR08].

In the [LMR08], they also show the sufficient condition for (2.19) to happen. The sub-additivity inequality (2.19) is satisfied if either

21

(i) j is a polytrope

j (f) = fp, for pcrit < p < +∞,

or

(ii) there exists 32 < p1 ≤ p2 < +∞ such that

∀t ≥ 0, ∀b ≥ 1, bp1j(t) ≤ j (bt) ≤ bp2j(t).

Having the above mentioned relative compactness result, now it is possible to show theorbital stability in the energy space E of the minimizers of (2.18). The derivation of this resultis analogous to the one given in the Theorem 2.8 and it is stated in the following theorem from[LMR08].

Theorem 2.10. Let pcrit < p < +∞. The polytrope f0 defined by (2.16) is orbitally stable, i.e.for all ε > 0 there exists δ > 0 such that for any initial data fin ∈ E with

H (fin)−H(f0)≤ δ, ‖fin‖L1 ≤ ‖f0‖L1 + δ and ‖fin‖Lp ≤ ‖f0‖Lp + δ,

there exists a translation shift x(t) ∈ RN , such that the solution f (t, x, v) of (1.5)–(1.6) withthe initial data fin satisfies

‖f0 − f (t, x+ x(t), v) ‖E < ε,

for all t ≥ 0.

3 Nonlinear stability by rearrangements

In this section, we give the results from the work of [LMR11a], where they prove the nonlin-ear stability of spherically-symmetric, isotropic and decreasing galaxy models, under generalperturbations. Previously, in [LMR11b], they resolved this question of stability in the caseof anisotropic models under spherical perturbations. While in [LMR11b], they consider gen-eral perturbations but only for the isotropic case. The question of the nonlinear stability ofanisotropic, spherically-symmetric and decreasing galaxy models, under general perturbations,remains open. They deal with the solutions of (1.5)–(1.6) which are in the energy space E , wherethe energy conservation is replaced by the inequality (1.12). In this space, we have already seenthat nonlinear stability of specific subclasses of steady states, such as polytropes, can be provedas a direct consequence of concentration-compactness principle. However, here, the more generalmodels are considered. The theorem they give in [LMR11b] is the following one.

Theorem 3.1. Suppose f0 = F (E) ≥ 0 is a continuous stationary solution, nonzero, withcompact support, of the system (1.5)–(1.6), for which there exists E0 < 0 such as F (E) = 0 forE ≥ E0, F is C1 on (−∞, E0) and F ′ < 0 on (−∞, E0). Then f0 is orbitally stable, i.e. for allM > 0 and ε > 0 there is ν > 0 such as, for any initial data

fin ∈ E ={g ≥ 0, g ∈ L1 ∩ L∞

(R6), |v|2g ∈ L1

(R6)}

such that

‖fin − f0‖L1(R6) ≤ ν, H (fin) ≤ H(f0)

+ ν, ‖fin‖L∞(R6) ≤ ‖f0‖L∞(R6) +M,

then any weak solution for given initial data like this verifies

∀t ≥ 0,

∫R6

∣∣(1 + |v|2) (f (t, x, v)− f0 (x− z(t), v)

)∣∣ dxdv ≤ ε.22

We will give the detail outline of the proof of this theorem as it is given in [Mou12]. However,before going into details, let us give just the strategy of the proof. The proof is based onthe notion of rearrangements, which we discuss in the next subsection. The first step of theproof is to show that the Hamiltonian has a monotonicity property under the general symmetricrearrangements. Then, the reduced Hamiltonian is introduced and for the corresponding reducedminimization problem, certain coercivity inequality is shown. This coercivity inequality is thenused to prove compactness of «generalized» minimizing sequences. Finally, the compactnessargument gives the base for the contradiction used to prove Theorem 3.1, which we give in thelast subsection.

3.1 Rearrangement according to the microscopic energy

First, let us define the symmetric rearrangements of a set and of a function, as they are definedin [LL01, Chapter 3].

Definition 3.2. Given a measurable set A ∈ R6, its symmetric rearrangement A∗ is an openball centered at zero

A∗ = {x : |x| < r}

with r chosen such that they are of the same volume, i.e. |A| = |A∗|.

Definition 3.3. Given a non-negative integrable function f : R6 → R+, its symmetric rear-rangement f∗ is a non-negative function whose upper level sets are obtained by the symmetricalrearrangements of the corresponding level sets of f , which gives the following formula

f∗ (x, v) =

∫ +∞

01∗{f(x,v)≥s} (x, v) ds, where 1∗A = 1A∗ . (3.1)

The rearrangement f∗ is radially symmetric, decreasing and equimeasurable to f . Next, theelementary property of the symmetrical rearrangements is given.

Lemma 3.4. For f ≥ 0 integrable on R6 we have∫R6

f∗ (x, v) | (x, v) |R6dxdv ≤∫R6

f (x, v) | (x, v) |R6dxdv,

where | (x, v) |R6 denotes the norm on R6.

Proof. Since for every measurable set A ∈ R6 one has |A|R6 =∫R6 1Adxdv, if there is another

measurable set B ∈ R6, such that |A| ≤ |B|, the following inequality holds∫R6

1A1Bdxdv = |A ∩B|R6 ≤ |A|R6 = |A∗|R6 = |A∗ ∩B∗|R6 =

∫R6

1∗A1∗Bdxdv.

By using this, the definition of the rearrangement (3.1) and the layer-cake representation of afunction given by the formula

f (x, v) =

∫ ∞0

1{f(x,v)≥s} (x, v) ds, (3.2)

one gets ∫R6

f1|(x,v)|R6≤sdxdv ≤∫R6

f∗1∗|(x,v)|R6≤sdxdv,

23

and since 1∗|(x,v)|R6≤s= 1|(x,v)|R6≤s for every s ≥ 0 given, one also has∫

R6

f1|(x,v)|R6≤sdxdv ≤∫R6

f∗1|(x,v)|R6≤sdxdv.

From∫R6 fdxdv =

∫R6 f

∗dxdv and by rewriting the expression∫R6 f

∗1|(x,v)|R6>sdxdv as∫R6

f∗(

1− 1|(x,v)|R6≤s

)dxdv,

one obtains ∫R6

f∗1|(x,v)|R6>sdxdv ≤∫R6

f1|(x,v)|R6>sdxdv.

Finally, by integrating this with respect to s ∈ [0,+∞] and using (3.2), one deduces the result.

Now, in the similar manner, we introduce the rearrangement with respect to the microscopicenergy

Eφ :=

(|v|2

2+ φ (x)

),

for a given potential φ on R3.

Definition 3.5. Given a measurable set A ⊂ R6, its symmetric rearrangement A∗φ with respectto the microscopic energy Eφ is the open (energy) ball

A∗φ = {(x, v) : Eφ (x, v) < EA},

where EA is selected such that |A∗φ| = |A|.

Definition 3.6. Given a non-negative integrable function f with a compact support on R6,its symmetric rearrangement f∗φ with respect to the microscopic energy Eφ is the non-negativefunction f∗φ on R6 whose level sets are obtained by rearrangement with respect to the microscopicenergy of corresponding upper level set of f , which gives the following formula

f∗φ (x, v) =

∫ +∞

01∗φ{f≥s}ds with 1∗φA = 1A∗φ .

From the definition, the function f∗φ is a function of Eφ, with the compact support, decreas-ing in the microscopic energy Eφ, and equimeasurable to f . In the following lemma, given isthe property which proof follows by the similar reasoning as we had in the previous lemma.

Lemma 3.7. For the function f ≥ 0 integrable on its compact support in R6, the following istrue ∫

R6

f∗φEφ (x, v) dxdv ≤∫R6

f (x, v)Eφ (x, v) dxdv.

We will use the rearrangement of f with respect to the microscopic energy created by thefunction f itself, which we denote f̂ = f∗φf where, as before, φf = − (1/ (4π|x|)) ∗ ρf . We seethat we obtain an rearrangement operation f → f̂ highly non-linear. Note immediately that thespherical stationary solution f0 is a fixed point of this non-linear rearrangement, i.e.

f̂0 =(f0)∗φf0 = f0.

24

3.2 Monotonicity of the Hamiltonian and the reduced Hamiltonian

Firstly, the following functional is introduced

Jf (φ) := H(f∗φ) +1

2‖∇xφ−∇xφf∗φ‖2L2(R3)

and then the following monotonicity of Hamiltonian with respect to rearrangements is proved.

Theorem 3.8. If f ∈ E and f̂ = f∗φf , then

H (f) ≥ Jf (φf ) ≥ H(f̂)

with equality if and only if f = f̂ .

Proof. The right inequality follows from the definition of the reduced Hamiltonian. By thesimple calculation, for any two functions f, g ∈ E we have

H (f) = H (g) +1

2‖∇xφf −∇xφg‖2L2(R3) +

∫R6

(|v|2

2+ φf

)(f − g) dxdv. (3.3)

Therefore, for g = f̂

H (f) = Jf (φ) +

∫R6

(|v|2

2+ φf

)(f − f̂)dxdv. (3.4)

The equality case is clear and the left inequality is concluded just by applying the previouslemma to ∫

R6

(|v|2

2+ φf

)(f − f̂)dxdv,

which then has to be non-negative, since every rearrangement is non-negative function.

We introduce now the functional J (φ) := Jf0 (φ), which we call the reduced Hamiltonian.Note that it just depends on the potential. It turns out that

J(φf0)

= H(f0),

and that

H (f)−H(f0)

+ C‖f∗ −(f0)∗ ‖L1 ≥ J (φ)− J

(φf0). (3.5)

Hence, it is reasonable to pursuit the local minimum φf0 of the reduced Hamiltonian J , in orderto show that f0 is the local minimum of H when

∥∥f∗ − (f0)∗∥∥

L1 = 0. Therefore, we have thereduced variational problem, which involves the functional J depending on the potential only.

3.3 Coercivity inequality for the reduced Hamiltonian

Now, given is the study of convexity properties of the functional J in the vicinity of φf0 . Bythe following theorem, it is shown that φf0 is a local minimum of J . But first, let us define aspace of eligible potentials in which the statement will hold.

X :=

{φ ∈ C0

(R3)

: φ ≤ 0, lim∞φ = 0,∇xφ ∈ L2

(R3), infx∈R3

(1 + |x|) |φ (x) | > 0

}.

25

Theorem 3.9. There exist the constants c0, δ0 > 0 and a continuous map φ→ zφ from H1(R3)

into R3 such that for φ ∈ X with

infz∈R3

[‖φ− φf0 (· − z) ‖L∞(R3) + ‖∇xφ−∇xφf0 (· − zφ) ‖L2(R3)

]< δ0,

the following holds

J (φ)− J(φf0)≥ c0‖∇xφ−∇xφf0 (· − zφ) ‖2L2(R3).

Proof. Here, given is the outline of the proof. For more details see [LMR11b].First step of the proof is to write the Taylor expansion of the functional J in the vicinity

of the potential φf0 . It turns out that the first variation of J is zero at φf0 and that Taylorexpansion of order two is as follows

J (φ)− J(φf0)

=1

2D2J

(φ− φf0 , φ− φf0

)+ o

(‖φ− φf0‖L∞(R3)

)‖∇xφ−∇xφf0‖2L2(R3)

where

D2J(φf0)

(h, h) =

∫R6

|∇xh|2dx−∫R6

|F ′ (E) |[h (x)− (Ph) (x, v)]2dxdv,

with E = Eφf0 = |v|22 + φf0 (x) and the projection operator P defined as

(Ph) (x, v) :=

∫R3

(E − φf0 (y)

)1/2+

h (y) dy∫R3

(E − φf0 (y)

)1/2+

dy.

Next step of the proof is to show that D2J(φf0)

(h, h) is coercive, up to the degeneracyinduced by the translation invariance. To do so, the following operator is defined

Lh := −∆xh−∫R3

|F ′ (E) | (1− P) dv,

for which one has

〈Lh, h〉L2(R3) = D2J(φf0)

(h, h) .

Therefore, the problem is reduced to the study of the operator L. There is the following Lemmadescribing this operator.

Lemma 3.10. The operator L is positive, it is a compact perturbation of laplacian on H1(R3),

its kernel is given by

Ker (L) = Span{∂xiφf0 , 1 ≤ i ≤ 3

}and the inequality

〈Lh, h〉L2(R3) ≥ c0‖∇xh‖2L2(R3) −1

c0

3∑i=1

(∫R3

h∆x

(∂xiφf0

)dx)2

(3.6)

holds for all h ∈ H1(R3)and some constant c0 > 0.

This lemma is proved by the decomposition of h into radial part and its orthogonal comple-ment

h = h0 + h1, h0 ∈ H1rad

(R3), h1 ∈

(H1rad

(R3))⊥

.

Note that in this case when L depends on the radial part, Ph1 = 0 and L reduces to Schrödingeroperator.

The completion of the proof of Theorem 3.9 is then done by canceling the defects of coercivity,i.e. the negative terms in (3.6).

26

3.4 Compactness of the minimizing sequence

It is possible to show the compactness of certain generalized minimizing sequences fn in thefollowing sense.

Theorem 3.11. If (fn)n≥0 is the sequence in E satisfying

supn≥0

infz∈R3

[∥∥φfn − φf0 (· − zφfn)∥∥L∞(R3)+∥∥∇xφfn −∇xφf0 (· − zφfn)∥∥L2(R3)

]< δ0

and

limn→∞

‖f∗n − (f0)∗‖L1(R6) = 0, lim infn→∞

H (fn) ≤ H(f0),

then

limn→∞

∫R6

∣∣(1 + |v|2) (fn (x, v)− f0

(x− zφfn , v

))∣∣ dxdv = 0.

Proof. First of all, notice that from the coercivity inequality proved in the Theorem 3.9, fromthe assumptions made here and the inequality (3.5), it follows that

limn→∞

‖∇xφfn −∇xφf0(· − zφfn )‖2L2(R3) = 0. (3.7)

We introduce the notation f̄n (x, v) = fn(x+ zφfn , v

)and return to the complete Hamiltonian

by using the equality we already had

H (f)−H(f0)

+1

2‖∇xφf̄n −∇xφf0‖2L2(R3) =

∫R6

Eφf0(f̄n − f0

)dxdv. (3.8)

Since the hypothesis on the potential and the convergence (3.7) hold, it is possible to conclude

lim supn→∞

(H(f̄n)−H

(f0))

= lim supn→∞

(H (fn)−H

(f0))≤ 0

and

‖∇xφf̄n −∇xφf0‖2L2(R3) = ‖∇xφfn −∇xφf0(· − zφfn )‖2L2(R3) → 0, as n→∞.

Now, from (3.8) one derives

lim supn→∞

∫R6

Eφf0(f̄n − f0

)dxdv ≤ 0.

By using the monotonicity of the rearrangement from the Lemma 3.7, the fact that the rear-rangement is non-negative function and that f0 is a fixed point of the operator f̂ , we get

limn→∞

∫R6

Eφf0

(f0 − f̄

∗φf0n

)dxdv = 0

from where, by the monotonicity of the rearrangement, we deduce

lim supn→∞

∫R6

Eφf0

(f̄n − f̄

∗φf0n

)dxdv ≤ 0.

27

By using the monotonicity of the rearrangement once again, it is possible to conclude that thequantity above is also greater or equal to zero, which gives us equality

limn→∞

∫R6

Eφf0

(f̄n − f̄

∗φf0n

)dxdv = 0.

Now, by the assumption limn→∞ ‖f∗n − (f0)∗‖L1(R6) = 0 and by using again that f0 is a fixedpoint of the operator f̂ , we obtain

limn→∞

‖fn − f0‖L1(R6) = 0.

To complete the claim of the proof, we still need to prove the convergence of the kinetic energy,i.e. to prove that

limn→∞

∫R6

|v|2fndxdv =

∫R6

|v|2f0dxdv.

But this follows from the convergence of the potential (3.7) and assumption on the Hamiltonian.

3.5 Completion of the proof of Theorem 3.1

Having all the lemmas and theorems from the previous subsections, finally we can prove theclaim form the Theorem 3.1. The proof is given by the contradiction. Let us suppose opposite,i.e. f0 = F (E) is not orbitally stable. Then, there would exist M > 0, ε > 0, tn > 0 andfnin ∈ E , such that for all n ∈ N

‖fnin − f0‖L1(R6) ≤1

n, H (fnin) ≤ H

(f0)

+1

n, ‖fnin‖L∞(R6) ≤ ‖f0‖L∞(R6) +M (3.9)

and the weak solution fn for the given initial data fnin verifies∫R6

∣∣(1 + |v|2) (fn (tn, x, v)− f0 (x− z(tn), v)

)∣∣ dxdv > ε. (3.10)

Since the conservation laws of Casimir functionals and Hamiltonian H hold for the solutionof Vlasov–Poisson system (1.5)–(1.6), we have constructed the sequence (fn)n∈N in E whichsatisfies the conditions of the Theorem 3.11. The condition on ‖ (fn)∗− (f0)∗‖L1(R6) is satisfiedbecause the first inequality in (3.9) and since ‖ (fn)∗ − (f0)∗‖L1(R6) ≤ ‖fn − f0‖L1(R6) holds.The condition on Hamiltonian is satisfied because of the second inequality in (3.9), while thecondition on the potential holds because of third inequality in (3.9). Therefore, we have theconvergence stated in the Theorem 3.11 which is the contradiction with (3.10).

Let us note here, what also can be noted at the end of the Theorem 2.10, that we proved theorbital stability of f0 in the sense given in the Theorem 3.1, respectively in the Theorem 2.10.However, the orbital stability given by (1.15) also holds, since the interpolation inequality fromthe Theorem 1.1 gives the control of the potential energy by the kinetic energy.

28

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