Jeans’s Theorem in Kinetic Theory Min Harold... · widely used in such scienti c disciplines as plasma physics or stellar dynamics. The Vlasov-Poisson system consists of a Vlasov

Jeans’s Theorem in Kinetic Theory

Student: Zhuo Min ‘Harold’ LimSupervisor: Jonathan Ben-Artzi

30 May, 2014

Contents

1 Introduction 11.1 A Heuristic Derivation of the Vlasov-Poisson System . . . . . . . . . . . . . . . 21.2 The Cauchy Problem for the Vlasov-Poisson System . . . . . . . . . . . . . . . 31.3 Scope of the Essay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Jeans’s Theorem 42.1 Existence of Steady Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 A Rigorous Version of Jeans’s Theorem . . . . . . . . . . . . . . . . . . . . . . 82.3 Failure of Jeans’s Theorem for the Vlasov-Einstein System . . . . . . . . . . . . 10

3 Steady Solutions from Casimir Functionals 133.1 The Energy-Casimir Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 An Early Stability Result of Camm Steady States . . . . . . . . . . . . . . . . 153.3 A Reduction Approach for Isotropic Steady States . . . . . . . . . . . . . . . . 183.4 An Unsuccessful Attempt at Anisotropic Steady States . . . . . . . . . . . . . . 243.5 More Recent Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A Rearrangements of Functions 35

1 Introduction

The Vlasov-Poisson system is one of the fundamental systems in kinetic theory. It describesa very large, dilute collection of particles interacting with each other only via a Newtonianforce; the effects of collisions or other body forces are assumed to be negligible. As such, it iswidely used in such scientific disciplines as plasma physics or stellar dynamics.

The Vlasov-Poisson system consists of a Vlasov equation defining the evolution of a phase-space density function f = f(t, x, v), with the forcing term being the negative gradient of theNewtonian potential of the spatial density function ρ[f ]. It reads

∂f

∂t+ v · ∇xf −∇U[f(t)] · ∇vf = 0

where, for g = g(x, v) ∈ L1(R6), U[g] : R3 → R satisfies Poisson’s equation

4U[g] = 4πγρ[g] , lim|x|→∞

U[g](x) = 0 .

with ρ[g] : R3 → [0,∞) denoting the spatial density function of g, i.e.

ρ[g](x) :=

∫R3

g(x, v) dv .

1

1 INTRODUCTION Page 2 of 38

Here, γ ∈ ±1 defines the nature of the force. The case γ = +1 describes an attractive forcesuch as that of Newtonian gravitation; we henceforth refer to this as the gravitational case.The case γ = −1 describes a repulsive force such as Coulomb repulsion between like charges,and will henceforth be called the plasma physics case.

Various generalisations and alternatives of the Vlasov-Poisson system are possible, andsome have been intensively studied as well. For example, it may be desired to study plasmacomprising more than one species of particles, with differing charges, in which case one couldconsider a Vlasov-Poisson system comprising a Vlasov equation for each particle species, anda Poisson equation describing the interaction between different particle species. If collisionsare not longer negligible, it may be desired to include a Boltzmann collision term in thesystem. Finally, there also exist other Vlasov-type system, such as the Vlasov-Maxwell systemdescribing collisionless systems with electromagnetic interaction, or the Vlasov-Einstein systemin case of a spacetime with nontrivial curvature. With the exception of Subsection 2.3, we donot consider these systems in this essay.

1.1 A Heuristic Derivation of the Vlasov-Poisson System

In this subsection we sketch a very formal and heuristic derivation of the Vlasov-Poisson systemfrom Hamiltonian dynamics. The reader is warned that this subsection contains no rigorousmathematics, and is included only to motivate the study of the Vlasov-Poisson system. Indeed,the rigorous derivation of kinetic equations from particle dynamics is a difficult subject, inwhich there has been and still is active research.

Consider a system of N 1 point particles of equal mass (which we may assume tobe 1

N , by making a suitable choice of units), interacting with each other via a potential ψand also with an external potential φ. A state of the system can be described by a point(X,V ) ∈ R3N × R3N , where X = (x1, . . . , xN ) ∈ (R3)N are the positions of the particles andV = (v1, . . . , vN ) ∈ (R3)N are the velocities of the particles. The Hamiltonian of the systemis

H(X,V ) =

N∑i=1

1

2|vi|2 +

1

2

∑i6=j

ψ (xi − xj) +

N∑i=1

φ (xi) ,

and the system evolves according to Hamilton equations of motions

dxidt

=∂H

∂vi(X,V ) ,

dvidt

= −∂H∂xi

(X,V ) .

Kinetic theory involves replacing Hamiltonian dynamics by time-evolution of a probabilitydensity function on phase space R6N . Specifically, the phase-space density function for Nparticles, fN (t) : R6N → [0,∞), should evolve according to

∂fN∂t

+

N∑i=1

(∂H

∂vi· ∂fN∂xi

− ∂H

∂xi· ∂fN∂vi

)= 0 .

For our choice of Hamiltonian, this gives

∂fN∂t

+

N∑i=1

vi ·∂fN∂xi

−N∑i=1

∇φ(xi) ·∂fN∂vi−∑i 6=j

∇ψ(xi − xj) ·∂fN∂vi

= 0 .

To get further, we now make the assumption of chaos: for very large N , the particles arealmost independently distributed; we mention that this is the usual practice in kinetic theory,and that the rigorous derivation of the chaos property in kinetic equations from microscopicparticle dynamics can be performed for many models, but is by no means straightforward.Thus we plug in an ansatz

fN = f⊗N = f ⊗ · · · ⊗ f

1 INTRODUCTION Page 3 of 38

for some single-particle phase-space density function f(t) : R6 → [0,∞). With this ansatz,taking the limit N →∞ in an appropriate way gives

∂f

∂t+ v · ∂f

∂x−∇φ(x) · ∂f

∂v−∫R3

∫R3

∇ψ(x− y)f(y, w) · ∂f∂v

(x) dw dy = 0 .

In particular, if there is no external field, i.e. φ ≡ 0, and ψ is the Newtonian potential

ψ := − γ

| · |,

then we recover the Vlasov-Poisson system.

1.2 The Cauchy Problem for the Vlasov-Poisson System

A starting point for the mathematical investigation of any evolutionary PDE or PDE system isthe study of its Cauchy problem, i.e. issues of existence and uniqueness. The Vlasov-Poissonsystem is one of the few nonlinear systems of mathematical physics for which there is a reason-ably complete theory for its Cauchy problem. Global existence results were obtained around1989, by Pfaffelmoser [14] and independently by Lions and Perthame [12]. Pfaffelmoser’s proofwas later considerably simplified by Schaeffer [23]. A very general uniqueness result has alsobeen obtained by Loeper [13] based on the theory of optimal transport.

Theorem 1.1 (Lions, Perthame). Fix a choice of γ. Let 0 ≤ f0 ∈ L1 ∩ L∞(R6) be such thatthere exists s0 > 3 so that∫

R3

∫R3

|v|sf0(x, v) dv dx < +∞ , for all s ∈ [0, s0) .

Then there exists a solution f ∈ C0([0,∞), Lp(R6)) ∩ L∞([0,∞), L∞(R6)) to the Vlasov-Poisson system with f(0) = f0, and satisfying

supt∈[0,T ]

∫R3

∫R3

|v|sf(t, x, v) dv dx < +∞ , for all s ∈ [0, s0)

for any T > 0.

The result of Schaeffer assumes an initial datum of class C1c (R6), but guarantees the existence

of a global classical solution which at every time is also of class C1c (R6), and moreover provides

an explicit quantitative bound on how quickly the velocity support can grow. More specifically,we have the following.

Theorem 1.2 (Schaeffer). Fix a choice of γ. Any initial datum 0 ≤ f0 ∈ C1c (R6) will launch

a global classical solution f : [0,∞) × R6 → [0,∞) to the Vlasov-Poisson system such thatf(t) ∈ C1

c (R6) for every t ≥ 0.Moreover, for any p > 33

17 , the solution f satisfies

1 + sup|v|∣∣∣ ∃τ ∈ [0, t], x ∈ R3 : f(τ, x, v) > 0

≤ C(f0, p) (1 + t)

p

where C(f0, p) > 0 is some constant depending only on f0 and p.

The impressive result of Loeper asserts uniqueness in a very general class of solutions, namelythe class of measure-valued solutions in the sense of distribution. In the following statement ofLoeper’s result, (M+(R6),w∗) denotes the set of bounded positive measures on R6, endowedwith the weak σ(C0

b(R6),M+(R6)) topology.

Theorem 1.3 (Loeper). Fix a choice of γ. Given f0 ∈ M+(R6) and T > 0, there can existat most one solution in C([0, T ), (M+(R6),w∗)), starting at f0, satisfying the Vlasov-Poissonsystem in the sense of distributions.

We omit the proofs of the above results as they are beyond the scope of this essay; theinterested reader can find them in the papers cited above.

2 JEANS’S THEOREM Page 4 of 38

1.3 Scope of the Essay

In this essay we are concerned with steady (i.e. time-indepedent) solutions to the Vlasov-Poisson system, of finite mass and compact support, satisfying the functional form

f(x, v) = F

(1

2|v|2 + U[f ](x) , |x ∧ v|2

)(1)

for some function F = F (E,L) : R2 → [0,∞).Our interest in this ansatz is due to the so-called “Jeans’s theorem” which is often quoted

in the astrophysics literature, as a starting point for any discussion. This is the statementthat steady solutions to Vlasov-Poisson system can be expressed in terms of first integralsof motion, i.e. quantities which are conserved along the characteristic curves (X,V ) to theVlasov-Poisson system,

X = V , V = −∇U[f ](X) .

Such claims possibly refer to a paper [7] by J. H. Jeans, pioneering the application of kineticmodels to the study of stellar dynamics. There (see pp. 79ff) he made the assertion that thephase-space density function is really a function of these first integrals of motion, and thenexamined a few examples where the phase-space density function is a function of energy density12 |v|

2+U[f ](x) and angular momentum density x∧v. Although he mentioned the example of an“ellipsoidal universe” where another first integral was used instead of the angular momentumdensity, he dismissed it as artificial and of no physical interest. In short, he suggested that allphysically relevant steady solutions to the Vlasov-Poisson system should be functions only oftotal energy density and angular momentum density.

The ansatz (1) is even more restrictive than the one considered by Jeans, since the phase-space density function f depends only on the magnitude of the angular momentum density,not on its direction. This suggests that our ansatz describes only particle systems with a veryhigh degree of symmetry, e.g. spherically-symmetric solutions.

This essay is divided into two parts. In the first, we will discuss Jeans’s theorem andprovide a proof in the case of spherically-symmetric solutions; we will also outline a proof thatthe analogous statement does not hold for the Vlasov-Einstein system. In the second part,we will follow a framework, originally due to G. Rein, to identify some of these solutions asminimizers of certain energy-Casimir functionals. No result listed outside of Subsection 3.4 isdue to original work by the author.

1.4 Acknowledgements

The author would like to thank his supervisor Dr. Jonathan Ben-Artzi for suggesting thismini-project, and for being a very helpful and encouraging mentor. This work was supportedby a graduate research scholarship at St. John’s College, Cambridge, and also by the UKEngineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for theUniversity of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis.

2 Jeans’s Theorem

2.1 Existence of Steady Solutions

Before embarking on studying steady solutions having the functional form (1), it is natural toask if steady solutions are known to exist in the first place. Energy estimates provide rathersevere constraints on whether steady states can exist.

Definition 2.1. Fix a choice of γ. For f ∈ L1+(R6), define

Ekin(f) :=

∫R3

∫R3

1

2|v|2f(x, v) dv dx

Epot(f) := − γ

8π

∫R3

∣∣∇U[f ](x)∣∣2 dx .


We call them the total kinetic energy and total potential energy of f respectively.

We now quote the following growth and decay estimates, whose proofs the reader can findin [20], Section 1.7.

Proposition 2.2. For the plasma physics case γ = −1, any (global classical) solution f withf(0) ∈ C1

c (R6) to the Vlasov-Poisson system satisfies, for t > 0, the estimates∥∥∇U[f(t)]

∥∥L2 ≤ C (1 + t)

− 12∥∥ρ[f(t)]

∥∥L5/3 ≤ C (1 + t)

− 35

where C > 0 is some constant depending only on f(0).

Proposition 2.3. For the gravitational case γ = 1, suppose f is a (global classical) solutionto the Vlasov-Poisson system with f(0) ∈ C1

c (R6) having positive total energy,

Ekin(f(0)) + Epot(f(0)) > 0 .

Then

C1t2 ≤

∫R3

∫R3

|x|2f(t, x, v) dv dx ≤ C2t2

where C1, C2 > 0 are constants depending only on f(0).

Using the above estimates we deduce that steady solutions to the Vlasov-Poisson system,in the plasma physics case, are impossible in the absence of external fields or backgroundradiation; and in the gravitational case they can exist only with negative total energy. Hencewe make the following convention.

Convention 2.4. Henceforth, in the rest of this essay, unless explicitly stated otherwise, wewill consider only the Vlasov-Poisson system in the gravitational case γ = 1.

On the other hand, families of steady solutions, necessarily of negative total energy, havebeen constructed in the gravitational case. The paper [1], aside from providing a rigorousproof of Jeans’s theorem, also provides a framework in which certain ansatzes can be shownto lead to steady states which are spherically symmetric, in the following sense.

Definition 2.5. We say a function f : R6 → R is spherically symmetric if f(Ax,Av) = f(x, v)for all rotations A ∈ SO(3).

Notice that a spherically symmetric function is therefore a function only of the variables

r := |x| , w :=x · vr

, L := |x ∧ v|2 .

For simplicity we will ignore the set r = 0 or L = 0 in phase space, i.e. we assume thephase-space density function f is defined only on (x, v) ∈ R6 | x ∧ v 6= 0.

We make the following conventions.

Convention 2.6. In this subsection and the next, we consider a steady, spherically symmetricclassical solution 0 ≤ f = f(x, v) ∈ C1(R6)∩L1(R6) to the Vlasov-Poisson system. We expresssuch a solution as f(x, v) = ϕ(r, w, L) on (r, w, L) ∈ (0,∞)× R× (0,∞).

We also write U[ϕ] = U[ϕ](r) for the solution of Poisson’s equation

1

r2

(r2U ′[ϕ](r)

)′= 4πρ[ϕ](r) , lim

r→∞U[ϕ](r) = 0 ,

where ρ[ϕ] = ρ[ϕ](r) is

ρ[ϕ](r) =π

r2

∫ ∞−∞

dw

∫ ∞0

dL ϕ(r, w, L) .


We now return to the framework described in [1] for constructing spherically symmetricsteady states. Suppose we postulate an ansatz satisfying the functional form (1), i.e. wepostulate that there exists a “nice” function F such that

ϕ(r, w, L) = F

(1

2|w|2 +

L

2r2+ U(r), L

).

Then the function U[ϕ] satisfies the differential equation

1

r2

(r2U ′[ϕ](r)

)′=

4π2

r2

∫ ∞−∞

dw

∫ ∞0

dL F

(1

2|w|2 +

L

2r2+ U[ϕ](r), L

).

This motivates the more general study of the differential equation

1

r2

(r2U ′(r)

)′= h (r, U(r)) . (∗)

The main idea of the framework in [1] is to prove the existence of a solution U to (∗) given h,and to construct the corresponding steady solution ϕ such that U[ϕ] = U + const (the additiveconstant is necessary to ensure U[ϕ](r)→ 0 as r →∞. We state the relevant theorems below.We omit the rather technical proofs, for which we refer the reader to the paper [1].

Theorem 2.7. Let α ∈ R and let h : (0,∞)× R→ R be a measurable function such that

(i) The function s 7→ sh(s, α) is L1loc([0,∞)).

(ii) For all (r0, u0) ∈ (0, α) ∪ (0,∞)× R, there exist δ > 0 and a function

L[r0,u0] : (r0, r0 + δ)→ [0,∞]

such that (r − r0)L[r0,u0] ∈ L1([r0, r0 + δ]), and∣∣h(r, u1)− h(r, u2)∣∣ ≤ L[r0,u0](r) |u1 − u2| ∀ r ∈ (r0, r0 + δ), u1, u2 ∈ [u0 − δ, u0 + δ] .

(iii) There exists 0 ≤ H ∈ L1loc((0,∞)) such that

|h(r, u)| ≤ H(r) (1 + |u|) ∀ r > 0, u ∈ R .

Then (∗) has a unique solution U : [0,∞)→ R with U(0) = α.

Theorem 2.8. Let α, β ∈ R, let r∗ > 0 and let h : (0,∞)×R→ R be a measurable function,such that

(i) For all (r0, u0) ∈ (0, α) ∪ (0,∞)× R, there exist δ > 0 and a function

L[r0,u0] : (r0 − δ, r0 + δ)→ [0,∞]

such that (r − r0)L[r0,u0] ∈ L1([r0 − δ, r0 + δ]), and∣∣h(r, u1)− h(r, u2)∣∣ ≤ L[r0,u0](r) |u1 − u2| ∀ r ∈ (r0−δ, r0 +δ), u1, u2 ∈ [u0−δ, u0 +δ] .

(ii) There exists 0 ≤ H ∈ L1loc((0,∞)) such that

|h(r, u)| ≤ H(r) (1 + |u|) ∀ r > 0, u ∈ R .

Then (∗) has a unique solution U : (0,∞)→ R with U(r∗) = α,U ′(r∗) = β.


These theorems allows us to reverse our previous heuristic argument. Specifically, supposethat F0 = F0(E,L) is given, such that

h(r, u) :=4π2

r2

∫ ∞−∞

dw

∫ ∞0

dL F0

(1

2|w|2 +

L

2r2+ u, L

)satisfies the conditions of either Theorem 2.7 or Theorem 2.8. Having constructed the solutionU to (∗), we may set

ϕ(r, w, L) := F0

(1

2|w|2 +

L

2r2+ U(r), L

).

Then (∗) simply says that U satisfies Poisson’s equation

1

r2

(r2U ′(r)

)′= h(r, U(r)) =

1

4πρ[ϕ](r)

and therefore the potential energy density of ϕ satisfies

U[ϕ](r) = U(r)− U(∞) .

Hence ϕ is indeed a solution to the time-independent Vlasov-Poisson system.The following two families of solutions were constructed by this approach in [1].

Example 2.9. For k > −1, l > −1, E0 > 0 such that k + l + 32 > 0, the ansatz

f0(x, v) =

(E0 −

1

2|v|2 − U[f0](x)

)k+

|x ∧ v|2l

leads to a family of steady solutions to the Vlasov-Poisson system. These are known aspolytropic steady states. For k < 3l+ 7

2 , such steady states are known to have finite total massand compact support.

Example 2.10. For k > −1, l > −1, E0 > 0,Γ > 0 such that k + l + 32 > 0, the ansatz

f0(x, v) =

(E0 −

1

2|v|2 − U[f0](x)− Γ |x ∧ v|2

)k+

|x ∧ v|2l

leads to a family of steady solutions to the Vlasov-Poisson system (see [1]). These are knownas Camm steady states, after the paper [4] of G. L. Camm. For k < 3l+ 7

2 and Γ small enough,such steady states are known to have finite total mass and compact support.

As a final note, we mention that steady solutions to the Vlasov-Poisson system, that arenot spherically symmetric, are known to exist.

In [17], families of steady solutions, which are axially symmetric but are not sphericallysymmetric, are constructed by a perturbation about spherically symmetric solutions. Moreprecisely, an ansatz of the form f(x, v) = φ(E[f ])ψ(ϑ(x1v2 − x2v1)), where φ, ψ are functionssatisfying some given assumptions and ϑ is a small parameter, is shown to lead to a familyof steady solutions. The solutions in this family are axially symmetric about the x3-axis,and depend continuously on ϑ. Furthermore, the solutions with ϑ 6= 0 are not sphericallysymmetric. These solutions are also shown to have finite mass and finite radius.

More recently, in the related work [25], steady solutions to the Vlasov-Poisson system,which are axially but not spherically symmetric, have been constructed by postulating anansatz involving Jacobi’s integral. These solutions describe rotating systems, and are alsoshown to have finite mass and finite radius.

We will make no use of these constructions in the sequel.


2.2 A Rigorous Version of Jeans’s Theorem

As mentioned in the Introduction, the term “Jeans’s theorem” is generally used to refer toclaims that steady solutions to the Vlasov-Poisson system can be expressed as first integralsof the characteristic motion. A precise statement backed up by a rigorous proof does notseem to be available in the astrophysics literature except in certain trivial situations. Forexample, the version stated in the textbook [2] makes the very strong assumption that action-angle coordinates exist everywhere in phase space, from which Jeans’s theorem is an immediateconsequence (since we can simply write the phase-space density function in these coordinates).

To the author’s knowledge, the first rigorous proof of Jeans’s theorem was published byJ. Batt, W. Faltenbacher and E. Horst [1] for steady spherically symmetric solutions to theVlasov-Poisson system. Recently, G. Rein [21] obtained a proof of Jeans’s theorem for steadyspherically symmetric solutions to the Vlasov-MOND equation, a generalisation of the Vlasov-Poisson system based on M. Milgrom’s theory of modified Newtonian dynamics (MOND).

The goal of this subsection is to present the proof of Rein, specialised to the Vlasov-Poissonsystem. We remind the reader that Convention 2.6 applies to this subsection.

In the (r, w, L) coordinates, the steady (time-indepedent) Vlasov-Poisson system is

w∂ϕ

∂r−∂Υ[ϕ]

∂r(r, L)

∂ϕ

∂w= 0

where Υ[ϕ] : (0,∞)× (0,∞)→ R denotes the effective potential energy density,

Υ[ϕ](r, L) :=L

2r2+ U[ϕ](r) .

Let us define

m[ϕ](r) := 4π

∫ r

0

s2ρ[ϕ](s) ds .

Then we have an explicit formula for U[ϕ], namely

U[ϕ](r) = −∫ ∞r

m[ϕ](s)

s2ds .

Indeed, U[ϕ], given by this formula, satisfies Poisson’s equation, and the condition f ∈ L1(R6)guarantees that ρ[ϕ] is integrable, so that U[ϕ](r)→ 0 as r →∞.

Lemma 2.11. For any L > 0, the function Υ[ϕ](·, L) has a unique global minimum rmin(L),and Υ[ϕ](·, L) is strictly decreasing on (0, rmin(L)) and strictly increasing on (rmin(L),∞).

Moreover rmin is a continuous, strictly increasing function of L.

Proof. Let

r0 := supr > 0

∣∣∣ m(r) = 0.

We find (rm[ϕ](r))′ = m[ϕ](r) + 4πρ[ϕ](r), so rm[ϕ](r) is zero on (0, r0) and strictly increasing

on (r0,∞). Moreover, it is clear that rm[ϕ](r)→∞ as r →∞. Now,

∂Υ[ϕ]

∂r(r, L) =

rm[ϕ](r)− Lr3

.

Thus there is exactly one solution rmin(L) to∂Υ[ϕ]

∂r (·, L) = 0 in (0,∞). The behaviour of

Υ[ϕ](·, L) follows from noting that∂Υ[ϕ]

∂r (·, L) is negative on (0, r0) and positive on (r0,∞).We have characterised rmin(L) as the unique solution of rm[ϕ](r) = L in (r0,∞). Now m[ϕ]

is continuous, so rm[ϕ](r) gives a continuous increasing bijection, and hence a homeomorphism,(r0,∞)→ (0,∞). Its inverse rmin is therefore continuous and strictly increasing.


As an immediate consequence,

Υ[ϕ],min(L) := infr∈R

Υ[ϕ](r, L) = Υ[ϕ] (rmin(L), L)

defines a continuous function (0,∞)→ R, and

Ω :=

(E,L) ∈ R× (0,∞)∣∣∣ E > Υ[ϕ],min(L)

defines an open subset of R× (0,∞).

Lemma 2.12. For any L > 0,limr→∞

Υ[ϕ](r, L) =∞ .

As such, Υ[ϕ](·, L) defines a decreasing homeomorphism (0, rmin(L))→ (Υ[ϕ],min(L),∞).

Proof. Since

Υ[ϕ](r, L) =1

r2

(L

2− r2U[ϕ](r)

),

it suffices to show that r2U[ϕ](r)→ 0 as r → 0.If U(r) remains bounded as r → 0, we are done. Otherwise, by l’Hopital’s rule,

limr→0

r2U[ϕ](r) = limr→0

(− 1

2r−3U ′[ϕ](r)

)= limr→0

(−r

3

2

m(r)

r2

)= 0

as required.

We are finally ready to state and prove the following version of Jeans’s theorem.

Theorem 2.13. Suppose f(x, v) = ϕ(r, w, L) is a steady, spherically symmetric classicalsolution to the Vlasov-Poisson system, such that ϕ is differentiable in the classical sense, andmoreover ρ[ϕ] ∈ L∞loc((0,∞)).

Then there exists a function F : Ω→ [0,∞) such that

f(x, v) = F

(1

2|v|2 + U[f ](x), |x ∧ v|2

)on

(x, v) ∈ R6

∣∣ x ∧ v 6= 0.

Proof. Define the total energy density by

E[ϕ](r, w, L) :=1

2|w|2 + Υ[ϕ](r, L) ,

which is numerically equal to 12 |v|

2 + U[f ](x).For any fixed L > 0, the characteristic system

R = W , W = −∂Υ[ϕ]

∂r(R,L) (∗)

has well-defined orbits, because ρ[ϕ] being locally bounded guarantees that∂Υ[ϕ]

∂r (·, L) is locallyLipschitz. By direct calculation, we see that both E[ϕ](·, ·, L) and ϕ(·, ·, L) are constant oneach orbit.

On the other hand, the monotonicity properties of Υ[ϕ], given in Lemma 2.11, guaranteesthat every nonempty level set of E[ϕ](·, ·, L) is a single connected curve. Such a curve is, by

definition, given by 12 |w|

2 + Υ[ϕ](r, L) = const. It is easy to see (by drawing these curves inthe (r, w)-plane) that each such curve is an orbit of the characteristic system (∗).

By Lemma 2.12, we may define Υ[ϕ](·, L)−1 : (0, rmin(L)) → (Υ[ϕ],min(L),∞) to be theinverse function of Υ[ϕ](·, L). The theorem is proved by simply setting

F (E,L) := ϕ((

Υ[ϕ](·, L))−1

(E), 0, L)

for E > Υ[ϕ],min(L), i.e. (E,L) ∈ Ω.


2.3 Failure of Jeans’s Theorem for the Vlasov-Einstein System

In this subsection, which is somewhat tangential to the rest of the essay, we consider theVlasov-Einstein system, a relativistic version of the gravitational Vlasov-Poisson system. Thegoal of this subsection is to outline the main idea of a construction, due to Schaeffer [24], of aclass of counterexamples to the statement of Jeans’s theorem for the Vlasov-Einstein system.Thus, Jeans’s theorem does not hold for the Vlasov-Einstein system.

We warn the reader that the actual construction is very long and technical, and here wegive a necessarily sketchy exposition of the main ideas of the construction. For the technicaldetails we refer the reader to the original paper [24].

Assume an asymptotically-flat spacetime on which (t, x) coordinates, have been chosen.As before, a spherically symmetric function can be expressed in the variables

r := |x| , w :=x · vr

, L := |x ∧ v|2 .

The Vlasov-Einstein system for a steady, spherically symmetric phase-space density functionf(x, v) = ϕ(r, w, L) ≥ 0 then reads

w(1 + w2 + L

r2

) 12

∂ϕ

∂r−(

1 + w2 +L

r2

) 12

µ′[ϕ](r)∂ϕ

∂w= 0

with λ[ϕ], µ[ϕ] : (0,∞)→ R being the space-time metric coefficients, given by

e−2λ[ϕ]

(2rλ′[ϕ] − 1

)+ 1 = 8πr2ρ[ϕ] , lim

r→∞λ[ϕ](r) = 0 ,

e−2λ[ϕ]

(2rµ′[ϕ] + 1

)− 1 = 8πr2p[ϕ] , lim

r→∞µ[ϕ](r) = 0 ,

where

ρ[ϕ](r) :=π

r2

∫ ∞−∞

dw

∫ ∞0

dL ϕ(r, w, L) ,

p[ϕ](r) :=π

r2

∫ ∞−∞

dw

∫ ∞0

dL w2ϕ(r, w, L)

(1 + w2 +

L

r2

)−2

.

The corresponding space time metric is

ds2 = −e2µ[ϕ](r) dt2 + e2λ[ϕ](r) dr2 + r2χ

where χ is the area 2-form of S2.As before, L has the physical meaning of angular momentum. The generalisation of the

total energy to the relativistic setting is the relativistic energy,

E[ϕ](r, w, L) := eµ[ϕ](r)

√1 + w2 +

L

r2.

The statement of Jeans’s theorem, in the context of the Vlasov-Einstein system, is that everysteady, spherically symmetric solution to the Vlasov-Einstein system satisfies the functionalform ϕ(r, w, L) = F (E[ϕ](r, w, L), L) for some function F .

For convenience, we also define

m[ϕ](r) :=

∫ r

0

4πs2ρ[ϕ](s) ds ,

Note that the equation for λ[ϕ] can be integrated immediately to give

e−2λ[ϕ](r) = 1− 2m[ϕ](r)

r.


Substituting into the equation for µ[ϕ] gives

µ′[ϕ](r) =4πrp[ϕ](r) + r−2m[ϕ](r)

1− 2r−1m[ϕ](r).

Now we make the ansatz

ϕ(r, w, L) = D0

(E0 − eµ(r)

√1 + w2 +

L

r2

)k+

(L− L0)l+ (∗)

with

D0 > 0 , E0 > 0 , L0 > 0 , k > 0 , l > −1

2(∗.1)

andµ = µ[ϕ] − const . (∗.2)

If ϕ is such a solution, then µ solves the integro-differential equation

µ′(r) =4πrp(r) + r−2m(r)

1− 2r−1m(r)(†)

with

m(r) := 4π2D0

∫ r

0

ds

∫ ∞−∞

dw

∫ ∞L0

dL

(E0 − eµ(s)

√1 + w2 +

L

s2

)k+

(L− L0)l , (†.1)

p(r) :=πD0

r2

∫ ∞−∞

dw

∫ ∞L0

dL w2

(E0 − eµ(r)

√1 + w2 +

L

r2

)k+

(L− L0)l(1 + w2 + L

r2

)2 . (†.2)

Remark 2.14. Suppose µ is a solution to (†), (†.1), (†.2). Then (†) implies that µ is anon-decreasing function. Therefore, we may define

r0 :=

(L0

E20e−2µ(0) − 1

) 12

, provided µ(0) < log(E0) ,

so that E0 = eµ(0)√

1 + L0r−20 .

The key idea of Schaeffer’s construction is the study of the integro-differential equation (†)with (†.1), (†.2). The following lemma is the main result from such a study, and is crucial toSchaeffer’s construction. However, its proof is very long and technical, so we shall omit it andrefer the interested reader to the original paper [24].

Lemma 2.15. Fix D0, E0, L0, k, l as above, and let

q := k + l +3

2.

Suppose µ solves (†), (†.1), (†.2) with initial condition satisfying

µ(0) < − log (E0) ,

so that r0 may be defined in Remark 2.14.Then there exist positive constants D1, D2, D3, D4 independent of µ(0), such that, if it

additionally holds that µ(0) < −D1, then there exists

R ∈(r0 +D2r

1+ 2q+1

0 , r0 +D3r1+ 2

q+1

0

)


such that

E ′(R) > 0 , E(R) = E0 ,m(R)

R<

1

2−D4 ,

where E is the function defined by

E(r) := eµ(r)

(1 +

L0

r2

) 12

.

Now, fix a solution f(x, v) = ϕ(r, w, L) of the Vlasov-Einstein equation of the form (∗),(∗.1), (∗.2), such that

• The constant is chosen so that µ(0) is sufficiently negative, such that r0, as defined inRemark 2.14, is small enough so that

r0 +D3r1+ 2

q+1

0√2D4L0

< 1 .

Observe also that the definitions (†.1) and (†.2) give m = m[ϕ] and p = p[ϕ].

• With R > 0 chosen corresponding to µ in Lemma 2.15, ϕ ≡ 0 for r ≥ R.

This is known to be possible; see [16]. Also, observe that, by definition of r0, ϕ ≡ 0 for r ≤ r0.In summary,

ϕ(r, w, L) = 0 for r ≤ r0 or r ≥ R .

We therefore have

m[ϕ](r) = m[ϕ](R) , ρ[ϕ](r) = p[ϕ](R) = 0 for r ≥ R.

Since µ solves (†), we have

µ′(r) =r−2m[ϕ](R)

1− 2r−1m[ϕ](R)for r ≥ R,

which may be integrated directly to give

eµ(r) = eµ(R)

(1− 2r−1m[ϕ](R)

1− 2R−1m[ϕ](R)

) 12

for r ≥ R.

By Lemma 2.15, we have

E(r) = E0

(1− 2r−1m[ϕ](R)

1− 2R−1m[ϕ](R)

) 12(

1 + L0r−2

1 + L0R−2

) 12

≤ E0

(1− 2r−1m[ϕ](R)

2D4

) 12

1 + L0r−2

L0

(r0 +D3r

1+ 2q+1

0

)−2

12

= E0r0 +D3r

1+ 2q+1

0√2D4L0

(1−

2m[ϕ](R)

r

) 12(

1 +L0

r2

) 12

for r ≥ R.

By our choice of µ(0), so that r0 is small, we have

E(r) < E0 for r sufficiently large.

Schaeffer [24] asserts that it is possible to choose r1 ∈ (r0, R), w1 ∈ R, L1 ∈ (L0,∞) so that

ϕ(r1, w1, L1) > 0 , E0r0 +D3r

1+ 2q+1

0√2D4L0

< E(r1) < E0 = E(R) .

3 STEADY SOLUTIONS FROM CASIMIR FUNCTIONALS Page 13 of 38

By the intermediate value theorem, we may pick r2 > R such that

E(r2) = E(r1) ,

that is

eµ(r2)

(1 +

L0

r22

) 12

= eµ(r1)

(1 +

L0

r21

) 12

.

Since L1 > L0, the function s 7→ (1 + L1s−2)/(1 + L0s

−2) is decreasing on (0,∞), so

1 + L1r−22

1 + L0r−22

<1 + L1r

−21

1 + L0r−21

.

The preceding two displayed equations give

eµ(r2)

(1 +

L1

r22

) 12

< eµ(r1)

(1 +

L1

r21

) 12

.

Therefore we may choose w2 ∈ R, with |w2| > |w1|, so that

E[ϕ](r2, w2, L1) = eµ(r2)

(1 + w2

2 +L1

r22

) 12

= eµ(r1)

(1 + w2

1 +L1

r21

) 12

= E[ϕ](r1, w1, L1) .

Thus, with L2 := L1, we have E[ϕ](r2, w2, L2) = E[ϕ](r1, w1, L1) and L2 = L1, but on theother hand ϕ(r2, w2, L2) = 0 (since r2 > R) and yet ϕ(r1, w1, L1) > 0.

We conclude that Jeans’s theorem fails to hold for this solution f(x, v) = ϕ(r, w, L).

3 Steady Solutions from Casimir Functionals

Various families of steady, spherically symmetric solutions to the Vlasov-Poisson system, withfinite mass and compact support, have been constructed by the application of Theorems 2.7and 2.8 (see Subsection 2.2).

The present section is concerned with an alternative approach for the construction of suchsolutions, which involves the use of Casimir functionals. The advantage of this approach isthat it provides a framework to investigate the issue of dynamical stability of the resultingsolutions: given a time-dependent solution f to the Vlasov-Poisson system in the gravitationalcase, whose initial datum is close to a steady solution f0, will f(t) stay close to f0? This andsimilar questions have important ramifications to the stability of models of stellar structure.

We remind the reader that throughout this section, Convention 2.4 will apply: unless oth-erwise stated, we only consider the Vlasov-Poisson system in the gravitational case. Moreover,throughout this section we will use the notation

E[f ](x, v) :=1

2|v|2 + U[f ](x)

3.1 The Energy-Casimir Method

The paper [6] contains a mostly heuristic discussion of a general abstract framework, theenergy-Casimir method, for the investigation of nonlinear stability of stationary solutionsto infinite-dimensional, degenerate Hamiltonian systems, of which many interesting physicalsystems are examples. Though this method cannot be applied verbatim to the Vlasov-Poissonsystem, it does inspire an alternative method which can be used to prove some interestingstability results for the Vlasov-Poisson system. We shall therefore give, in this subsection,an overview of the energy-Casimir method, and heuristically motivate how it can be used todemonstrate nonlinear stability.


Suppose the system under consideration has the equation of motion

du

dt= A(u) (∗)

on some state space X, which should be a Banach space, and A : D(A) → X is a nonlinearoperator. Assume there exists a steady solution u0 whose stability we wish to investigate. Theprincipal steps are

1. Find an energy functional, i.e. a smooth function H : X → R which is conserved alongflows of the system,

d

dtH(u(t)) = 0 .

2. Find a Casimir functional, i.e. a smooth function C : X → R which is also conservedalong flows of the system, so that u0 is a critical point of the energy-Casimir functionalHC := H+ C, that is DHC(u0) = 0.

3. Show that the quadratic part in the Taylor expansion of HC at u0 is positive-definite, i.e.

D2HC(u0) (v, v) ≥ C‖v‖2X for all v ∈ X

for some constant C > 0.

The heuristic underlying the last step is the Taylor expansion

HC(u) = HC(u0) +1

2D2HC(u0) (u− u0, u− u0) + . . .

so that the quadratic part is the main contribution to HC(u)−HC(u0) when u− u0 is small.If the above steps could be carried out, then for a flow u : [0,∞)→ X of the above dynamicalsystem (∗), we would have

‖u(t)− u0‖2X ≤ C |HC (u(0))−HC (u0)|

for ‖u(0)− u0‖X sufficiently small; therefore the system (∗) is dynamically stable.Let us illustrate why the above framework cannot be directly applied to the Vlasov-Poisson

system. We proceed formally and heuristically, ignoring rigorous issues such as the choice ofstate space to work in. For the Vlasov-Poisson system, the natural energy functional is

H(f) := Ekin(f) + Epot(f)

=

∫R3

∫R3

1

2|v|2f(x, v) dv dx− 1

8π

∫R3

∣∣∇U[f ](x)∣∣2 dx .

Suppose we have a Casimir functional C, and a steady state f0 which is a critical point of theenergy-Casimir functional HC := H+ C. Formally expanding, we have

HC(f)−HC(f0) =

∫R3

∫R3

E[f0](x, v) (f(x, v)− f0(x, v)) dv dx+ DC(f0) (f − f0)

− 1

8π

∫R3

∣∣∇U[f ](x)−∇U[f0](x)∣∣2 dx+

1

2D2C(f0) (f − f0, f − f0)

+O(‖f − f0‖3

)where

E[f ](x, v) :=1

2|v|2 + U[f ](x) .

Notice that the quadratic term is not necessarily positive-definite. Hence the above frameworkfails, and we need to modify our strategy.


Remark 3.1. Notice that the only reason for the lack of positive-definiteness, in the quadraticterm in the above expansion, is our choice to work in the gravitational case. This is becausethe potential energy is negative in the gravitational case.

For the plasma physics case, the potential energy is positive and the above framework canbe applied to certain situations where there are steady solutions, for example if a fixed ionbackground is present or if the problem is posed in a bounded domain rather than in the wholeof R3. Some of these problems are treated in [15].

3.2 An Early Stability Result of Camm Steady States

The paper [5] is a pioneering work in the construction of steady, spherically symmetric solutionsto the Vlasov-Poisson system, and the investigation of their dynamical stability, by minimisingenergy-Casimir functionals. The method used is reminiscent of the direct method of thecalculus of variations.

The goal of this subsection is to give an overview of the proof that these minimisers areactually steady solutions to the Vlasov-Poisson equation, satisfying the functional form (1).We warn the reader that we will be employing notation that is different from that of theoriginal paper, but aims to be consistent with the rest of this essay.

Guo and Rein considered the Casimir functional

C(f) :=

∫R3

∫R3

[Q(f(x, v)|x ∧ v|2λ

)|x ∧ v|−2λ

+ Γf(x, v) |x ∧ v|2]

dv dx

where Γ ≥ 0 and λ < 1, and Q ∈ C1([0,∞)) ∩ C2((0,∞)) is assumed to satisfy

(i) There exist C0 > 0, κ0 ∈ (0, 32 − λ), such that

Q(h) ≥ C0hκ0+1κ0 .

(ii) There exist C1 > 0, κ1 ∈ (0, 32 − λ), such that

Q(h) ≤ C1hκ1+1κ1 for sufficient small h ≥ 0 .

(iii) There exists κ0 ∈ (0, 32 − λ), such that

Q(ch) ≥ cκ0+1κ0 Q(h) for c ∈ [0, 1] .

(iv) Q′(0) = 0 and Q′′ > 0.

In particular this means that Q′ is an increasing homeomorphism [0,∞)→ [0,∞), whichis continuously differentiable on (0,∞).

For fixed M > 0, the associated energy-Casimir functional

HC := H+ C = Ekin + Epot + C

is to be minimised over

FMsym :=

f ∈ L1(R6)

∣∣∣ f ≥ 0 ,

∫R3

∫R3

f(x, v) dv dx = M, f is spherically symmetric

.

For convenience, we will denotehM := inf

FMsymHC .

Remark 3.2. The assumptions above are motivated by the applicability to the considerationof Camm steady states. A typical Casimir functional of the above form would have

Q(h) = C0hκ0+1κ0 + C1h

κ1+1κ1

for C0 > 0, C1 ≥ 0. Camm steady states would result from setting C1 = 0.


The first issue to verify is that HC does not take arbitrarily large negative values, i.e. thatit is bounded below. The following result, whose proof by a scaling argument the reader canfind in [5], Lemma 4, guarantees this.

Lemma 3.3. Fix M > 0. Then for Γ ≥ 0 sufficiently small, we have

0 > hM > −∞ .

With this reassuring fact, we can now pick a minimising subsequence fnn∈N ⊂ FMsym.One then hopes, as in the direct method of the calculus of variations, to extract a convergentsubsequence (with respect to some topology), and show that the limit is a minimiser of theenergy-Casimir functional.

The natural function space to work in turns out to be

Fκ0,λsym :=

f : R6 → R

∣∣∣ f measurable, spherically symmetric , ‖f‖Fκ0,λ <∞

equipped with the norm

‖f‖Fκ0,λ :=

(∫R3

∫R3

|f(x, v)|κ0+1κ0 |x ∧ v|

2λκ0 dv dx

) κ0κ0+1

,

which is simply the Lκ0+1κ0 (|x∧v|

2λκ0 dv dx) norm. Under this norm, Fκ0,λ

sym becomes a separable,reflexive Banach space. This choice of function space is motivated by the following estimate,whose proof the reader can find in [5], Lemma 1.

Lemma 3.4. There exists a constant C = C(M) > 0 such that∫R3

∫R3

|f(x, v)|κ0+1κ0 |x ∧ v|

2λκ0 dv dx ≤ C (1 + Ekin(f) + C(f))

for every f ∈ FMsym.

Using Lemma 3.4 and some other estimates, it can be shown that a minimising sequencein FMsym for HC must be a bounded sequence in Fκ0,λ

sym . Thus a subsequence, which is weakly

convergent in Fκ0,λsym , can be extracted. The limit can then be shown to be a minimiser of

the energy-Casimir functional HC. Specifically, the following existence result is proved in [5],Theorem 1.

Theorem 3.5. Let M > 0, and let Γ ≥ 0 be sufficiently small so that Lemma 3.3 applies.Suppose fnn∈N ⊂ FMsym is a minimising sequence for HC.

Then there is a subsequence fn(k)k∈N ⊆ fnn∈N which converges weakly in Fκ0,λsym to a

minimiser f0 ∈ FMsym of HC. The corresponding spatial density ρ[f0] has compact support, and∇U[fn(k)] → ∇U[f0] strongly in (L2(R3))3.

We now arrive at the main result in this section: identifying minimisers of HC over Fκ0,λsym

as steady, spherically symmetric solutions to the Vlasov-Poisson system. Although we do notobtain an explicit formula for a minimiser f0, we show that f0 is of the functional form (1);therefore f0 does indeed solve the Vlasov-Poisson system. The idea of the proof is almostidentical to that used to derive Euler-Lagrange equations for minimisation problems in thecalculus of variations.

Theorem 3.6. Let f0 ∈ FMsym be a minimiser of HC over FMsym. Then there exists E0 ∈ Rsuch that f0 satisfies

f0(x, v) =[(Q′)−1

((E0 − E[f0](x, v)− Γ|x ∧ v|2

)+

)]|x ∧ v|−2λ for a.e. (x, v) ∈ R6 .


Proof. Let ε > 0 be sufficiently small so that

Sε :=

(x, v) ∈ R6

∣∣∣ ε ≤ f0(x, v) ≤ 1

ε

has finite positive measure. Then let ζ : R6 → R be a bounded, measurable, sphericallysymmetric, compactly supported function, such that ζ ≥ 0 on R6 \ Sε. Set

η(x, v) := ζ(x, v)− 1

|Sε|

∫∫Sεζ(y, w) dw dy 1Sε(x, v) .

Note that η ∈ L∞(R6) is spherically symmetric and has compact support.For sufficiently small τ > 0 we have f0 + τη ∈ FMsym. As f0 is a minimiser of HC over FMsym,

we have

0 ≤ limτ→0+

1

τ(HC (f0 + τη)−HC (f0))

=

∫R3

∫R3

[E[f0](x, v) +Q′

(f0(x, v)|x ∧ v|2λ

)+ Γ|x ∧ v|2

]η(x, v) dv dx

=

∫R3

∫R3

[E[f0](x, v) +Q′

(f0(x, v)|x ∧ v|2λ

)+ Γ|x ∧ v|2

]ζ(x, v) dv dx

− 1

|Sε|

∫∫Sε

[E[f0](y, w) +Q′

(f0(y, w)|y ∧ w|2λ

)+ Γ|y ∧ w|2

]dy dw

∫∫Sεζ(x, v) dv dx .

In particular, since this holds for ζ vanishing outside Sε but otherwise arbitrary (subject tobeing bounded and spherically symmetric) in Sε, we see that, for a.e. (x, v) ∈ Sε,

E[f0](x, v) +Q′(f0(x, v)|x ∧ v|2λ

)+ Γ|x ∧ v|2

=1

|Sε|

∫∫Sε

[E[f0](y, w) +Q′

(f0(y, w)|y ∧ w|2λ

)+ Γ|y ∧ w|2

]dy dw .

But the left-hand side of this equation does not depend on ε, so both sides are equivalent toa constant E0 independent of ε, everywhere in Sε. Since ε > 0 could be chosen arbitrarilysmall, we deduce

E[f0](x, v) +Q′(f0(x, v)|x ∧ v|2λ

)+ Γ|x ∧ v|2 = E0 for a.e. (x, v) ∈ f0 > 0 .

On the other hand, taking ζ to vanish inside f0 > 0 but otherwise arbitrary non-negative,bounded, spherically symmetric on f0 = 0, we have that

E[f0](x, v) + Γ|x ∧ v|2 ≥ E0 for a.e. (x, v) ∈ f0 = 0 .

The two preceding displayed equations prove the theorem.

An advantage of constructing solutions using the above energy-Casimir method is thatstability results can be proved for minimisers of the energy-Casimir functional. For this specificproblem, we have the following dynamical stability result.

Theorem 3.7. Let f0 ∈ FMsym be a minimiser of HC over FMsym. Then for any ε > 0 there

exists δ > 0 such that whenever f(0) ∈ C1c (R6) ∩ FMsym satisfies

HC(f(0))−HC(f0) +1

4π

∥∥∇U[f(0)] −∇U[f0]

∥∥2

L2 < δ ,

the classical solution f to the Vlasov-Poisson system, launched by the initial data f(0), satisfies

HC(f(t))−HC(f0) +1

4π

∥∥∇U[f(t)] −∇U[f0]

∥∥2

L2 < ε for all t ≥ 0 .


3.3 A Reduction Approach for Isotropic Steady States

The preceding stability result, Theorem 3.7, suffers from the defect that it caters only for theunphysical situation spherically symmetric perturbations about the minimiser f0. Much workhas been done in attempting to remove this restriction. The paper [18] presents an argumentthat proves dynamical stability, with respect to general perturbations, for isotropic steadystates, i.e. steady states satisfying the functional form f(x, v) = F (E[f ](x, v)) with no explicitdependence on |x ∧ v|2.

An interesting alternative approach [19], also only for isotropic steady states, involvesreducing the minimisation problem over phase-space densities f to a “reduced” minimisationproblem over spatial densities ρ. A comprehensive exposition of the argument can be found in[20], whose presentation we follow. In this subsection we focus only on showing the equivalenceof the two problems, demonstrating how isotropic steady states to the Vlasov-Poisson systemcan be recovered from the reduced problem, and stating the dynamical stability result.

The Casimir functional considered in [19] is

C(f) :=

∫R3

∫R3

Φ(f(x, v)) dv dx

where Φ ∈ C1([0,∞)) is assumed to satisfy

(i) There exist C0 > 0, κ0 ∈ (0, 32 ), such that

Φ(h) ≥ C0hκ0+1κ0 for sufficiently large h ≥ 0 .

(ii) There exist C1 > 0, κ1 ∈ (0, 32 ), such that

Φ(h) ≤ C1hκ1+1κ1 for sufficiently small h ≥ 0 .

(iii) Φ(0) = Φ′(0) = 0, and Φ is strictly convex on [0,∞).

In particular Φ′ is an increasing homeomorphism [0,∞)→ [0,∞),

We also define Φ ≡ +∞ on (−∞, 0]. This allows us to define the convex conjugate of Φ by

Φ∗(θ) := suph∈R

(hθ − Φ(h)) .

Since Φ is strictly convex wherever it is finite, we have that Φ∗ is differentiable on R (by[22], Theorem 26.3) and the inverse homeomorphism of Φ′ : [0,∞) → [0,∞) is given by(Φ′)−1 = (Φ∗)′|[0,∞) (by [22], Theorem 26.5). Since Φ∗ ≡ 0 on (−∞, 0], we see that (Φ∗)′ iscontinuous on R, hence Φ∗ ∈ C1(R).

Original Minimisation Problem. Minimise the energy-Casimir functional

HC := Ekin + Epot + C

over

FM :=

f ∈ L1(R6)

∣∣∣ f ≥ 0 ,

∫R3

∫R3

f(x, v) dv dx = M

.

Convention 3.8. In the following, for ρ ∈ L1(R3), we will write U[ρ] for the solution of

4U[ρ] = 4πρ , lim|x|→∞

U[ρ](x) = 0 ,

and use the notation

Epot(ρ) := − 1

8π

∫R3

∣∣∇U[ρ](x)∣∣2 dx =

1

2

∫R3

U[ρ](x)ρ(x) dx .

This abuse of notation should not cause any confusion. With this notation, we have, forexample, Epot(f) = Epot(ρ[f ]) for any f ∈ L1(R6).


To formulate the reduced problem, we define, for s ≥ 0,

G[s] :=

g ∈ L1(R3)

∣∣∣ g ≥ 0 ,

∫R3

g(v) dv = s

,

Ψ(s) := infg∈G[s]

∫R3

(1

2|v|2g(v) + Φ(g(v))

)dv .

It is immediate from the definition above that Ψ is convex and non-decreasing on [0,∞). Forsimplicity, we define Ψ ≡ +∞ on (−∞, 0).

Reduced Minimisation Problem. Minimise the reduced energy-Casimir functional

Hr(ρ) :=

∫R3

Ψ(ρ(x)) dx+ Epot(ρ)

over

RM :=

ρ ∈ L1(R3)

∣∣∣ f ≥ 0 ,

∫R3

ρ(x, v) dx = M

.

Our first goal in this subsection is to show that these two problems are equivalent. Wefirst make the following straightforward observation.

Lemma 3.9. For every f ∈ FM , we have

HC(f) ≥ Hr

(ρ[f ]

).

In particular,infFMHC ≥ inf

RMHr .

Proof. If f ∈ FM then ρ[f ] ∈ RM , and

Ekin(f) + C(f) =

∫R3

∫R3

(1

2|v|2f(x, v) + Φ (f(x, v))

)dv dx

≥∫R3

infg∈G[ρ[f](x)]

[∫R3

(1

2|v|2g(v) + Φ (g(v))

)dv

]dx =

∫R3

Ψ(ρ[f ](x)

)dx .

Adding Epot(f) = Epot(ρ[f ]) to both sides gives the desired result.

We next derive a relation between the convex conjugates of Φ and Ψ.

Lemma 3.10. The following relation holds:

Ψ∗(θ) =

∫R3

Φ∗(θ − 1

2|v|2)

dv .

Proof. From the definitions we have

Ψ∗(θ) = sups≥0

[θs− inf

g∈G[s]

∫R3

(1

2|v|2g(v) + Φ (g(v))

)dv

]= sup

s≥0supg∈G[s]

[θs−

∫R3

(1

2|v|2g(v) + Φ (g(v))

)dv

]= sup

0≤g∈L1(R3)

∫R3

((θ − 1

2|v|2)g(v)− Φ (g(v))

)dv

≤∫R3

supz≥0

[(θ − 1

2|v|2)z − Φ (z)

]dv

=

∫R3

Φ∗(θ − 1

2|v|2)

dv .


Therefore it suffices to prove the reverse inequality.For θ ≤ 0, note that the left-hand side is non-negative, while the right-hand side is zero.

For θ > 0, set

g1(v) :=

(Φ′)−1

(θ − 1

2 |v|2)

, if |v| <√

2θ

0 , if |v| ≥√

2θ.

Then 0 ≤ g1 ∈ L1(R3), and

supz≥0

[(θ − 1

2|v|2)z − Φ (z)

]=

(θ − 1

2|v|2)g1(v)− Φ (g1(v)) ,

so a direct integration yields∫R3

supz≥0

[(θ − 1

2|v|2)z − Φ (z)

]dv =

∫R3

((θ − 1

2|v|2)g1(v)− Φ (g1(v))

)dv

≤ sup0≤g∈L1(R3)

∫R3

((θ − 1

2|v|2)g(v)− Φ (g(v))

)dv

which is the desired reverse inequality.

Lemma 3.11. Ψ∗ ∈ C1(R), Ψ ∈ C1([0,∞)) and Ψ is strictly convex. In particular Ψ′ definesan increasing homeomorphism [0,∞)→ [0,∞), with inverse (Ψ′)−1 = (Ψ∗)′|[0,∞).

Proof. Differentiating the formula in Lemma 3.10, we get

(Ψ∗)′(θ) =

∫R3

(Φ∗)′(θ − 1

2|v|2)

dv .

From this we see that Ψ∗ is continuously differentiable and (Φ∗)′ is strictly increasing.Since Ψ is convex lower-semicontinuous, we have Ψ = Ψ∗∗ (by [3], Theorem 4.2.1). Thus,

by [22], Theorems 26.3 and 26.5 again, we have that Ψ ∈ C1([0,∞)) and (Ψ∗)′ = (Ψ′)−1.Hence Ψ′ is strictly increasing, so Ψ is strictly convex.

We are finally ready to prove the following theorem, which shows that the two minimisationproblems stated above are equivalent, in the sense that equality holds in Lemma 3.9, i.e.

infFMHC = inf

RMHr ,

and minimisers of HC over FM can be constructed from minimisers of Hr over RM .

Theorem 3.12. Let ρ0 ∈ RM be a minimiser of Hr over RM . Then there exists E0 ∈ R suchthat ρ0 satisfies

ρ0(x) = (Ψ∗)′(E0 − U[ρ0](x)

)for a.e. x ∈ R3 .

Moreover, defining

f0(x, v) := (Φ∗)′(E0 −

1

2|v|2 − U[ρ0](x)

),

we have ρ0 = ρ[f0] and HC(f0) = Hr(ρ0). Consequently, f0 is a minimiser of HC over FM .

Proof. The first part is proved in an identical manner to Theorem 3.6. Let ε > 0 be sufficientlysmall so that

Sε :=

x ∈ R3

∣∣∣ ε ≤ ρ0(x) ≤ 1

ε

has finite positive measure. Then let ζ : R3 → R be a bounded, measurable, compactlysupported function, such that ζ ≥ 0 on R3 \ Sε.


For sufficiently small τ > 0 we have that

ρτ := ρ0 + τ

(ζ − 1

|Sε|

∫Sεζ(y) dy 1Sε

)belongs to RM . As ρ0 is a minimiser of Hr over RM , we have

0 ≤ limτ→0+

1

τ(Hr (ρτ )−Hr (ρ0))

=

∫R3

[U[ρ0](x) + Ψ′ (ρ0(x))

] [ζ(x)− 1

|Sε|

∫Sεζ(y) dy 1Sε(x)

]dx

=

∫R3

[U[ρ0](x) + Ψ′ (ρ0(x))− 1

|Sε|

∫Sε

[U[ρ0](y) + Ψ′ (ρ0(y))

]dy 1Sε(x)

]ζ(x) dx .

In particular, since this holds for ζ vanishing outside Sε but otherwise an arbitrary boundedmeasurable function in Sε, we see that, for a.e. x ∈ Sε,

U[ρ0](x) + Ψ′ (ρ0(x)) =1

|Sε|

∫Sε

[U[ρ0](y) + Ψ′ (ρ0(y))

]dy .

But the left-hand side of this equation does not depend on ε, so both sides are equivalent toa constant E0 independent of ε, everywhere in Sε. Since ε > 0 could be chosen arbitrarilysmall, we deduce

U[ρ0](x) + Ψ′ (ρ0(x)) = E0 for a.e. x ∈ ρ0 > 0 .

On the other hand, taking ζ to vanish inside ρ0 > 0 but otherwise arbitrary boundednon-negative on ρ0 = 0, we have that

U[ρ0](x) + Ψ′ (ρ0(x)) ≥ E0 for a.e. x ∈ ρ0 = 0 .

Recalling from Lemma 3.11 that (Ψ′)−1 = (Ψ∗)′ on [0,∞), and that (Φ∗)′ ≡ 0 on (−∞, 0),we see that the two preceding displayed equations prove the first part of the theorem.

Now let f0 be defined as in the statement of the theorem. By Lemma 3.10,

ρ0(x) = (Ψ∗)′ (E0 − U[ρ0](x)

)=

∫R3

(Φ∗)′(E0 −

1

2|v|2 − U[ρ0](x)

)dv =

∫R3

f0(x, v) dv ,

that is ρ[f0] = ρ0. Thus, U[f0] = U[ρ0] also.Next observe the formulae

Φ∗(θ) = θ(Φ′)−1(θ)− Φ((Φ′)−1(θ)

)Ψ∗(θ) = θ(Ψ′)−1(θ)−Ψ

((Ψ′)−1(θ)

) for θ ≥ 0 .


Using these, and Lemma 3.10, we find∫R3

Ψ (ρ0(x)) dx =

∫ρ0>0

Ψ((Ψ′)−1

(E0 − U[f0](x)

))dx

=

∫ρ0>0

[(E0 − U[f0](x)

)(Ψ∗)′

(E0 − U[f0](x)

)−Ψ∗

(E0 − U[f0](x)

)]dx

=

∫ρ0>0

∫R3

[ (E0 − U[f0](x)

)(Φ∗)′

(E0 −

1

2|v|2 − U[f0](x)

)

− Φ∗(E0 −

1

2|v|2 − U[f0](x)

)]dv dx

=

∫ρ0>0

∫R3

[1

2|v|2(Φ∗)′

(E0 −

1

2|v|2 − U[f0](x)

)

+ Φ

((Φ′)−1

(E0 −

1

2|v|2 − U[f0](x)

))]dv dx

=

∫ρ0>0

∫R3

[1

2|v|2f0(x, v) + Φ (f0(x, v))

]dv dx

= Ekin (f0) + C (f0) .

Adding Epot(f0) to both sides completes the proof of the second part of the theorem.

We have therefore proven the equivalence of the two minimisation problems. It remains toshow that the reduced minimisation problem admits minimisers in RM . This is guaranteedby the following theorem, whose proof we shall omit; the reader can find it in the papers wehave cited above.

Theorem 3.13. For any M > 0, the reduced energy-Casimir functional Hr is bounded frombelow on RM .

If ρjj∈N ⊂ RM is a minimising sequence for Hr over RM , then ρjj∈N is bounded in

Lν0+1ν0 (R3), where ν0 := κ0 + 3

2 . Moreover there exists a subsequence ρjj∈N (not relabelled),and a sequence zjj∈N ⊂ R3, such that

ρj (·+ zj) ρ0 weakly in Lν0+1ν0 (R3) ,

∇U[ρj ] (·+ zj)→ ∇U[ρ0] strongly in(L2(R3)

)3,

where ρ0 ∈ RM is a minimiser of Hr over RM .

We shall focus only on one aspect of this theorem, namely that weak convergence in

Lν0+1ν0 (R3) implies strong convergence in (L2(R3))3 of the Newtonian force. For this, we will

need the following inequality, whose proof the reader can find in [11], Theorem 4.2.

Proposition 3.14. Let p, q, r ≥ 1 be such that 1p + 1

q + 1r = 2. Then there exists a constant

C = C(d, p, q, r) > 0 such that∣∣∣∣∫Rd

∫Rdf(x)g(x− y)h(y) dx dy

∣∣∣∣ ≤ C‖f‖Lp(Rd)‖g‖Lq(Rd)‖h‖Lr(Rd)

for every f ∈ Lp(Rd), g ∈ Lq(Rd), h ∈ Lr(Rd).

Proposition 3.15. Let 1 < ν < 5, and suppose ρjj∈N ⊂ Lν+1ν (R3) is such that

ρj ρ0 weakly in Lν+1ν (R3) ,

∀ ε > 0 ∃ R > 0 : lim supj→∞

∫R3\RB3

|ρj(x)| dx < ε .

Then ∇U[ρj ] → ∇U[ρ0] strongly in (L2(R3))3.


Proof. For any r > 0, Holder’s inequality gives∫rB3

|ρ(x)| dx ≤∣∣rB3

∣∣ 1ν+1

(∫R3

|ρ(x)|ν+1ν dx

) νν+1

≤∣∣rB3

∣∣ 1ν+1 ‖ρ‖

Lν+1ν (R3)

.

Therefore the hypotheses guarantee that ρjj∈N is bounded in L1(R3), and weak convergenceguarantees that ρ0 ∈ L1(R3) as well.

The sequence σj := ρj − ρ0 converges weakly to 0 in Lν+1ν (R3), and is bounded in L1(R3).

We need to show that ∇U[σj ] → 0 strongly in L2(R3), which is equivalent to showing∫R3

∫R3

σj(x)σj(y)

|x− y|dx dy → 0 . (∗)

For convenience, denote

P := supj∈N‖σj‖L1(R3) , Q := sup

j∈N‖σj‖

Lν+1)ν (R3

.

Let δ > 0 be small and R > 0 large, to be chosen later. We partition the domain of integration,(x, y) ∈ R3 × R3, into

A1 := |x− y| ≥ δ ∩ (|x| ≥ R ∪ |y| ≥ R) ,A2 := |x− y| ≥ δ ∩ |x| < R ∩ |y| < R ,A3 := |x− y| < δ .

We will estimate the integral (∗) over each of these sets.The integral over A1 is estimated by∫∫A1

|σj(x)||σj(y)||x− y|

dx dy ≤ 2

δ

∫R3

∫R3\RB3

|σj(x)||σj(y)| dx dy ≤ 2P

δ

∫R3\RB3

|σj(y)| dy .

We next consider the integral over A2. Let

hj(x) := 1RB3(x)

∫R3

1R3\δB3(x− y)

|x− y|1RB3(y)σj(y) dy .

For any fixed x ∈ R3, note that

1R3\δB3(x− ·)|x− ·|

1RB3 ∈ Lν+1(R3)

defines a continuous linear functional on Lν+1ν (R3). Since σj 0 in L

ν+1ν (R3), we see that

hj → 0 pointwise on R3\0. On the other hand, |hj | ≤ Pδ 1RB3 , so the dominated convergence

theorem guarantees hj → 0 in Lν+1(R3). Hence,∫∫A2

σj(x)σj(y)

|x− y|dx dy =

∫R3

hj(x)σj(x) dx

tends to 0 as j →∞.The integral over A3 is estimated using Proposition 3.14 to obtain∫∫

A3

|σj(x)||σj(y)||x− y|

dx dy ≤ C(ν) ‖σj‖2Lν+1ν (R3)

∥∥∥∥1δB3

| · |

∥∥∥∥Lν+12 (R3)

≤ C(ν)Q2δ5−νν+1 .

We collect the above estimates to conclude∣∣∣∣∫R3

∫R3

σj(x)σj(y)

|x− y|dx dy

∣∣∣∣ ≤ C(ν)Q2δ5−νν+1 +

2P

δ

∫R3\RB3

|σj(y)| dy

+

∣∣∣∣∫∫A2

σj(x)σj(y)

|x− y|dx dy

∣∣∣∣ .


Now, given ε > 0, choose δ > 0 small enough, and then choose R > 0 large enough, so that

C(ν)Q2δ5−νν+1 <

ε

2and

2P

δlim supj→∞

∫R3\RB3

|σj(y)| dy < ε

2.

Then, since the integral over A2 vanish in the limit j →∞, we obtain

lim supj→∞

∣∣∣∣∫R3

∫R3

σj(x)σj(y)

|x− y|dx dy

∣∣∣∣ < ε .

Since ε > 0 was arbitrary, we have proven the desired result (∗).

In the next subsection, we will prove a similar result in the next subsection, using a slightlydifferent technique.

We end this subsection with the statement of a dynamical stability result, whose proof weshall omit; the reader can find the proof in the papers we have cited above.

Theorem 3.16. Let f0 ∈ FM be a minimiser of HC over FM . Then for any ε > 0 thereexists δ > 0 such that whenever f(0) ∈ C1

c (R6) ∩ FM satisfies

HC(f(0))−HC(f0) +1

4π

∥∥∇U[f(0)] −∇U[f0]

∥∥2

L2 < δ ,

the classical solution f to the Vlasov-Poisson system, launched by the initial data f(0), satisfiesthe condition that for any t ≥ 0, there exists zt ∈ R3, such that

HC(f(t, ·+ zt))−HC(f0) +1

4π

∥∥∇U[f(t)](·+ zt)−∇U[f0]

∥∥2

L2 < ε .

3.4 An Unsuccessful Attempt at Anisotropic Steady States

Inspired by the previous results, the author has attempted, as part his research mini-projectin the first year of his doctoral studies, to construct anisotropic steady states (i.e. steadystates satisfying the functional form (1) with explicit dependence on the angular momentumdensity |x∧ v|2) by identifying them as minimisers of certain energy-Casimir functionals. TheCasimir part is taken to be

C(f) :=

∫R3

∫R3

Φ(f(x, v) , |x ∧ v|2

)dv dx

where Φ should satisfy certain assumptions.Unfortunately the attempt has yet to be successful, and this subsection is a record of the

results that the author has managed to obtain. The reader is advised that the results inthis subsection do not in any way represent a complete theory, nor do they lead up to anyinteresting theorem.

In the sequel, we will work with the function spaces

Fκ,λ := Lκ+1κ

(R6, |x ∧ v|

2λκ dL3(x)dL3(v)

)for κ, λ > 0 ,

andRν,λ := L

ν+1ν

(R3, |x|

2λν dL3(x)

)for ν, λ > 0 .

The assumptions on Φ : [0,∞)× [0,∞)→ R used by the author are as follows.

(i) There exist C0 > 0, κ0 > 0, λ0 ∈ (0, 1), such that

ν0 :=3− 2κ0 − 2λ0

2satisfies 1 < ν0 < 3 and ν0 + 4λ0 < 5 ,

and

Φ(h, L) ≥ C0hκ0+1κ0 L

λ0κ0 .


(ii) There exist C1 > 0, κ1 > 1, λ1 ∈ (− 32κ1,

32 − κ1), such that

Φ(h, L) ≤ C1hκ1+1κ1 L

λ1κ1 for all sufficiently small h ≥ 0.

(iii) Φ(ch, L) ≥ c3Φ(h, L) for all c ∈ [0, 1].

(iv) Φ should satisfy some strict convexity assumption with respect to h.

As before, for M > 0, denote

FM :=

f ∈ Fκ0,λ0

∣∣∣ f ≥ 0 ,

∫R3

∫R3

f(x, v) dv dx = M

and

hM := infFMHC .

These assumptions are similar to the ones imposed in the paper [5] by Guo and Rein. Theoriginal hope of the author was to prove similar results to those in that paper. The author’sapproach, however, is more similar to that employed in Rein’s review paper [20] to solve theminimisation problem referred to in that paper as “Version 2 of the variational problem”.

The first step is to show that hM > −∞ for all M > 0, so that the minimisation problemmakes sense. The following three propositions show that 0 > hM > −∞ for all M > 0. Wehave indicated where and how the assumptions made above are used.

Proposition 3.17. Assume

Φ(ch, L) ≥ c3Φ(h, L) for all c ∈ [0, 1] .

Then hcM ≥ c3hM for c ∈ (0, 1).

Proof. We employ a scaling argument. Fix c ∈ (0, 1), and let f ∈ FM . Put

f(x, v) := cf(cx, c−1v

).

Then f ∈ FcM while Ekin(f) = c3Ekin(f) and Epot(f) = c3Epot(f). Thus,

HC(f) = c3 (Ekin(f) + Epot(f)) +

∫R3

∫R3

Φ(cf(x, v) , |x ∧ v|2

)dv dx

≥ c3(Ekin(f) + Epot(f) +

∫R3

∫R3

Φ(f(x, v) , |x ∧ v|2

)dv dx

)≥ c3hM .

Since any g ∈ FcM can be constructed as f for some f ∈ FM , taking the infimum over theleft-hand side proves our assertion.

Proposition 3.18. Assume there exist C1 > 0, κ1 > 1, λ1 ∈ (− 32κ1,

32 − κ1), such that

Φ(h, L) ≤ C1hκ1+1κ1 L

λ1κ1 for all sufficiently small h > 0 .

Then hM < 0 for all sufficiently small M > 0.

Proof. Since 3− 2λ1 − κ1 > κ1 > 0, we may pick α ∈ R such that

−2 < α < −1− κ1

3− 2λ1 − κ1.

We may then pick β ∈ R such that

0 < β < min

(α+ 2 ,

(3− 2λ1 − κ1)|α| − (3− 2λ1)

κ1 − 1

).


Let ε > 0 be sufficiently small. Consider the phase-space density function

f(x, v) := 1[0,εα) (|x|) 1[0,ε) (|v|)(

4π

3

)−2

εβ−3α−3 .

which has total mass M = εβ . Clearly,

Ekin(f) ≤ 1

2εβ+2 .

We calculate

ρ[f ](x) = 1[0,εα) (|x|)(

4π

3

)−1

εβ−3α .

Thus,

Epot(f) = −1

2

∫R3

∫R3

ρ[f ](x)ρ[f ](y)

|x− y|dx dy ≤ −1

2

(εβ−3α

)2(2εα)

−1(εα)

6= −1

4ε2β−α

Since ε was chosen sufficiently small and β − 3α− 3 > 0, the hypothesis gives

C(f) =

∫εαB3

dx

∫εB3

dv Φ

((4π

3

)−2

εβ−3α−3 , |x ∧ v|2)

≤ C1

(4π

3

)−2κ1+1κ1 (

εβ−3α−3)κ1+1

κ1

∫εαB3

dx

∫εB3

dv |x ∧ v|2λ1κ1

= C (C1)(εβ−3α−3

)κ1+1κ1

∫ εα

0

d|x|∫ ε

0

d|v|∫ π

0

dϑ |x|2+2λ1κ1 |v|2+

2λ1κ1 (sin(ϑ))

1+2λ1κ1

= C(κ1, λ1, C1)(εβ−3α−3

)κ1+1κ1 ε(1+α)(3+

2λ1κ1

)

= C(κ1, λ1, C1)εβκ1+1κ1− 2λ1−3

κ1(α+1) .

The condition λ1 > − 32κ1 guarantees that the above integrals converge.

Putting the above estimates together, we have

HC(f) ≤ 1

2εβ+2 − 1

4ε2β−α + C(κ1, λ1, C2)εβ

κ1+1κ1− 2λ1−3

κ1(α+1) . (∗)

Now,β < α+ 2 implies 2β − α < β + 2 ,

β <(3− 2λ1 − κ1)|α| − (3− 2λ1)

κ1 − 1implies 2β − α < β

κ1 + 1

κ1− 2λ1 − 3

κ1(α+ 1) .

Hence, if ε > 0 is taken sufficiently small, the negative term in (∗) dominates the positiveterms, so that HC(f) < 0. Thus hM < 0 for M > 0 sufficiently small.

It turns out that the various terms in HC can be estimated by the Fκ0,λ0 norm, and theRν0,λ0 norm of the corresponding spatial density. The specific estimates are provided in thenext few lemmas.

Lemma 3.19. Assume κ0 > 0 and λ0 ∈ (0, 1). Set

ν0 :=3 + 2κ0 − 2λ0

2.

Then for any f ∈ Fκ0,λ0 , we have

∥∥ρ[f ]

∥∥Rν0,λ0

≤ C(κ0, λ0)Ekin(|f |)3−2λ02(ν0+1) ‖f‖

κ0+1ν0+1

Fκ0,λ0.


Proof. In this proof, C always denotes generic positive constants, whose values may changefrom line to line, but depending only on κ0, λ0.

For any S > 0, we have

|ρ[f ](x)| ≤∫|v|≤S

|f(x, v)| |x ∧ v|2λ0κ0+1 |x ∧ v|−

2λ0κ0+1 dv +

∫|v|>S

|f(x, v)| dv

≤(∫

R3

|f(x, v)|κ0+1κ0 |x ∧ v|

2λ0κ0 dv

) κ0κ0+1

(∫|v|≤S

|x ∧ v|−2λ0 dv

) 1κ0+1

+1

S2

∫R3

|f(x, v)||v|2dv

= C|x|−2λ0κ0+1

(∫R3

|f(x, v)|κ0+1κ0 |x ∧ v|

2λ0κ0 dv

) κ0κ0+1

S3−2λ0κ0+1

+1

S2

∫R3

|f(x, v)||v|2dv

where the condition λ0 < 1 guarantees

C =

(2π

∫ 1

0

s2−2λ0ds

∫ π

0

(sin(ϑ))1−2λ0 dϑ

) 1κ0+1

<∞

We optimise the preceding estimate by setting

S5+2κ0−2λ0

κ0+1 = C|x|2λ0κ0+1

(∫R3

|f(x, v)|κ0+1κ0 |x ∧ v|

2λ0κ0 dv

)− κ0κ0+1

(∫R3

|f(x, v)||v|2dv

),

which gives

|ρ[f ](x)| ≤ C|x|−2λ0ν0+1

(∫R3

|f(x, v)|κ0+1κ0 |x ∧ v|

2λ0κ0 dv

) κ0ν0+1

(∫R3

|f(x, v)||v|2dv

) 3−2λ02(ν0+1)

.

Therefore, using Holder’s inequality,∫R3

|ρ[f ](x)|ν0+1ν0 |x|

2λ0ν0 dx

≤ C∫R3

(∫R3

|f(x, v)|κ0+1κ0 |x ∧ v|

2λ0κ0 dv

)κ0ν0(∫

R3

|f(x, v)||v|2dv

) 3−2λ02ν0

dx

≤ C(∫

R3

∫R3

|f(x, v)|κ0+1κ0 |x ∧ v|

2λ0κ0 dv dx

)κ0ν0(∫

R3

∫R3

|f(x, v)||v|2dv dx

) 3−2λ02ν0

= CEkin(|f |)3−2λ02ν0

(∫R3

∫R3

|f(x, v)|κ0+1κ0 |x ∧ v|

2λ0κ0 dv dx

)κ0ν0

which proves the result.

In order to estimate the potential energy term, we will need to relate the usual Lp normswith the Rν0,λ0 norm. In general these two norms are not equivalent on spaces of functionsdefined over R3. However, when we restrict to bounded neighbourhoods of the origin, we havethe following relations.

Lemma 3.20. Let 1 ≤ p <∞ and s > 0.

(i) There is an inclusionLploc

(Rd,Ld

)→ Lploc

(Rd, | · |sLd

)which is continuous as a linear map between Frechet spaces: For R > 0,

‖ · ‖Lp(RBd,|·|sLd) ≤ Rsp ‖ · ‖Lp(RBd,Ld) .


(ii) If 1 ≤ q < p and qsp−q < d, then there is an inclusion

Lploc

(Rd, | · |sLd

)→ Lqloc

(Rd,Ld

)which is continuous as a linear map between Frechet spaces: For R > 0,

‖ · ‖Lq(RBd,Ld) ≤(∣∣Sd−1

∣∣ p− q(p− q)d− qs

) p−qpq

R(p−q)d−qs

pq ‖ · ‖Lp(RBd,|·|sLd) .

Proof. The first statement is obvious from

‖f‖pLp(RBd,|·|sLd)

=

∫RBd|f(x)|p|x|s dx ≤ Rs

∫RBd|f(x)|p dx = Rs‖f‖p

Lp(RBd,Ld).

The second statement is an easy exercise in Holder’s inequality:

‖f‖qLq(RBd,Ld)

=

∫RBd|x|−

sqp (|f(x)|p|x|s)

qp dx

≤(∫

RBd|x|−

sqp−q dx

) p−qp(∫

RBd|f(x)|p|x|s dx

) qp

=

(∣∣Sd−1∣∣ (d− sq

p− q

)−1

Rd−sqp−q

) p−qp

‖f‖qLp(RBd,|·|sLd)

.

This completes the proof.

Lemma 3.21. Let 1 < ν < 3 and 0 < λ < 1 be such that ν + 4λ < 5. Then there exists aconstant τ = τ(ν, λ) ∈ (0, 1) such that any ρ ∈ L1(R3) ∩Rν,λ satisfies∣∣∣∣∫

R3

∫R3

ρ(x)ρ(y)

|x− y|dy dx

∣∣∣∣ ≤ C (ν, λ, ‖ρ‖L1(R3)

)‖ρ‖

(ν+1)ν τ

Rν,λ+ C

(‖ρ‖L1(R3)

).

The constants can be chosen to be non-decreasing in ‖ρ‖L1(R3).

Proof. Without loss of generality we may assume that ρ ≥ 0. We partition the domain ofintegration, (x, y) ∈ R3 × R3, into

A1 := |x− y| ≥ 1 ∩(ρ(x) ≤ 1 ∪ ρ(y) ≤ 1

),

A2 := |x− y| ≥ 1 ∩ ρ(x) > 1 ∩ ρ(y) > 1 ,A3 := |x− y| < 1 ,

and estimate the integral over each of these regions.The integrals over A1 and A3 are easily estimated:∫∫

A1

ρ(x)ρ(y)

|x− y|dy dx ≤ 2

∫R3

ρ(y)

∫B3

dx

|x− y|dy = 4π‖ρ‖L1(R3) ,∫∫

A3

ρ(x)ρ(y)

|x− y|dy dx ≤

∫R3

∫R3

ρ(x)ρ(y) dy dx = ‖ρ‖2L1(R3) .

We now turn to the integral over A2. Fix α, σ ∈ R such that

6

5< α <

3(ν + 1)

3ν + 2λ< σ <

ν + 1

ν

and

τ :=2ν

ν + 1

σ

σ − 1

α− 1

α< 1 .


We claim that this is possible. Indeed, the hypothesis ν + 4λ < 5 guarantees 65 <

3(ν+1)3ν+2λ , so

there exist α, σ in the ranges specified. Moreover, σσ−1

α−1α decreases when σ increases or α

decreases, and so τ can be made slightly larger than ν3 , by choosing α slightly larger than 6

5and σ slightly smaller than ν+1

ν .Define β > 0 by

2

α+

1

β:= 2 .

The condition 65 < α < ν+1

ν < 2 then guarantees 1 < β < 3.Let ρ be the symmetric decreasing rearrangement of ρ1ρ≥1. By Markov’s inequality, ρ

is supported in C‖ρ‖L1(R3)B3 for some constant C independent of ν, λ, ρ. Moreover, we have,from Corollary A.5, that

‖ρ‖L1(R3) = ‖ρ‖L1(R3) , ‖ρ‖Rν,λ ≤ ‖ρ‖Rν,λ .

Using Holder’s inequality and Lemma 3.20,

‖ρ‖αLα(R3) ≤ ‖ρ‖

σ−ασ−1

L1(R3) ‖ρ‖

α−1σ−1σ

Lσ(R3)

≤ ‖ρ‖σ−ασ−1

L1(R3)

[C(ν, λ, ‖ρ‖L1(R3)

)‖ρ‖Rν,λ

] σσ−1 (α−1)

= C(ν, λ, ‖ρ‖L1(R3)

)‖ρ‖

σσ−1 (α−1)

Rν,λ

where the constant C = C(ν, λ, ‖ρ‖L1(R3)) can be chosen to be of the form C(ν, λ)‖ρ‖θL1(R3)

for some θ > 0. Using Riesz’s rearrangement inequality (Theorem A.6) and Proposition 3.14,we have ∫∫

A2

ρ(x)ρ(y)

|x− y|dy dx ≤

∫R3

∫R3

ρ(x)1B3(x− y)

|x− y|ρ(y) dy dx

≤∥∥∥∥1B3

| · |

∥∥∥∥Lβ(R3)

‖ρ‖2Lα(R3)

≤ C(ν, λ, ‖ρ‖L1(R3)

)‖ρ‖2

σσ−1

α−1α

Rν,λ

≤ C(ν, λ, ‖ρ‖L1(R3)

)‖ρ‖

(ν+1)ν τ

Rν,λ.


Lemma 3.22. If α, β ≥ 0 satisfy α+β < 1, then there exist a ∈ (α, 1), b ∈ (β, 1), C > 0, suchthat

AαBβ ≤ α

aAa +

β

bBb + C for all A,B ∈ [0,∞) .

Proof. Pick p, q > 1 such that α < 1p , β < 1

q and 1p + 1

q < 1. The usual Young’s inequalitythen gives

AαBβ ≤ 1

pApα +

1

qBqβ +

(1− 1

p− 1

q

)for all A,B ∈ [0,∞) .

So the lemma follows from setting a := pα and b := qβ.

With the preceding lemmas in place, we are now ready to prove an estimate which showsthat hM > −∞ for M > 0. In other words, our minimisation problem makes sense.

Proposition 3.23. Assume κ0 > 0 and λ0 ∈ (0, 1) are such that ν0 := 3+2κ0−2λ0

2 satisfies1 < ν0 < 3 and ν0 + 4λ0 < 5. Assume also

Φ(h, L) ≥ C0hκ0+1κ0 L

λ0κ0 .


Then there exist a = a(κ0, λ0) ∈ (0, 1), b = b(κ0, λ0) ∈ (0, 1) such that for any fixed M > 0,there exist constants C = C(κ0, λ0, C0,M) which satisfies the property that whenever f ∈ FMhas Ekin(f) <∞, it holds that

HC(f) ≥ C [Ekin(f)− CEkin(f)a] + C

[‖f‖

κ0+1κ0

Fκ0,λ0− C

(‖f‖

κ0+1κ0

Fκ0,λ0

)b]− C .

In particular, hM > −∞.

Proof. In this proof, C always denotes generic positive constants, which may have differentvalues, but depending only on κ0, λ0, C0,M .

Pick τ for (ν0, λ0) as in Lemma 3.21. Let f ∈ FM . Then, using Lemmas 3.21 and 3.19,

HC(f) = Ekin(f) +

∫R3

∫R3

Φ(f(x, v), |x ∧ v|2

)dv dx− |Epot(f)|

≥ Ekin(f) + C0‖f‖κ0+1κ0

Fκ0,λ0− C

∥∥ρ[f ]

∥∥ (ν0+1)ν0

τ

Rν0,λ0− C

≥ Ekin(f) + C0‖f‖κ0+1κ0

Fκ0,λ0− C

[Ekin(f)

3−2λ02(ν0+1) ‖f‖

κ0+1ν0+1

Fκ0,λ0

] (ν0+1)ν0

τ

− C

= Ekin(f) + C2‖f‖κ0+1κ0

Fκ0,λ0− CEkin(f)

3−2λ02ν0

τ

(‖f‖

κ0+1κ0

Fκ0,λ0

)κ0ν0τ

− C .

Since 3−2λ0

2ν0τ + κ0

ν0τ = τ < 1, we may use Lemma 3.22 to split the middle term as

Ekin(f)3−2λ02ν0

τ

(‖f‖

κ0+1κ0

Fκ0,λ0

)κ0ν0τ

≤ CEkin(f)a + C

(‖f‖

κ0+1κ0

Fκ0,λ0

)b+ C

with a > 3−2λ0

2ν0τ and b > κ0

ν0τ , but a+ b < 1. Hence,

HC(f) ≥ C [Ekin(f)− CEkin(f)a] + C

[‖f‖

κ0+1κ0

Fκ0,λ0− C

(‖f‖

κ0+1κ0

Fκ0,λ0

)b]− C

where the right-hand side is bounded below. Since f ∈ FM was arbitrary, we are done.

Corollary 3.24. Let assumptions (i)–(iii) hold for Φ. Then

0 > hM > −∞ for any M > 0 .

Moreover, if fjj∈N ⊂ FM is a minimising sequence for HC over FM , then

• The sequences Ekin(fj)j∈N, |Epot(fj)|j∈N, C(fj)j∈N are bounded.

• fjj∈N is a bounded sequence in Fκ0,λ0 .

• ρ[fj ]j∈N is a bounded sequence in Rν0,λ0 .

With these boundedness properties for a minimising sequence, it is now possible to provethe following compactness result, which is an analogue of Proposition 3.15. Notice, however,that since we now have control over the Rν0,λ0 instead of a Lebesgue space norm, one of theintegrals has to be estimated using a different technique.

Proposition 3.25. Let ν > 1 and 0 < λ < 1 satisfy ν + 4λ < 5. Let ρjj∈N ∪ ρ0 ⊂ Rν,λ

be such thatρj ρ0 weakly in Rν,λ ,

∀ ε > 0 ∃ R > 0 : lim supj→∞

∫R3\RB3

|ρj(x)| dx < ε ,

Then ∇U[ρj ] → ∇U[ρ0] strongly in (L2(R3))3.


Proof. For any r > 0, Holder’s inequality gives∫rB3

|ρ(x)| dx ≤(∫

rB3

|x|−2λdx

) 1ν+1(∫

rB3

|ρ(x)|ν+1ν |x| 2λν dx

) νν+1

≤ C(ν, λ)‖ρ‖Rν,λ .

Therefore the hypotheses guarantee that ρjj∈N is bounded in L1(R3), and weak convergenceguarantees that ρ0 ∈ L1(R3) as well.

The sequence σj := ρj − ρ0 converges weakly to 0 in Rν,λ, and is bounded in L1(R3). Weneed to show ∇U[σj ] → 0 strongly in (L2(R3))3, which is equivalent to showing∫

R3

∫R3

σj(x)σj(y)

|x− y|dy dx→ 0 . (∗)

For convenience, denote

P := supj∈N‖σj‖L1(R3) , Q := sup

j∈N‖σj‖Rν,λ .

In addition, fix some

α ∈(

6

5,

3(ν + 1)

3ν + 2λ

).

This is possible since the hypothesis ν + 4λ < 5 guarantees 65 <

3(ν+1)3ν+2λ . Define β > 0 by

2

α+

1

β:= 2 .

The condition 65 < α < ν+1

ν < 2 then guarantees 1 < β < 3.Let δ > 0 be small and R > 0 be large, to be chosen later. We partition the domain of

integration, (x, y) ∈ R3 × R3, into

A1 := |x− y| ≥ δ ∩(|x| ≥ R ∪ |y| ≥ R

),

A2 := |x− y| ≥ δ ∩ |x| < R ∩ |y| < R ,Bj,1 := |x− y| < δ ∩ (|σj(x)| ≤ P ∪ |σj(y)| ≤ P) ,Bj,2 := |x− y| < δ ∩ |σj(x)| > P ∩ |σj(y)| > P .

We will estimate the integral (∗) over each of these sets.The integral over A1 is estimated by∫∫A1

|σj(x)| |σj(y)||x− y|

dx dy ≤ 2

δ

∫R3

∫R3\RB3

|σj(x)| |σj(y)| dx dy ≤ 2P

δ

∫R3\RB3

|σj(y)| dy .

We next consider the integral over A2. Let

hj(x) := |x|− 2λν 1RB3(x)

∫R3

1R3\δB3(x− y)

|x− y|1RB3(y)σj(y) dy

= |x|− 2λν 1RB3(x)

∫R3

(1R3\δB3(x− y)

|x− y||y|− 2λ

ν 1RB3(y) σj(y)

)|y| 2λν dy .

For any x ∈ R3, a straightforward estimate yields

1R3\δB3(x− · )|x− · |

| · |− 2λν 1RB3 ∈ Lν+1

(R3, | · | 2λν L3

)=(Rν,λ

)′.

Since σj 0 in Rν,λ, we see that hj → 0 pointwise on R3 \ 0. On the other hand, we have

the uniform bound |hj | ≤ Pδ | · |

− 2λν 1RB3 , so the dominated convergence theorem guarantees

that hj → 0 in Rν,λ. Thus∫∫A2

σj(x)σj(y)

|x− y|dx dy =

∫R3

hj(x)σj(x)|x| 2λν dx


tends to 0 as j →∞.The integral over Bj,1 is estimated by∫∫

Bj,1

|σj(x)| |σj(y)||x− y|

dx dy ≤ 2P

∫R3

|σj(y)|∫δB3

dx

|x− y|dy = CP 2δ2 .

We now turn to the rather tricky task of estimating the integral over Bj,2. Let σj bethe symmetric decreasing rearrangement of |σj |1|σj |>P. Since Markov’s inequality gives

L3|σj | > P ≤ C, we see that each σj is supported in CB3. Since α < 3ν+33ν+2λ , Lemma 3.20

applies to give ∥∥σj∥∥Lα(L3)≤ C(ν, λ, α)

∥∥σj∥∥Rν,λ .

Moreover, by Corollary A.5, ∥∥σj∥∥Rν,λ ≤ ‖σj‖Rν,λ ≤ Q .

Using Riesz’s rearrangement inequality (Theorem A.6) and Proposition 3.14, we obtain thefollowing estimate for the integral (∗) over Bj,2.∫∫

Bj,2

|σj(x)| |σj(y)||x− y|

dx dy ≤∫R3

∫R3

σj (x)1δB3(x− y)

|x− y|σj (y) dx dy

≤ C(α)

∥∥∥∥1δB3

| · |

∥∥∥∥Lβ(R3)

∥∥σj∥∥2

Lα(R3)

≤ C(ν, λ, α)Q2δ3−ββ .

Finally, we collect our estimates above to conclude∣∣∣∣∫R3

∫R3

σj(x)σj(y)

|x− y|dx dy

∣∣∣∣ ≤ CP 2δ2 + C(ν, λ, α)Q2δ3−ββ

+2P

δ

∫R3\RB3

|σj(y)| dy +

∣∣∣∣∫∫A2

σj(x)σj(y)

|x− y|dx dy

∣∣∣∣Now, given ε > 0, choose δ > 0 small enough so that

CP 2δ2 + C(ν, λ, α)Q2δ3−ββ <

ε

2.

Then choose R > 0 large enough so that

2P

δlim supj→∞

∫R3\RB3

|σj(y)| dy <ε

2.

Since the integral over A2 tends to 0 as j →∞, we find

lim supj→∞

∣∣∣∣∫R3

∫R3

σj(x)σj(y)

|x− y|dx dy

∣∣∣∣ < ε

2.

Since ε > 0 was arbitrary, we have proved the desired result (∗).

The last result the author managed to prove says that, along a minimising sequence for HC

over FM , some minimal mass must remain within a ball of sufficiently large radius. Specifically,we have the following result. The proof uses a similar strategy to that employed in the proofof the preceding Proposition 3.25.

Proposition 3.26. Let assumptions (i)–(iii) hold for Φ.Let fjj∈N be a minimising sequence for HC over FM . Then there exist m0 > 0, R0 > 0

and a sequence zjj∈N ⊂ R3, such that∫zj+R0B3

ρ[fj ](x) dx ≥ m0 for sufficiently large j ∈ N .


Proof. Recalling from Corollary 3.24 that hM < 0, we may, after discarding the first few termsof the minimising sequence fjj∈N if necessary, assume without loss of generality that

hM2

> HC(fj) for all j ∈ N .

Since also ρ[fj ]j∈N is bounded in Rν0,λ0 , we may let

P := supj∈N

∥∥ρ[fj ]

∥∥Rν0,λ0

.

Fix some

α ∈(

6

5,

3(ν0 + 1)

3ν0 + 2λ0

).

This is possible since ν0 + 4λ0 < 5 guarantees 65 <

3(ν0+1)3ν0+2λ0

. Define β > 0 by

2

α+

1

β:= 2 .

The condition 65 < α < ν0+1

ν0< 2 then guarantees 1 < β < 3.

Let R > 1 be large, to be chosen later. We partition (x, y) ∈ R3 × R3 into

A1 := |x− y| ≥ R , A2 :=R−1 ≤ |x− y| ≤ R

,

Bj,1 :=|x− y| < R−1

∩(ρ[fj ](x) ≤ 1

∪ρ[fj ](y) ≤ 1

),

Bj,2 :=|x− y| < R−1

∩ρ[fj ](x) > 1

∪ρ[fj ](y) > 1

.

We have the estimates∫∫A1

ρ[fj ](x)ρ[fj ](y)

|x− y|dx dy ≤ 1

R

∫∫A1

ρ[fj ](x)ρ[fj ](y) dx dy =M2

R,

and ∫∫A2


|x− y|dx dy ≤ R

∫R3

∫y+RB3

ρ[fj ](x)ρ[fj ](y) dx dy

≤MR supy∈R3

∫y+RB3

ρ[fj ](x) dx ,

and ∫∫Bj,1


|x− y|dx dy ≤ 2

∫R3

ρ[fj ](y)

∫y+R−1B3

dx

|x− y|dy =

4πM

R.

Let ρj be the symmetric decreasing rearrangement of ρ[fj ]1ρ[fj ]>1. By Markov’s inequality,

each ρj is supported in CPB3 for some C > 0 independent of j. Using Lemma 3.20 andCorollary A.5, we have∥∥ρj∥∥Lα(R3)

≤ C(ν0, λ0, α, P )∥∥ρj∥∥Rν0,λ0 ≤ C(ν0, λ0, α, P )

∥∥ρ[fj ]

∥∥Rν0,λ0

≤ C(ν0, λ0, α, P ) .

Therefore, by Riesz’s rearrangement inequality (Theorem A.6) and Proposition 3.14,∫∫Bj,2


|x− y|dx dy ≤

∫R3

∫R3

ρj (x)1R−1B3(x− y)

|x− y|ρj (y) dx dy

≤ C(α)

∥∥∥∥1R−1B3

| · |

∥∥∥∥Lβ(R3)

∥∥ρj∥∥2

Lα(R3)

≤ C(ν0, λ0, α, P )R−3−ββ .


Combining the estimates above, we find

−Epot(fj) =

∫R3

∫R3


|x− y|dx dy

≤ C(M)

R+ C(ν0, λ0, α, P )R−

3−ββ +MR sup

y∈R3

∫y+RB3

ρ[fj ](x) dx .

Since hM2 > Epot(fj) for all j, we have

hM2

> −C(M)

R− C(ν0, λ0, α, P )R−

3−ββ −MR sup

y∈R3

∫y+RB3

rho[fj ](x) dx ,

which on rearranging gives

supy∈R3

∫y+RB3

rho[fj ](x) dx >1

MR

[|hM |

2− C(M)

R− C(ν0, λ0, α, P )R−

3−ββ

].

Taking R = R0 > 1 sufficiently large, so that the right-hand side is strictly positive, provesthe assertion.

One might hope that using Proposition 3.26, a minimising sequence could, after translation,be made to satisfy the hypothesis of the compactness result, Proposition 3.25. This is, in fact,the strategy employed in [20], pages 70–72.

Unfortunately, this turns out not to work here: for our minimisation problem, a minimisingsequence would not remain minimising after translation, so one cannot employ the directmethod to construct minimisers. It might be possible to salvage the situation by showing thatthe sequence zjj∈N in Proposition 3.26 can be chosen to be bounded, but the author hasnot been successful in doing so.

3.5 More Recent Stability Results

The proofs of the stability results presented earlier in this section relied on the fact that theCasimir functional is conserved along flows of the Vlasov-Poisson system (this means thatC(f(t)) = C(f(0)) for a time-dependent solution f to the Vlasov-Poisson system) in eachof the situations considered. This holds true when considering radial perturbations aboutanisotropic, spherically symmetric steady states (Subsection 3.2) or arbitrary perturbationsabout isotropic steady states (Subsection 3.3).

Toward the end of this mini-project, the author was introduced to a series of papers byLemou, Mehats and Raphael proving stability results by instead exploiting monotonicity ofthe Hamiltonian H under generalised notions of symmetric rearrangement. Their methodsapply to large classes of steady states, including those not yet treated by the energy-Casimirmethods; one such model is the so-called King model, with

f(x, v) =(exp

(E0 − E[f ](x, v)

)− 1)

+.

One of their results involves the stability of anisotropic, spherically symmetric modelsunder radial perturbations, as stated in the following result. The full proof can be found in[9], while a summary of the method of proof can be found in [8].

Theorem 3.27. Let f0(x, v) be a continuous, non-negative, compactly supported steady solu-tion to the Vlasov-Poisson system, satisfying the functional form

f0(x, v) = F(E[f0](x, v), |x ∧ v|2

).

Assume that there exists E0 < 0 such that F > 0 ⊆ (−∞, E0)×[0,∞),0 F is C1 on F > 0,and

∂F

∂E< 0 on F > 0 .

A REARRANGEMENTS OF FUNCTIONS Page 35 of 38

Then f0 is stable under radial pertubations in the following sense: For all L > 0, ε > 0, thereexists δ > 0 such that whenever f(0) ∈ C1

c (R6) is spherically symmetric with

‖f(0)− f0‖L1(R6) ≤ δ , ‖f(0)‖L∞(R6) ≤ ‖f0‖L∞(R6) + L and |H(f(0))−H(f0)| ≤ δ ,

the corresponding global solution f satisfies∥∥(1 + |v|2)

(f(t)− f0)∥∥L1(R6)

≤ ε .

The strategy of proof was later adapted to obtain the following stability result for isotropicsteady states under arbitrary perturbations. Its full proof can be found in [10].

Theorem 3.28. Let f0(x, v) be a continuous, non-negative, compactly supported steady solu-tion to the Vlasov-Poisson system, satisfying the functional form

f0(x, v) = F(E[f0](x, v)

).

Assume that there exists E0 < 0 such that F > 0 ⊆ (−∞, E0), F is C1 on F > 0, and

F ′ < 0 on F > 0 .

Then f0 is stable in the following sense: For all L > 0, ε > 0, there exists δ > 0 such thatwhenever 0 ≤ f(0) ∈ L1(R6) ∩ L∞(R6) with |v|2f(0) ∈ L1(R6) is such that

‖f(0)− f0‖L1(R6) ≤ δ , ‖f(0)‖L∞(R6) ≤ ‖f0‖L∞(R6) + L and H(f(0)) ≤ H(f0) + δ ,

there is a translation shift z = z(t) such that the corresponding global solution f satisfies∥∥(1 + |v|2)

(f(t, x, v)− f0(x− z(t), v))∥∥L1(R6)

≤ ε .

A Rearrangements of Functions

The preceding portions of this essay have made heavy use of rearrangements of functions. Inthis short appendix, we remind the reader of the basic notions and results. Our treatment isbased on [11], Chapter 3.

Throughout this appendix, Ld denotes the Lebesgue measure on Rd.

Definition A.1. If A ⊂ Rd is a Borel set with Ld(A) <∞, its symmetric rearrangement is

A :=

(Ld(A)

Ld(Bd)

) 1d

Bd ,

i.e. the open ball centred at the origin with the same Lebesgue measure as A.

Definition A.2. We say a Borel measurable function f : Rd → C vanishes at infinity if

Ld|f | > λ <∞ for every λ > 0 .

For a function f : Rd → C which vanishes at infinity, we define its symmetric decreasingrearrangement to be the function f : Rd → [0,∞) given by

f(x) :=

∫ ∞0

1(|x|d,∞)

(Ld|f | > λLd(Bd)

)dλ .

Remark A.3. Observe that

x ∈ |f | > λ ⇐⇒ |x|d < Ld|f | > λLd(Bd)

.

Therefore the symmetric decreasing rearrangement of f satisfies

f(x) =

∫ ∞0

1|f |>λ(x) dλ .

This is, in fact, the definition provided in [11], Chapter 3. The equivalent Definition A.2 is,however, easier to work with.

A REARRANGEMENTS OF FUNCTIONS Page 36 of 38

The symmetric decreasing rearrangement enjoys the following nice properties, whose proofs,when not given, are straightforward from Definition A.2.

(i) f(x) <∞ for every x ∈ Rd \ 0 (since f vanishes at infinity).

Moreover, f(0) <∞ if and only if f ∈ L∞(Rd).

(ii) f is radially symmetric and non-increasing.

(iii) f is a lower-semicontinuous function.

Proof. If xkk∈N ⊂ Rd converges to x ∈ Rd, then

1(|x|d,∞)

(Ld|f | > λLd(Bd)

)≤ lim inf

k→∞1(|xk|d,∞)

(Ld|f | > λLd(Bd)

)for every λ > 0 .

Therefore, Fatou’s lemma gives∫ ∞0

1(|x|d,∞)

(Ld|f | > λLd(Bd)

)dλ ≤ lim inf

k→∞

∫ ∞0

1(|xk|d,∞)

(Ld|f | > λLd(Bd)

)dλ

which proves the claim.

(iv) f > λ = |f | > λ.

Proof. We have

f(x) > t ⇐⇒ ∃k ∈ N : 1(|x|d,∞)

(Ld|f | > t+ 1

k

Ld(Bd)

)= 1

⇐⇒ x ∈⋃k∈N

(Ld |f | > t+ 1k

Ld(Bd)

) 1d

Bd

and observe this last set is

⋃k∈N

(Ld |f | > t+ 1k

Ld(Bd)

) 1d

Bd =

limk→∞

(Ld|f | > t+ 1

k

Ld(Bd)

) 1d

Bd

=

(Ld |f | > tLd(Bd)

) 1d

Bd

= |f | > t .


Theorem A.4. Let f, g : Rd → [0,∞) be non-negative functions vanishing at infinity, and letf, g be their respective symmetric decreasing rearrangements. Then∫

Rdf(x)g(x) dx ≤

∫Rdf(x)g(x) dx .

Proof. Firstly, observe that if A,B ⊂ Rd are Borel subsets of finite Lebesgue measure, then

Ld(A ∩B) ≤ Ld(A ∩B) .

Indeed, without loss of generality we may assume Ld(A) ≤ Ld(B); then A ⊆ B, and hencewe have Ld(A ∩B) ≤ Ld(A) = Ld(A) = Ld(A ∩B).

In particular, for any s, t > 0, we have

Ld (f > s ∩ g > t) ≤ Ld(f > s ∩ g > t

)= Ld (f > s ∩ g > t) ,

REFERENCES Page 37 of 38

that is ∫Rd1f>s(x)1g>t(x) dx ≤

∫Rd1f>s(x)1g>t(x) dx .

Therefore, using the layer-cake representation and Tonelli’s theorem, we find that∫Rdf(x)g(x) dx =

∫ ∞0

∫ ∞0

∫Rd1f>s(x)1g>t(x) dx dt ds

≤∫ ∞

0

∫ ∞0

∫Rd1f>s(x)1g>t(x) dx dt ds =

∫Rdf(x)g(x) dx .


Corollary A.5. Let ϕ : Rd → [0,∞) be a radially symmetric non-decreasing function. Let0 ≤ f ∈ L1(Rd), and let f be the symmetric decreasing rearrangement of f . Then∫

Rdϕ(x)f(x) dx ≥

∫Rdϕ(x)f(x) dx .

Proof. Using the monotone convergence theorem, we may assume without loss of generalitythat ϕ is bounded. The claim then follows from applying Theorem A.4 to the functions‖ϕ‖L∞ − ϕ and f , which are both vanishing at infinity.

Symmetric decreasing rearrangements also satisfy Riesz’s rearrangement inequality, whichwe have used repeatedly in the essay. Its proof, which we omit, is very long and technical,and can be found in [11], Theorem 3.7.

Theorem A.6 (Riesz’s Rearrangement Inequality). Let f, g, h : Rd → [0,∞) be non-negativefunctions vanishing at infinity, and let f, g, h be the symmetric decreasing rearrangementsof f, g, h respectively. Then∫

Rd

∫Rdf(x)g(x− y)h(y) dx dy ≤

∫Rd

∫Rdf(x)g(x− y)h(y) dx dy ,

References

[1] J. Batt, W. Faltenbacher, E. Horst: Stationary Spherically Symmetric Models in StellarDynamics. Arch. Rational Mech. Anal. 93 (1986), 159–183.

[2] J. Binney, S. Tremaine: Galactic Dynamics (2nd edition). Princeton University Press,Princeton, NJ (2008).

[3] J. M. Borwein, A. S. Lewis: Convex Analysis and Nonlinear Optimization – Theory andExamples (2nd edition). Springer Science+Business Media, New York, NY (2006).

[4] G. L. Camm: Self-Gravitating Star Systems. II. Mon. Not. R. Astr. Soc. 112 (1952),155–176.

[5] Y. Guo, G. Rein: Existence and stability of Camm type steady states in galactic dynamics.Indiana Univ. Math. J. 48 (1999), 1237–1255.

[6] D. D. Holm, J. E. Marsden, T. Ratiu, A. Weinstein: Nonlinear Stability of Fluid andPlasma Equilibria. Phys. Reports 123 (1985), 1–116.

[7] J. H. Jeans: On the Theory of Star-Streaming and the Structure of the Universe. Mon.Not. R. Astr. Soc. 76 (1915), 70–84.

[8] M. Lemou, F. Mehats, P. Raphael: A new variational approach to the stability of gravi-tational systems. C. R. Acad. Sci. Paris, Ser. I 347 (2009), 979–984.

REFERENCES Page 38 of 38

[9] M. Lemou, F. Mehats, P. Raphael: A New Variational Approach to the Stability of Grav-itational Systems. Comm. Math. Phys. 302 (2011), 161–224.

[10] M. Lemou, F. Mehats, P. Raphael: Orbital stability of spherical galactic models. Invent.Math. 187 (2012), 145–194.

[11] E. H. Lieb, M. Loss: Analysis (2nd edition). American Mathematical Society, Providence,RI (2001).

[12] P.-L. Lions, B. Perthame: Propagation of moments and regularity for the 3-dimensionalVlasov-Poisson system. Invent. Math. 105 (1991), 415–430.

[13] G. Loeper: Uniqueness of the solution to the Vlasov-Poisson system with bounded density.J. Math. Pures. Appl. 86 (2006), 68–79.

[14] K. Pfaffelmoser: Global classical solutions of the Vlasov-Poisson system in three dimen-sions for general initial data. J. Diff. Eqns 95 (1992), 281–303.

[15] G. Rein: Non-Linear Stability for the Vlasov-Poisson System – The Energy-CasimirMethod. Math. Meth. Appl. Sci. 17 (1994), 1129–1140.

[16] G. Rein: Static shells for the Vlasov-Poisson and Vlasov-Einstein systems. Indiana Univ.Math. J. 48 (1999), 335–346.

[17] G. Rein: Stationary and static stellar dynamic models with axial symmetry. NonlinearAnalysis 41 (2000), 313–344.

[18] G. Rein: Stability of Spherically Symmetric Steady States in Galactic Dynamics againstGeneral Perturbations. Arch. Rational Mech. Anal. 161 (2002), 27–42.

[19] G. Rein: Reduction and a Concentration-Compactness Principle for Energy-CasimirFunctionals. SIAM J. Math. Anal. 33 (2002), 896–912.

[20] G. Rein: Collisionless Kinetic Equations from Astrophysics – The Vlasov-Poisson System,in Handbook of Differential Equations, Evolutionary Equations. Vol. 3. (C. Dafermos, E.Feireisl eds.) Elsevier / North-Holland, Amsterdam (2007), 383–476.

[21] G. Rein: Galactic dynamics in MOND – Existence of quilibria with finite mass andcompact support. arXiv:1312.3765 (2013)

[22] R. T. Rockafellar: Convex Analysis. Princeton University Press, Princeton, NJ (1970).

[23] J. Schaeffer: Global existence of smooth solutions to the Vlasov-Poisson system in threedimensions. Comm. Partial Diff. Eqns. 16 (1991), 1313–1335.

[24] J. Schaeffer: A Class of Counterexamples to Jeans’ Theorem for the Vlasov-EinsteinSystem. Comm. Math. Phys. 204 (1999), 313–327.

[25] A. Schulze: Existence of Axially Symmetric Solutions to the Vlasov-Poisson System De-pending on Jacobi’s Integral. Comm. Math. Sci. 6 (1999), 711–727.

Documents

Jeans’s Theorem in Kinetic Theory Min Harold... · widely used in such scienti c disciplines as plasma physics or stellar dynamics. The Vlasov-Poisson system consists of a Vlasov