UNSTABLE AND STABLE GALAXY MODELS
YAN GUO AND ZHIWU LIN
Abstract. To determine the stability and instability of a given steady galaxy
configuration is one of the fundamental problems in the Vlasov theory for
galaxy dynamics. In this article, we study the stability of isotropic spherical
symmetric galaxy models f0(E), for which the distribution function f0 depends
on the particle energy E only. In the first part of the article, we derive the first
sufficient criterion for linear instability of f0(E) : f0(E) is linearly unstable if
the second-order operator
A0 ≡ −∆+ 4π
(E){I −P}dv
has a negative direction, where P is the projection onto the function space
{g(E,L)}, L being the angular momentum [see the explicit formulae (27) and
(26)]. In the second part of the article, we prove that for the important King
model, the corresponding A0 is positive definite. Such a positivity leads to the
nonlinear stability of the King model under all spherically symmetric pertur-
bations.
1. Introduction
A galaxy is an ensemble of billions of stars, which interact by the gravitational
field which they create collectively. For galaxies, the collisional relaxation time is
much longer than the age of the universe ([8]). The collisions can therefore be
ignored and the galactic dynamics is well described by the Vlasov - Poisson system
(collisionless Boltzmann equation)
(1) ∂tf + v · ∇xf −∇xU · ∇vf = 0, ∆U = 4π
f(t, x, v)dv,
where (x, v) ∈ R3 × R3, f(t, x, v) is the distribution function and Uf (t, x) is its
gravitational potential. The Vlasov-Poisson system can also be used to describe the
dynamics of globular clusters over their period of orbital revolutions ([11]). One of
the central questions in such galactic problems, which has attracted considerable
attention in the astrophysics literature, of [7], [8], [11], [31] and the references there,
is to determine dynamical stability of steady galaxy models. Stability study can be
used to test a proposed configuration as a model for a real stellar system. On the
other hand, instabilities of steady galaxy models can be used to explain some of the
striking irregularities of galaxies, such as spiral arms as arising from the instability
of an initially featureless galaxy disk ([7]), ([32]).
In this article, we consider stability of spherical galaxies, which are the simplest
elliptical galaxy models. Though most elliptical galaxies are known to be non-
spherical, the study of instability and dynamical evolution of spherical galaxies
could be useful to understand more complicated and practical galaxy models . By
Jeans’s Theorem, a steady spherical galaxy is of the form
f0(x, v) ≡ f0(E,L2),
http://arxiv.org/abs/0704.1012v2
2 YAN GUO AND ZHIWU LIN
where the particle energy and total momentum are
|v|2 + U0(x), L2 = |x× v|2 ,
and U0(x) = U0 (|x|) satisfies the self-consistent Poisson equation. The isotropic
models take the form
f0(x, v) ≡ f0(E).
The cases when f ′0(E) < 0 has been widely studied and these models are known
to be linearly stable to both radial ([9]) and non-radial perturbations ([2]). The
well-known Casimir-Energy functional (as a Liapunov functional)
(2) H(f) ≡
Q(f) +
|v|2f − 1
|∇xUf |2,
is constant along the time evolution. If f ′0(E) < 0, we can choose the Casimir
function Q0 such that
Q′0(f0(E)) ≡ −E
for all E. By a Taylor expansion of H(f)−H(f0), it follows that formally the first
variation at f0 is zero, that is, H(1)(f0(E)) = 0 (on the support of f0(E)), and the
second order variation of H at f0 is
(3) H(2)f0 [g] ≡
{f0>0}
−f ′0(E)
dxdv − 1
|∇xUg|2dx
where Q′′(f0) =
, g = f − f0 and ∆Ug =
gdv. In the 1960s, Antonov ([1],
[2]) proved that
(4) H(2)f0 [Dh] =
∫ ∫ |Dh|2
|f ′0(E)|
dxdv −
|∇ψh|2 dx
is positive definite for a large class of monotone models. Here
D = v · ∇x −∇xU0 · ∇v,
h(x, v) is odd in v and −∆ψ =
Dhdv. He showed that such a positivity is
equivalent to the linear stability of f0(E). In [9], Doremus, Baumann and Feix
proved the radial stability of any monotone spherical models. Their proof was
further clarified and simplified in [10], [37], [22], and more recently in [33], [21].
In particular, this implies that any monotone isotropic models are at least linearly
stable.
Unfortunately, despite its importance and a lot of research (e.g., [20], [5], [6],
[13]), to our knowledge, no rigorous and explicit instability criterion of non-monotone
models has been derived. When f ′0(E) changes sign, functional H
is indefinite
and it gives no stability information, although it seems to suggest that these mod-
els are not energy minimizers under symplectic perturbations. In this paper, we
first obtain the following instability criterion for general spherical galaxies. For
any function g with compact support within the support of f0(E), we define the
|f ′0(E)| −weighted L2
R3 ×R3
space L2|f ′0|
with the norm ‖·‖|f ′0| as
(5) ||h||2|f ′
|f ′0(E)|h2dxdv.
UNSTABLE AND STABLE GALAXY MODELS 3
Theorem 1.1. Assume that f0(E) has a compact support in x and v, and f
bounded. For φ ∈ H1, define the quadratic form
(6) (A0φ, φ) =
|∇φ|2dx + 4π
f ′0(E) (φ− Pφ)
dxdv,
where P is the projector of L2|f ′0|
kerD =
and more explicitly Pφ is given by (18) for radial functions and (26) for general
functions. If there exists φ0 ∈ H1 such that
(7) (A0φ0, φ0) < 0,
then there exists λ0 > 0 and φ ∈ H2, f (x, v) given by (14), such that eλ0t[f, φ] is
a growing mode to the Vlasov-Poisson system (1) linearized around [f0(E), Uf0 ] .
A similar instability criterion can be obtained for symmetry preserving pertur-
bations of anisotropic spherical models f0
, see Remark 2. We note that the
term Pφ in the instability criterion is highly non-local and this reflects the collective
nature of stellar instability. The proof of Theorem 1.1 is by extending an approach
developed in [25] for 1D Vlasov-Poisson, which has recently been generalized to
Vlasov-Maxwell systems ([26], [28]). There are two elements in this approach. One
is to formulate a family of dispersion operators Aλ for the potential, depending
on a positive parameter λ. The existence of a purely growing mode is reduced to
find a parameter λ0 such that the Aλ0 has a kernel. The key observation is that
these dispersion operators are self-adjoint due to the reversibility of the particle
trajectories. Then a continuation argument is applied to find the parameter λ0
corresponding to a growing mode, by comparing the spectra of Aλ for very small
and large values of λ. There are two new complications in the stellar case. First,
the essential spectrum of Aλ is [0,+∞) and thus we need to make sure that the
continuation does not end in the essential spectrum.This is achieved by using some
compactness property due to the compact support of the stellar model. Secondly,
it is more tricky to find the limit of Aλ when λ tends to zero. For that, we need an
ergodic lemma (Lemma 2.4) and use the integrable nature of the particle dynamics
in a central field to derive an expression for the projection Pφ appeared in the limit.
In the second part of the article, we further study the nonlinear (dynamical)
stability of the normalized King model:
(8) f0 = [e
E0−E − 1]+
motivated by the study of the operator A0. The famous King model describes
isothermal galaxies and the core of most globular clusters [24]. Such a model
provides a canonical form for many galaxy models widely used in astronomy. Even
though f ′0 < 0 for the King model, it is important to realize that, because of the
Hamiltonian nature of the Vlasov-Poisson system (1), linear stability fails to imply
nonlinear stability (even in the finite dimensional case). The Liapunov functional is
usually required to prove nonlinear stability. In the Casimir-energy functional (2), it
is natural to expect that the positivity of such a quadratic formH(2)f0 [g] should imply
stability for f0(E). However, there are at least two serious mathematical difficulties.
First of all, it is very challenging to use the positivity of H(2)f0 [g] to control higher
4 YAN GUO AND ZHIWU LIN
order remainder in H(f)−H(f0) to conclude stability [38]. For example, one of the
remainder terms is f3 whose L2 norm is difficult to be bounded by a power of the
stability norm. The non-smooth nature of f0(E) also causes trouble here. Second of
all, even if one can succeed in controlling the nonlinearity, the positivity of H
is only valid for certain perturbation of the form g = Dh [22]. It is not clear at all
if any arbitrary, general perturbation can be reduced to the form Dh. To overcome
these two difficulties, a direct variational approach was initiated by Wolansky [39],
then further developed systematically by Guo and Rein in [14], [15], [17], [18], [19].
Their method avoids entirely the delicate analysis of the second order variation
H(2)f0 in (3), which has led to first rigorous nonlinear stability proof for a large class
of f0(E). The high point of such a program is the nonlinear stability proof for every
polytrope [18] f0(E) = (E0 − E)k+. Their basic idea is to construct galaxy models
by solving a variational problem of minimizing the energy under some constraints
of Casimir invariants. A concentration-compactness argument is used to show the
convergence of the minimizing sequence. All the models constructed in this way
are automatically stable.
Unfortunately, despite its success, the King model can not be studied by such a
variational approach. The Casimir function for a normalized King model is
(9) Q0(f) = (1 + f) ln(1 + f)− 1− f,
which has very slow growth for f → ∞. As a result, the direct variational method
fails. Recently, Guo and Rein [21] proved nonlinear radial stability among a class
of measure-preserving perturbations
Sf0 ≡
f(t, r, vr, L) ≥ 0 :
Q(f, L) =
Q(f0, L), for Q ∈ C∞c and Q(0, L) ≡ 0.
The basic idea is to observe that for perturbations in the class Sf0 , one can write
g = f − f0 as Dh = {h,E}. Therefore, H(2)f0 [g] = H
[Dh], for which the positivity
was proved in [22] for radial perturbations. To avoid the difficulty of controlling the
remainder term by H(2)f0 [g], an indirect contradiction argument was used in [21].
As our second main result of this article, we establish nonlinear stability of King’s
model for general perturbations with spherical symmetry:
Theorem 1.2. The King’s model f0 = [e
E0−E − 1]+ is nonlinearly stable under
spherically symmetric perturbations in the following sense: given any ε > 0 there
exists ε1 > 0 such that for any compact supported initial data f(0) ∈ C1c with
spherical symmetry, if d (f (0) , f0) < ε1 then
0≤t<∞
d (f (t) , f0) < ε,
where the distance functional d (f, f0) is defined by (35).
For the proof, we extended the approach in [27] for the 1 1
D Vlasov-Maxwell
model. To prove nonlinear stability, we study the Taylor expansion ofH(f)−H(f0).
Two difficulties as mentioned before are: to prove the positivity of the quadratic
form and to control the remainder. We use two ideas introduced in [27]. The first
idea is to use any finite number of Casimir functional Qi
f, L2
as constraints.
The difference from [21] is that we do not impose Qi
f, L2
f0, L
in the
perturbation class, but expand the invariance equationQi
f (t) , L2
f0, L
UNSTABLE AND STABLE GALAXY MODELS 5
f (0) , L2
f0, L
to the first order. In this way, we get a constraint for
g = f −f0 in the form that the coefficient of its projection to ∂1Qi
f0, L
is small.
Putting these constraints together, we deduce that a finite dimensional projection of
g to the space spanned by
f0, L
is small. To control the remainder term,
we use a duality argument. Noting that it is much easier to control the potential φ,
we use a Legendre transformation to reduce the nonlinear term in g to a new one
in φ only. The key observation is that the constraints on g in the projection form
are nicely suited to the Legendre transformation and yields a non-local nonlinear
term in φ only with the projections kept. By performing a Taylor expansion of this
non-local nonlinear term in φ, the quadratic form becomes a truncated version of
(A0φ, φ) defined by (6), whose positivity can be shown to be equivalent to that of
Antonov functional. The the remainder term now is only in terms of φ and can
be easily controlled by the quadratic form. The new complication in the stellar
case is that the steady distribution f0 (E) is non-smooth and compactly supported.
Therefore, we split the perturbation g into inner and outer parts, according to the
support of f0. For the inner part, we use the above constrainted duality argument
and the outer part is estimated separately.
2. An Instability Criterion
We consider a steady distribution
f0 (x, v) = f0(E)
has a bounded support in x and v and f ′0 is bounded, where the particle energy
E = 1
|v|2 + U0(x). The steady gravitational potential U0(x) satisfies a nonlinear
Poisson equation
∆U0 = 4π
f0dv.
The linearized Vlasov-Poisson system is
(11) ∂tf + v · ∇xf −∇xU0 · ∇vf = ∇xφ · ∇vf0, ∆φ = 4π
f(t, x, v)dv.
A growing mode solution (eλtf(x, v), eλtφ(x)) to (1) with λ > 0 satisfies
(12) λf + v · ∇xf −∇xU0 · ∇vf = f ′0v · ∇xφ.
We define [X(s;x, v), V (s;x, v)] as the trajectory of
dX(s;x,v)
= V (s;x, v)
dV (s;x,v)
= −∇xU0
such that X(0;x, v) = x, and V (0;x, v) = v. Notice that the particle energy E is
constant along the trajectory. Integrating along such a trajectory for −∞ ≤ s ≤ 0,
we have
f(x, v) =
eλsf ′0(E)V (s;x, v) · ∇xφ(X(s;x, v))ds(14)
= f ′0(E)φ(x) − f ′0(E)
λeλsφ(X(s;x, v))ds.
6 YAN GUO AND ZHIWU LIN
Plugging it back into the Poisson equation, we obtain an equation for φ
−∆φ+ [4π
f ′0(E)dv]φ − 4π
f ′0(E)
λeλsφ(X(s;x, v))dsdv = 0.
We therefore define the operator Aλ as
Aλφ ≡ −∆φ+ [4π
f ′0(E)dv]φ − 4π
f ′0(E)
λeλsφ(X(s;x, v))dsdv.
Lemma 2.1. Assume that f0(E) has a bounded support in x and v and f
bounded. For any λ > 0, the operator Aλ : H
2 → L2 is self-adjoint with the
essential spectrum [0,+∞) .
Proof. We denote
Kλφ = −4π[
f ′0(E)dv]φ + 4π
f ′0(E)
λeλsφ(X(s;x, v))dsdv.
Recall that f0 (x, v) = f0(E) has a compact support ⊂ S ⊂ R3x × R3v. We may
assume S = Sx×Sv, both balls in R3. Let χ = χ (|x|) be a smooth cut-off function
for the spatial support of f0 in the physical space Sx; that is, χ ≡ 1 on the spatial
support of f0 and has compact support inside Sx. Let Mχ be the operator of
multiplication by χ. Then Kλ = KλMχ =MχKλ =MχKλMχ. Indeed,
f ′0 (x, v) = f
0 (X(s;x, v), V (s;x, v))
because of the invariance of E under the flow. So
(Kλφ) (x) = −4π[
f ′0(E)dv]φ + 4π
f ′0(E)
λeλsφ(X(s;x, v))dsdv(15)
= −4π[
f ′0(E)dv]φ + 4π
∫ ∫ 0
λeλs (f ′0(E)φ) (X(s;x, v))dsdv
= (MχKλMχφ)(x).
First we claim that
‖Kλ‖L2→L2 ≤ 8π
|f ′0(E)| dv
Indeed, the L2 norm for the first term in Kλ is easily bounded by 4π
f ′0(E)dv
For the second term, we have for any ψ ∈ L2,
4πλeλsf ′0(E)φ(X(s;x, v))dsdvψ(x)dx|
|f ′0(E)|φ2(X(s;x, v))dvdx
|f ′0(E)|ψ2(x)dvdx
|f ′0(E)|φ2(x)dvdx
|f ′0(E)|ψ2(x)dvdx
|f ′0(E)|φ2(x)dvdx
|f ′0(E)|ψ2(x)dvdx
|f ′0(E)| dv
‖φ‖2 ‖ψ‖2 .
UNSTABLE AND STABLE GALAXY MODELS 7
Moreover, we have that Kλ is symmetric Indeed, for fixed s, by making a change of
variable (y, w) → (X(s;x, v), V (s;x, v)), so that (z, v) = (X(−s; y, w), V (−s; y, w)),
we deduce that
4πf ′0(E)
λeλsφ(X(s;x, v))dsdvψ(x)dx
4πf ′0(E)φ(y)ψ(X(−s; y, w))dydwds
4πf ′0(E)
λeλsψ(X(−s; y,−w))φ(y)dydwds
4πf ′0(E)
λeλsψ(X(s;x, v))φ(x)dvdxds.
Here we have used the fact [X(s; y, w), V (s; y, w)] = [X(−s; y,−w),−V (s; y,−w)]
in the last line. Hence
(Kλφ, ψ) = (φ,Kλψ).
Since Kλ = KλMχ and Mχ is compact from H
2 into L2 space with support in Sx,
so Kλ is relatively compact with respect to −∆. Thus by Kato-Relich and Weyl’s
Theorems, Aλ : H
2 → L2 is self-adjoint and σess(Aλ) = σess(−∆). �
Lemma 2.2. Assume that f ′0(E) has a bounded support in x and v and f
bounded. Let
k(λ) = inf
φ∈D(Aλ),||φ||2=1
(φ,Aλφ),
then k(λ) is a continuous function of λ when λ > 0. Moreover, there exists 0 <
Λ <∞ such that for λ > Λ
(17) k(λ) ≥ 0.
Proof. Fix λ0 > 0, φ ∈ D(Aλ), and ||φ||2 = 1. Then
k(λ0) ≤ (φ,Aλ0φ)
≤ (φ,Aλφ) + |(φ,Aλ0φ)− (φ,Aλφ)|
≤ (φ,Aλφ) + 4π
|f ′0(E)|
[λeλs − λ0eλ0s]φ(X(s;x, v))φ(x)dsdvdx
≤ (φ,Aλφ) + 4π
|f ′0(E)|
[λ̃|s|eλ̃s + eλ̃s]dλ̃φ(X(s;x, v))φ(x)dsdvdx
≤ (φ,Aλφ) + C
[λ̃|s|eλ̃s + eλ̃s]dλ̃ds
≤ (φ,Aλφ) + C| lnλ− lnλ0|.
We therefore deduce that by taking the infimum over all φ,
k(λ0) ≤ k(λ) + C| lnλ− lnλ0|.
Same argument also yields k(λ) ≤ k(λ0) + C| lnλ − lnλ0|.Thus |k(λ0)− k(λ)| ≤
C| lnλ− lnλ0| and k(λ) is continuous for λ > 0.
To prove (17), by (14), we recall from Sobolev’s inequality in R3
8 YAN GUO AND ZHIWU LIN
|(Kλφ, ψ)| =
4πf ′0(E)e
λs∇φ(X(s;x, v))V (s)dsdvψ(x)dx
|ψ|2|f ′0(E)|dvdx
|∇φ(X (s))|2|f ′0(E)||V (s) |2dxdv]1/2ds
|ψ|2|f ′0(E)|dvdx
)1/2 ∫ ∫
v2|∇φ(x)|2|f ′0(E)|dxdv]1/2ds
||ψ||6||∇φ||2 ≤
||∇ψ||2||∇φ||2,
since f0 has compact support. Therefore,
(Aλφ, φ) = ||∇φ||2 − (Kλφ, φ) ≥ (1 −
)||∇φ||2 ≥ 0
for λ large. �
We now compute limλ→0+Aλ. We first consider the case when the test function
φ is spherically symmetric.
Lemma 2.3. For spherically symmetric function φ(x) = φ (|x|) , we have
(Aλφ, φ) = (A0φ, φ) ≡
|∇φ|2dx+ 4π
f ′0(E)dvφ
− 32π3
minU0
f ′0(E)
∫ r2(E,L)
r1(E,L)
2(E−U0−L2/2r2)
∫ r2(E,L)
r1(E,L)
2(E−U0−L2/2r2)
|∇φ|2 + 32π3
f ′0(E)
∫ r2(E,L)
r1(E,L)
(φ− φ̄)2 drdEdL√
2(E − U0 − L2/2r2)
Proof. Given the steady state f0(E), U0(|x|) and any radial function φ (|x|) . To
find the limit of
(Aλφ, φ) =
|∇φ|2dx+ 4π
f ′0(E)dvφ
2dx(19)
f ′0(E)
λeλsφ(X(s;x, v))ds
φ (x) dxdv,
we study the following
(20) lim
λeλsφ(X(s;x, v))ds.
Note that we only need to study (20) for points (x, v) with E = 1
|v|2+U0| (x|) < E0
and L = |x× v| > 0, because in the third integral of (19) f ′0(E) has support in
UNSTABLE AND STABLE GALAXY MODELS 9
{E < E0} and the set {L = 0} has a zero measure. We recall the linearized Vlasov-
Poisson system in the r, vr, L coordinates takes the form
∂tf + vr∂rf +
− ∂rU0
∂vrf = ∂rUf∂vrf0,
∂rrUf +
∂rUf = 4π
For the corresponding linearized system, for points (x, v) with E < E0 and L > 0,
the trajectory of (X(s;x, v), V (s;x, v)) in the coordinate (r, E, L) is a periodic
motion described by the ODE (see [8])
dr(s)
= vr(s),
dvr(s)
= −U ′0(r) +
with the period
T (E,L) = 2
∫ r2(E,L)
r1(E,L)
2(E − U0 − L2/2r2)
where 0 < r1(E,L) ≤ r2(E,L) < +∞ are zeros of E − U0 − L2/2r2.So by Lin’s
lemma in [[25]],
λeλsφ(X(s;x, v))ds =
φ(X(s;x, v))ds.
Since φ(X(s;x, v) = φ(r(s)), a change of variable from s→ r(s) leads to
φ(X(s;x, v))ds = 2
φ(r)dr
2(E − U0 − L2/2r2)
For any function g(r, E, L), we define its trajectory average as
ḡ(E,L) ≡
∫ r2(E,L)
r1(E,L)
g(r,E,L)dr√
2(E−U0−L2/2r2)
∫ r2(E,L)
r1(E,L)
2(E−U0−L2/2r2)
λeλsφ(X(s;x, v))ds = 2
φ(r)dr
2(E − U0 − L2/2r2)
/T (E,L) = φ̄ (E,L)
10 YAN GUO AND ZHIWU LIN
and the integrand in third term of (19) converges pointwise to f ′0(E)φ̄φ. Thus by
the dominated convergence theorem, we have
(Aλφ, φ) =
|∇φ|2dx+ 4π
f ′0(E)φ
2dxdv − 4π
f ′0(E)φ̄φ dxdv
|∇φ|2dx+ 4π
f ′0(E)φ
2dxdv
− 32π3
minU0
f ′0(E)
∫ r2(E,L)
r1(E,L)
φ̄ (E,L)φ (r)
drdEdL
2(E − U0 − L/2r2)
|∇φ|2dx+ 4π
f ′0(E)φ
2dxdv
− 32π3
minU0
f ′0(E)
∫ r2(E,L)
r1(E,L)
2(E−U0−L/2r2)
∫ r2(E,L)
r1(E,L)
2(E−U0−L/2r2)
|∇φ|2 + 32π3
f ′0(E)
∫ r2(E,L)
r1(E,L)
(φ− φ̄)2
drdEdL
2(E − U0 − L/2r2)
This finishes the proof of the lemma. �
To compute limλ→0+(Aλφ, φ) for more general test function φ, we use the fol-
lowing ergodic lemma which is a direct generalization of the result in [26].
Lemma 2.4. Consider the solution (P (s; p, q) , Q (s; p, q)) to be the solution of a
Hamiltonian system
Ṗ = ∂qH (P,Q)
Q̇ = −∂pH (P,Q)
with (P (0) , Q (0)) = (p, q) ∈ Rn ×Rn. Denote
Qλm =
λeλsm (P (s) , Q (s)) ds.
Then for anym (p, q) ∈ L2 (Rn ×Rn), we have Qλm→ Pm strongly in L2 (Rn ×Rn).
Here P is the projection operator of L2 (Rn ×Rn) to the kernel of the transport
operator D = ∂qH∂p − ∂pH∂q and Pm is the phase space average of m in the set
traced by the trajectory.
Proof. Denote U (s) : L2 (Rn ×Rn) → L2 (Rn ×Rn) to be the unitary semigroup
U (s)m = m (P (s) , Q (s)). By Stone Theorem ([40]), U (s) is generated by iR = D,
where R = −iD is self-adjoint and
U (s) =
eiαsdMα
where
Mα;α ∈ R1
is spectral measure of R. So
λeλsm(P (s), Q(s))ds =
eiαsdMαm ds =
λ+ iα
dMαm.
UNSTABLE AND STABLE GALAXY MODELS 11
On the other hand, the projection is P = M{0} =
ξdMα where ξ(α) = 0 for
α 6= 0 and ξ(0) = 1. Therefore
λeλsm(P (s), Q(s))ds − Pm
λ+ iα
− ξ(α)
d‖Mαm‖2L2
by orthogonality of the spectral projections. By the dominated convergence theo-
rem this expression tends to 0 as λ→ 0+, as we wished to prove. The explaination
of Pm as the phase space average of m is in our remark below. �
Remark 1. Since
λeλsds = 1, the function
(x, v) =
λeλsm (P (s), Q(s)) ds
is a weighted time average of the observable m along the particle trajectory. By the
same proof of Lemma 2.4, we have
(22) lim
m (P (s), Q(s)) ds = Pm.
But from the standard ergodic theory ([3]) of Hamiltonian systems, the limit of the
above time average in (22) equals the phase space average of m in the set traced
by the trajectory. Thus Pm has the meaning of the phase space average of m and
Lemma 2.4 states that the limit of the weighted time average (21) yields the same
phase space average. In particular, if the particle motion is ergodic in the invariant
set SI determined by the invariants E1, · · · , Ik, and if dσI denotes the induced
measure of Rn ×Rn on SI , then
(23) Pm =
σI (SI)
m (p, q) dσI (p, q) .
For integral systems, using action angle variables (J1, · · · , Jn;ϕ1, · · · , ϕn) we have
(Pm) (J1, · · · , Jn) = (2π)−n
· · ·
m (J1, · · · , Jn, ϕ1, · · · , ϕn) dϕ1, · · · dϕn
for the generic case with independent frequencies (see [4]).
Recall the weighted L2 space L2|f ′0|
in (5). Then U (s) : L2|f ′0|
→ L2|f ′0|
defined by
U (s)m = m (X(s;x, v), V (s;x, v)) is an unitary group, where (X(s;x, v), V (s;x, v))
is the particle trajectory (13). The generator of U (s) is D = v ·∂x−∇xU0 ·∇v and
R = −iD is self-adjoint by Stone Theorem. By the same proof, Lemma 2.4 is still
valid in L2|f ′0|
. In particular, for any φ (x) ∈ L2
we have
λeλsφ(X(s;x, v))ds → Pφ
in L2|f ′0|
, where P is the projector of L2|f ′0|
to kerD.
Now we derive an explicit formula for the above limit Pφ. Note that as in the
proof of lemma 2.3, we only need to derive the formula of Pφ for points (x, v) with
E < E0 and L > 0. Since U0 (x) = U0 (r), the particle motion (13) in such a center
field is integrable and has been well studied (see e.g. [8], [4]). For particles with
12 YAN GUO AND ZHIWU LIN
energy E < E0 < 0, L > 0 and momentum ~L = x× v, the particle orbit is a rosette
in the annulus
AE,L = {r1(E,L) ≤ r ≤ r2(E,L)} =
E − U0 − L2/2r2 ≥ 0
lying on the orbital plane perpendicular to ~L. So we can consider the particle
motion to be planar. For such case, the action-angle variables are as follows (see
e.g. [30]): the actions variables are
T (E,L)
, Jθ = L,
where
T (E,L) = 2
∫ r2(E,L)
r1(E,L)
2(E − U0 − L2/2r2)
is the radial period, the angle variable ϕr is determined by
dϕr =
T (E,L)
2(E − U0 − L2/2r2)
and ϕθ = θ −∆θ where
d (∆θ) =
Lr−2 − Ωθ
2(E − U0 − L2/2r2)
Ωθ (E,L) =
T (E,L)
∫ r2(E,L)
r1(E,L)
2(E − U0 − L2/2r2)
is the average angular velocity. For any function φ (x) ∈ H2
, we denote
φ~L (r, θ) to be the restriction of φ in the orbital plane perpendicular to
~L. Then by
(24), for the generic case when the radial and angular frequencies are independent,
we have
E, ~L
= (2π)
φ~Ldϕθdϕr(26)
πT (E,L)
∫ r2(E,L)
r1(E,L)
φ~L (r, θ) dθdr
2(E − U0 − L2/2r2)
In particular, for a spherically symmetric function φ = φ (r), we recover
(27) (Pφ) (E,L) = 2
T (E,L)
∫ r2(E,L)
r1(E,L)
φ(r)dr
2(E − U0 − L2/2r2)
We thus conclude the following
Lemma 2.5. Assume that f0(E) has a bounded support in x and v and f
bounded. For any φ ∈ H1
, we have
(Aλφ, φ) = (A0φ, φ)
|∇φ|2dx+ 4π
f ′0(E)dvφ
2dx− 4π
f ′0(E) (Pφ)
|∇φ|2dx+ 4π
f ′0(E) (φ− Pφ)
UNSTABLE AND STABLE GALAXY MODELS 13
where P is the projector of L2|f ′0|
to kerD and more explicitly Pφ is given by (26).
The limiting operator A0 is
(29) A0φ = −∆φ+ [4π
f ′0(E)dv]φ − 4π
f ′0(E)Pφdv.
Now we give the proof of the instability criterion.
Proof of Theorem 1.1. We define
λ∗ = sup
k(λ)<0
By Lemmas 2.1 and 2.5, we deduce that
−∞ < λ∗ ≤ Λ <∞.
Therefore, by the continuity of k(λ), we have
k(λ∗) = 0.
Hence, there exists an increasing sequence of λn < λn+1 < λ∗ so that λn → λ∗,
kn ≡ k(λn) < 0, and
kn → k(λ∗) = 0.
Therefore, kn are negative eigenvalues. By Lemma 2.2, we get a sequence φn ∈ H2
such that
(30) Aλnφn = knφn
with kn < 0, kn → 0 and λn → λ0 > 0, as n → ∞. Recall χ the cutoff function
of the support of f0(E) such that χ ≡ 1 for f0(E) > 0. We claim that χφn is a
nonzero function for any n. Suppose otherwise, χφn ≡ 0, then from the equation
(30) we have (−∆− kn)φn = 0 which implies that φn = 0, a contradiction.Thus
we can normalize φn by ‖χφn‖2 = 1. Taking inner product of (30) with φn and
integrating by parts, we have
‖▽φn‖22 ≤ −4π
f ′0(E)φ
n dvdx+
4πf ′0(E)
λnsφn(X(s;x, v))dsφn (x) dx
= −4π
f ′0(E) (χφn)
4πf ′0(E)
λns (χφn) (X(s;x, v))ds (χφn) (x) dx
f ′0(E)dv
‖χφn‖22 .
Here in the second equality above, we use the fact χ = 1 on the support of
f ′0(E) (f0(E)) and that (χφn) (X(s;x, v)) = φn(X(s;x, v)χ due to the invariance
of the support under the trajectory flow, as in (15). In the last inequality, we use
the same estimate as in (16). Thus,
||φn||L6 ≤ C sup
‖▽φn‖2 < C
for some constant C′ independent of n. Then there exists φ ∈ L6 and ∇φ ∈ L2
such that
φn → φ weakly in L6, and ∇φn → ∇φ weakly in L2.
14 YAN GUO AND ZHIWU LIN
This implies that χφn → χφ strongly in L2. Therefore ‖χφ‖2 = 1 and thus φ 6= 0.
It is easy to show that φ is a weak solution of Aλ0φ = 0 or
(31) −∆φ = −[4π
f ′0(E)dv]φ + 4πf
λ0sφ(X(s;x, v))dsdv = ρ.
We have that
ρdx = −4π
f ′0(E)φ (x) dxdv +
4πf ′0(E)φ(X(s;x, v))dxdvds
= −4π
f ′0(E)φ (x) dxdv +
4πf ′0(E)φ(x)dxdvds = 0
and by (31) ρ has compact support in Sx, the x−support of f0(E). Therefore from
the formula φ (x) =
|x−y|
dy, we have
φ (x) =
ρ (y)
|x− y|
ρ (y)
|x− y|
ρ (y)
dy = O
|x|−2
for x large, and thus φ ∈ L2. By elliptic regularity, φ ∈ H2. We define f (x, v) by
(14), then f ∈ L∞ with the compact support in S. Now we show that eλ0t[f, φ] is a
weak solution to the linearized Vlasov-Poisson system. Since φ satisfies the Poisson
equation (31), we only need to show that f satisfies the linearized Vlasov equation
(12) weakly. For that, we take any g ∈ C1c
3 × R3
, and
R3×R3
(Dg) fdxdv
R3×R3
(Dg) (f ′0(E)φ(x)) dxdv −
R3×R3
(Dg) f ′0(E)
λ0sφ(X(s;x, v))dsdxdv
= I + II.
Since D is skew-adjoint, the first term is
I = −
R3×R3
gD (f ′0(E)φ) dxdv = −
R3×R3
f ′0(E)gDφdxdv.
UNSTABLE AND STABLE GALAXY MODELS 15
For the second term,
II = −
R3×R3
f ′0(E) Dg(x, v) φ (X(s;x, v)) dxdvds
R3×R3
f ′0(E) (Dg) (X(−s), V (−s))φ (x) dxdvds
R3×R3
f ′0(E)
g (X(−s), V (−s))
ds φ (x) dxdv
R3×R3
f ′0(E)
λ0g (x, v)−
λ0sg (X(−s), V (−s)) ds
φ (x) dxdv
R3×R3
f ′0(E)λ0φ (x)− f ′0(E)
λ0sφ (X(s), V (s)) ds
g (x, v) dxdv
R3×R3
f ′0(E)φ (x)− f ′0(E)
λ0sφ (X(s), V (s)) ds
g dxdv
= .λ0
R3×R3
fgdxdv.
Thus we have
R3×R3
(Dg) fdxdv =
R3×R3
(λ0f − f ′0(E)Dφ) gdxdv
which implies that f is a weak solution to the linearized Vlasov equation
λ0f +Df = f
0 (E) v · ∇xφ.
Remark 2. Consider an anisotropic spherical galaxy with f0 (x, v) = f0
For a radial symmetric growing mode eλt (φ, f) with φ = φ (|x|) and f = f
|x| , E, L2
The linearized Vlasov equation (11) becomes
λf + v · ∇xf −∇xU0 · ∇vf
= ∇xφ · ∇vf0 = ∇xφ ·
|x× v|2
= φ′ (|x|) x
v + 2
[(x× v)× x]
v · ∇xφ,
which is of the same form as in the isotropic case (20). So by the same proof of The-
orem 1.1, we also get an instability criterion for radial perturbations of anisotropic
galaxy, in terms of the quadratic form (18) with f ′0(E) being replaced by
3. Nonlinear Stability of the King’s Model
In the second half of the article, we investigate the nonlinear stability of the King
model (8). We first establish:
Lemma 3.1. Consider spherical models f0 = f0 (E) with f
0 < 0. The operator
A0 : H
r → L2r
A0φ = −∆φ+ [4π
f ′0dv]φ − 4π
f ′0Pφdv
16 YAN GUO AND ZHIWU LIN
is positive, where H2r and L
r are spherically symmetric subspaces of H
2 and L2,
and the projection Pφ is defined by (27). Moreover, for φ ∈ H2r we have
(32) (A0φ, φ) ≥ ε
|∇φ|22 + |φ|
for some constant ε > 0.
Proof. Define k0 = inf (A0φ, φ) / (φ, φ) .We want to show that k0 > 0. First, by
using the compact embedding of H2r →֒ L2r it is easy to show that the minimum
can be obtained and k0 is the lowest eigenvalue. Let A0φ0 = k0φ0 with φ0 ∈ H2r
and ‖φ0‖2 = 1. The fact that k0 ≥ 0 follows immediately from Theorem 1.1 and the
nonexistence of radial modes ([9], [22]) for monotone spherical models. The proof
of k0 > 0 is more delicate. For that, we relate the quadratic form (A0φ, φ) to the
Antonov functional (4). We define D = v ·∂x−∇xU0 ·∇v to be the generator of the
unitary group U (s):L
|f ′0|
→ L2,r|f ′0|
defined by U (s)m = m (X(s;x, v), V (s;x, v)) .
Here L
|f ′0|
is the spherically symmetric subspace of L2|f ′0|
, which is preserved under
the flow mapping U (s). By the definition of Pφ, we have φ0 − Pφ0 ⊥ kerD. By
Stone theorem iD is self-adjoint and in particular D is closed. Therefore by the
closed range theorem ([40]), we have (kerD)
= R (D) , where R (D) is the range
of D. So there exists h ∈ L2,r|f ′0|
such that Dh = φ0−Pφ0. Moreover, since φ0−Pφ0
is even in v and the operator D reverses the parity in v, the function h is odd in v.
Define f− = f ′0h. We have
k0 = (A0φ0, φ0) =
|∇φ0|2 dx + 4π
f ′0 (φ0 − Pφ0)
|∇φ0|2 dx− 8π
|f ′0| (φ0 − Pφ0)φ0dxdv
|f ′0| (φ0 − Pφ0)
|Df−|2
|f ′0|
dxdv + 2
Df−dvdx +
|∇φ0|2 dx
|Df−|2
|f ′0|
dxdv +
|∇φ0|2 dx
|Df−|2
|f ′0|
dxdv +
|∇φ0|2 − 2∇φ0 · ∇φ−
|Df−|2
|f ′0|
dxdv − 1
where ∆φ− = 4π
Df−dv.Notice that the last expression above is the Antonov
functional 4πH (f−, f−). Since f− is spherical symmetric and odd in v,we have
H (f−, f−) > 0 by the proof in [22] which was further clarified in [33] and [21].
Therefore we get k0 > 0 as desired and (A0φ, φ) ≥ k0 |φ|22.
UNSTABLE AND STABLE GALAXY MODELS 17
To get the estimate (32), we rewrite
(A0φ, φ) = ε
|∇φ|2 dx + 4π
f ′0 (φ− Pφ)
+ (1− ε) (A0φ, φ)
|∇φ|2 dx− 4πε ‖φ− Pφ‖2L2
+ (1− ε) k0 |φ|22
|∇φ|2 dx− 8πε ‖φ‖2L2
+ (1− ε) k0 |φ|22 (since ‖P‖L2
|∇φ|2 dx+ ((1− ε) k0 − Cε) |φ|22 ≥ ε
|∇φ|2 dx+ |φ|22
if ε is small enough. �
Next, we will approximate the kerD by a finite dimensional approximation. Let
{ξi(E,L) = αi(E)βi(L)}∞i=1 be a smooth orthogonal basis for the subspace kerD =
{g(E,L)} ⊂ L2,r|f ′0|
.Define the finite-dimensional projection operator PN : L2,r|f ′0|
|f ′0|
(33) PNh ≡
(h, ξi)|f ′0|ξi
and the operator AN : H2r → L2r by
ANφ = −∆φ+ [4π
f ′0dv]φ− 4π
f ′0PNφdv.
Lemma 3.2. There exists K, δ0 > 0 such that when N > K we have
ANφ, φ
≥ δ0 |∇φ|22
for any φ ∈ H2r .
Proof. First we have AN → A0 strongly in L2. In deed, for any φ ∈ H2r ,
∥ANφ−A0φ
4πf ′0 (PNφ− Pφ) dv
≤ C ‖PNφ− Pφ‖L2
as N → ∞.We claim that for N sufficiently large, the lowest eigenvalue of AN
is at least k0/2 where k0 > 0 is the lowest eigenvalue of A0. Suppose otherwise,
then there exists a sequence {λn} and {φn} ⊂ H2r with λn < k0/2, ‖φn‖2 =
1 and Anφn = λnφn. This implies that ∆φn is uniformly bounded in L
2, by
elliptic estimate we have ‖φn‖H2 ≤ C for some constant C independent of n.
Therefore there exists φ0 ∈ H2r such that φn → φ0 weakly in H2r . By the compact
embedding of H2r →֒ L2r, we have φn → φ0 strongly in L2r and ‖φ0‖2 = 1. The
strong convergence of Anφ0 → A0φ0 implies that
Anφn → A0φ0
weakly in L2. Let λn → λ0 ≤ k0/2, then we have A0φ0 = λ0φ0, a contradiction.
Therefore we have
ANφ, φ
≥ k0/2 |φ|22 for φ ∈ H2r , when N is large enough. The
estimate (34) is by the same proof of (32) in Lemma 3.1. �
18 YAN GUO AND ZHIWU LIN
Recalling (8) with f0 = [e
E0−E−1]+ and Q0(f) = (f+1) ln(f+1)−f, we further
define functionals (related to the finite dimensional approximation of kerD) as
Ai(f) ≡
αi(− ln(s+ 1) + E0)ds,
Qi(f, L) ≡ Ai(f)βi(L), for 1 ≤ i ≤ N.
for 1 ≤ i ≤ N. Clearly,
∂1Qi(f0, L) = αi(− ln(f0 + 1) + E0)βi(L) = αi(E)βi(L) = ξi(E,L),
where {ξi(E,L)}Ni=1 are used to define PN in Lemma 3.2. Define the Casimir
functional (E0 < 0 )
I(f) =
[Q0(f) +
|v|2f − E0f ]dxdv −
|∇φ|2dx
which is invariant of the nonlinear Vlasov-Poisson system. We introduce additional
N invariants
Ji(f, L) ≡
Qi(f, L)dxdv.
for 1 ≤ i ≤ N . We define Ω to be the support of f0(E). We first consider
I(f)− I(f0) =
[Q0(f)−Q0(f0) +
|v|2(f − f0)− E0(f − f0)]dxdv
∇U0 · ∇(U − U0)−
|∇(U − U0)|2dx
[Q0(f)−Q0(f0) + (E − E0)(f − f0)]dxdv −
|∇(U − U0)|2dx.
We define
g = f − f0, φ = U − U0
gin ≡ (f − f0)1Ω, gout ≡ (f − f0)1Ωc , ∆φin ≡
gin, ∆φout ≡
gout .
And we define the distance function for nonlinear stability as
d(f, f0) ≡
[Q0(gin + f0)−Q0(f0) + (E − E0)gin]dxdv
|∇φin|2dx
Q0(gout)dxdv +
(E − E0)goutdxdv
= din +
|∇φin|2dx + dout,
for which each term is non-negative. We therefore split:
I(f)− I(f0)
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv −
|∇φin|2dx
Q0(gout)dxdv +
(E − E0)goutdxdv −
|∇φout|2dx−
∇φout · ∇φindx
= Iin + Iout .
UNSTABLE AND STABLE GALAXY MODELS 19
In the estimates below, we use C,C′, C′′ to denote general constants depending
only on f0 and quantities like ‖f (t)‖Lp (p ∈ [1,+∞]) which equals ‖f (0)‖Lp and
therefore always under control. We first estimate ‖∇φout‖22 to be of higher order of
d, which also implies that
∇φout · ∇φindx is of higher order of d.
Lemma 3.3. For ε > 0 sufficiently small, we have
|∇φout|2dx ≤ C
εd(f, f0) +
[d(f, f0)]
Proof. In fact, since
|∇φout|2dx ≤ C||
gout dv||2L6/5
≤ C||
gout 1E0≤E≤E0+εdv||2L6/5 + C||
gout 1E>E0+εdv||2L6/5 .
The first term is bounded by
g2out dv]
1E0≤E≤E0+εdv]
3/5dx
g2out dvdx]×
1E0≤E≤E0+εdv]
3/2dx
≤ Cε[
g2out dvdx] ≤ Cε[
g2out dvdx]
≤ Cεd(f, f0).
In the above estimates, we use that
Q0(gout)dvdx ≥ c
g2out dvdx and
1E0≤E≤E0+εdv ≤ Cε,
which can be checked by an explicit computation when ε > 0 is sufficiently small
such that E0 + ε ≤ 0.
On the other hand, by the standard estimates (see [12, P. 120-121])
gout 1E>E0+εdv||2L6/5
gout 1E>E0+εdxdv
|v|2gout 1E>E0+εdxdv
(E − E0)gout 1E>E0+εdxdv
(E − E0)gout 1E>E0+εdxdv + 2 sup |U0|
gout 1E>E0+εdxdv
2 sup |U0|
d5/3.
20 YAN GUO AND ZHIWU LIN
By Lemma 3.3, we have
∇φout · ∇φindx
≤ ‖∇φout‖2 ‖∇φin‖2
ε1/3d(f, f0) +
[d(f, f0)]
and therefore for ε sufficiently small,
(36) Iout ≥ dout − C
ε1/3d(f, f0) +
[d(f, f0)]
4/3 +
[d(f, f0)]
To estimate Iin, we split it into three parts:
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin + φingin]dxdv +
|∇φin|2dx
(1− τ)
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin + (I − PN )φingin]dxdv +
|∇φin|2dx
+ (1− τ)
PNφingindxdv
= I1in + I
in + I
where ∆φin = 4π
gin dv. We estimate each term in the following lemmas.
Lemma 3.4.
(38) I1
din − Cτ
|∇φin|2dx.
Proof. In fact, since the integration region Ω is finite, we have
I1in =τ
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin + φingin]dxdv +
|∇φin|2dx
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv − Cτ ||φin||L6 ||gin||L6/5
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv − C′τ ||∇φin||L2 ||gin||2
din − C′′τ ||∇φin||22,
since
din =
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv ≥ C||gin||22.
To estimate I2in, we need the following pointwise duality lemma from elementary
calculus.
Lemma 3.5. For any c, and any h, we have
gc,f0 (h) = Q0(h+ f0)−Q0(f0)−Q′0(f0)h− ch ≥ (f0 + 1)(1 + c− ec).
Proof. Direct computation yields that the minimizer fc of gc,f0 (h) satisfies the
Euler-Lagrange equation
ln (fc + f0 + 1)− ln (f0 + 1)− c = 0,
UNSTABLE AND STABLE GALAXY MODELS 21
fc = (f0 + 1) (e
c − 1) .
Thus by using the Euler-Lagrange equation, we deduce
min gc,f0 (h) = gc,d (fc)
= (fc + f0 + 1) ln(1 + fc + f0)
− (f0 + 1) ln(1 + f0)− [1 + ln(f0 + 1)]fc − cfc
= (fc + f0 + 1)[ln(1 + fc + f0)− ln(f0 + 1)− c]
+ fc ln(1 + f0) + c(f0 + 1)− [1 + ln(f0 + 1)]fc
= (f0 + 1)(1 + c− ec).
Lemma 3.6.
(39) I2
(1− τ) δ0
|∇φin|2dx− CeC
Proof. Recall (37). By using Lemma 3.5 for c = − (φin − PNφin) and using the
Taylor expansion, we have
I2in = (1 − τ)
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin + (φin − PNφin) fin]dxdv
(1− τ)
|∇φin|2dx
(1− τ)
|∇φin|2dx+ (1− τ)
(f0 + 1)1Ω(1 + φin − PNφin − eφin−PNφin)dxdv
≥ 1− τ
|∇φin|2dx− 4π
|f ′0 (E)| (φin − PNφin)
− Ce|φin−PNφin|∞
|f ′0 (E)| |φin − PNφin|
dxdv (Note (f0(E) + 1)1Ω = |f ′0(E)|)
≥ (1− τ) δ0
|∇φin|2dx− Ce|φin−PNφin|∞
|f ′0 (E)| |φin − PNφin|
dxdv.
In the last line, we have used Lemma 3.2. To estimate the last term above and
conclude our lemma, it suffices to show
|φin − PNφin|∞ ≤ CNd
This follows from the facts that for the fixed N smooth functions ξi, we have
|PNφin|∞ =
(φin, ξi)|f ′0|ξi
≤ CN |φin|∞ ,
and since φ is spherically symmetric,
|φin| (r) =
u2ρin (u) du+
uρin (u)du
R |ρin|2 ≤ C
′′ ‖gin‖2 ≤ CNd
where ρin =
gindv and R is the support radius of ρin. �
22 YAN GUO AND ZHIWU LIN
We now estimate the term
PNφinfindxdv, for which we use the additional
invariants.
Lemma 3.7. For any ε > 0, we have
∣ ≤ C(d1/2(0) + ε1/2d1/2 + 1
d)d1/2.
Proof. By the definition of I3in in (37), it suffices to estimate (gin, ξi). We expand
Ji(f, L)− Ji(f0, L)
= Ji(f0 + gin, L)− Ji(f0, L) + Ji(gout, L)
= (gin , ξi) +O(d) + Ji(gout, L).
Notice that
|Ji(gout, L)| ≤ C||gout||L1 ≤ C||1{E0≤E≤E0+ε}gout||L1 + C||1{E≥E0+ε}gout||L1
≤ ε1/2||gout||L2 +
||1{E≥E0+ε}(E − E0)gout||L1 ≤ C[ε
1/2d1/2 +
It thus follows that
|(gin , ξi)| ≤ |Ji(f(0), L)− Ji(f0, L)|+ C[ε1/2d1/2 +
≤ C[d1/2(0) + ε1/2d1/2 + 1
Therefore
∣I3in
∣ = (1− τ)
PNφingin dxdv
(φin, ξi)|f ′0|ξi
gin dxdv
(φin, ξi)|f ′0|
|(ξi, gin)| ≤ C′
|φin|∞ |(ξi, gin)|
≤ Cd1/2[d1/2(0) + ε1/2d1/2 +
Now we prove the nonlinear stability of King model.
Proof of Theorem 1.2. The global existence of classical solutions of 3D Vlasov-
Poisson system was shown in [34] for compactly supported initial data f (0) ∈ C1c .
Let the unique global solution be (f (t) , φ (t)). Let d (t) = d(f (t) , f0). Combining
estimates (36), (38), (39) and (40), we have
I(f (0))− I(f0) = I(f (t))− I(f0)
≥ dout +
din +
(1− τ) δ0
|∇φin|2dx
ε1/3d (t) +
d (t)
d (t)
− CeC
′d(t)
d (t)
− Cd (t)1/2 [d1/2(0) + ε1/2d (t)1/2 +
d (t)].
UNSTABLE AND STABLE GALAXY MODELS 23
Thus by choosing ε and τ sufficiently small, there exists δ′ > 0 such that
I(f (0))− I(f0) ≥ δ′d(t)− C
d (t)
+ d (t)
+ d (t)
− CeC
′d(t)
d (t)
− Cd (t)1/2 d1/2(0).
It is easy to show that I(f (0)) − I(f0) ≤ C′′d (0). Define the functions y1 (x) =
δ′x2 − CeC′xx3 − C
x8/3 + x10/3 + x3
and y2 (x) = Cd (0)
x + C′′d (0). Then
above estimates implies that y1
d (t)
d (t)
. The function y1 is in-
creasing in (0, x0) where x0 is the first maximum point. So if d (0) is sufficiently
small, the line y = y2 (x) intersects the curve y = y1 (x) at points x1, x2, · · · ,
with x1 (d (0)) < x0 < x2 (d (0)) < · · · . Thus the inequality y1 (x) ≤ y2 (x) is
valid in disjoint intervals [0, x1 (d (0))] and [x2 (d (0)) , x3 (d (0))], · · · . Because d (t)
is continuous, we have that d (t)
< x1 (d (0)) for all t < ∞, provided we choose
d (0)
< x0. Since x1 (d (0)) → 0 as d (0) → 0, we deduce the nonlinear stability
in terms of the distance functional d (t)
Acknowledgements
This research is supported partly by NSF grants DMS-0603815 and DMS-0505460.
We thank the referees for comments and corrections.
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[13] Goodman, Jeremy, An instability test for nonrotating galaxies, Astrophysical Journal, vol.
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[14] Guo, Y., Variational method for stable polytropic galaxies, Arch. Rational Mech. Anal., 147,
225-243, 1999.
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[15] Guo, Y., On generalized Antonov stablility criterion for polytropic steady states, Contem.
Math., 263, 85-107, 1999.
[16] Guo, Y., Rein, G., Stable steady states in stellar dynamics, Arch. Rational Mech. Anal., 147,
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[19] Guo, Y., Rein, G., Isotropic steady states in stellar dynamics revisited., Los Alamos Preprint,
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[20] Henon, M., Numerical Experiments on the Stability of Spherical Stellar Systems, Astronomy
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[21] Guo, Y., Rein, G., Stability of the King Model and Symmetric Measure-Preserving Pertur-
bations, Preprint.
[22] Kandrup, H.; Signet, J. F.; A simple proof of dynamical stability for a class of spherical
clusters. The Astrophys. J. 298, p. 27-33.(1985)
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[25] Lin, Zhiwu, Instability of periodicBG waves, Math. Res. Letts., 8, 521-534(2001).
[26] Lin, Zhiwu and Strauss, Walter, Linear stability and instability of relativistic Vlasov-Maxwell
systems, to appear in Comm. Pure Appl. Math.
[27] Lin, Zhiwu and Strauss, Walter, Nonlinear stability and instability of relativistic Vlasov-
Maxwell systems, to appear in Comm. Pure Appl. Math.
[28] Lin, Zhiwu and Strauss, Walter, A sharp stability criterion for the Vlasov-Maxwell systems,
submitted.
[29] Lynden-Bell, D., The Hartree-Fock exchange operator and the stability of galaxies, Monthly
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(Thessaloniki, 1993), 3–31, Lecture Notes in Phys., 433, Springer, Berlin, 1994.
[31] Merritt, David, Elliptical Galaxy Dynamics, The Publications of the Astronomical Society of
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[32] Palmer, P. L., Stability of collisionless stellar systems: mechanisms for the dynamical struc-
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[33] Perez, Jerome and Aly, Jean-Jacques, Stability of spherical stellar systems - I. Analytical
results, Monthly Notices of the Royal Astronomical Society, Volume 280, Issue 3, pp. 689-
699, 1996.
[34] Pfaffelmoser, K, Global classical solutions of the Vlasov-Poisson system in three dimensions
for general initial data, J. Differential Equations 95 (1992), no. 2, 281–303.
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no. 1, 1–19.
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[38] Wan, Y-H., On onlinear stability of isotropic models in stellar dynamics, Arch. Rational.
Mech. Anal., 147, (1999) 245-268.
[39] Wolansky, G., On nonlinear stability of polytropic galaxies. Ann. Inst. Henri Poincare. (1999),
16, 15-48.
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senschaften, 123. Springer-Verlag, 1980.
UNSTABLE AND STABLE GALAXY MODELS 25
Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown
University, Providence, RI 02912, USA
E-mail address: guoy@cfm.brown.edu
Mathematics Department, University of Missouri, Columbia, MO 65211 USA
E-mail address: lin@math.missouri.edu
	1. Introduction
	2. An Instability Criterion
	3. Nonlinear Stability of the King's Model
	References

  To determine the stability and instability of a given steady galaxy
configuration is one of the fundamental problems in the Vlasov theory for
galaxy dynamics. In this article, we study the stability of isotropic spherical
symmetric galaxy models $f_{0}(E)$, for which the distribution function $f_{0}$
depends on the particle energy $E$ only. In the first part of the article, we
derive the first sufficient criterion for linear instability of $f_{0}(E):$
$f_{0}(E)$ is linearly unstable if the second-order operator \[
A_{0}\equiv-\Delta+4\pi\int f_{0}^{\prime}(E)\{I-\mathcal{P}\}dv \] has a
negative direction, where $\mathcal{P}$ is the projection onto the function
space $\{g(E,L)\},$ $L$ being the angular momentum [see the explicit formula
(\ref{A0-radial})]. In the second part of the article, we prove that for the
important King model, the corresponding $A_{0}$ is positive definite. Such a
positivity leads to the nonlinear stability of the King model under all
spherically symmetric perturbations.

Introduction
A galaxy is an ensemble of billions of stars, which interact by the gravitational
field which they create collectively. For galaxies, the collisional relaxation time is
much longer than the age of the universe ([8]). The collisions can therefore be
ignored and the galactic dynamics is well described by the Vlasov - Poisson system
(collisionless Boltzmann equation)
(1) ∂tf + v · ∇xf −∇xU · ∇vf = 0, ∆U = 4π
f(t, x, v)dv,
where (x, v) ∈ R3 × R3, f(t, x, v) is the distribution function and Uf (t, x) is its
gravitational potential. The Vlasov-Poisson system can also be used to describe the
dynamics of globular clusters over their period of orbital revolutions ([11]). One of
the central questions in such galactic problems, which has attracted considerable
attention in the astrophysics literature, of [7], [8], [11], [31] and the references there,
is to determine dynamical stability of steady galaxy models. Stability study can be
used to test a proposed configuration as a model for a real stellar system. On the
other hand, instabilities of steady galaxy models can be used to explain some of the
striking irregularities of galaxies, such as spiral arms as arising from the instability
of an initially featureless galaxy disk ([7]), ([32]).
In this article, we consider stability of spherical galaxies, which are the simplest
elliptical galaxy models. Though most elliptical galaxies are known to be non-
spherical, the study of instability and dynamical evolution of spherical galaxies
could be useful to understand more complicated and practical galaxy models . By
Jeans’s Theorem, a steady spherical galaxy is of the form
f0(x, v) ≡ f0(E,L2),
http://arxiv.org/abs/0704.1012v2
2 YAN GUO AND ZHIWU LIN
where the particle energy and total momentum are
|v|2 + U0(x), L2 = |x× v|2 ,
and U0(x) = U0 (|x|) satisfies the self-consistent Poisson equation. The isotropic
models take the form
f0(x, v) ≡ f0(E).
The cases when f ′0(E) < 0 has been widely studied and these models are known
to be linearly stable to both radial ([9]) and non-radial perturbations ([2]). The
well-known Casimir-Energy functional (as a Liapunov functional)
(2) H(f) ≡
Q(f) +
|v|2f − 1
|∇xUf |2,
is constant along the time evolution. If f ′0(E) < 0, we can choose the Casimir
function Q0 such that
Q′0(f0(E)) ≡ −E
for all E. By a Taylor expansion of H(f)−H(f0), it follows that formally the first
variation at f0 is zero, that is, H(1)(f0(E)) = 0 (on the support of f0(E)), and the
second order variation of H at f0 is
(3) H(2)f0 [g] ≡
{f0>0}
−f ′0(E)
dxdv − 1
|∇xUg|2dx
where Q′′(f0) =
, g = f − f0 and ∆Ug =
gdv. In the 1960s, Antonov ([1],
[2]) proved that
(4) H(2)f0 [Dh] =
∫ ∫ |Dh|2
|f ′0(E)|
dxdv −
|∇ψh|2 dx
is positive definite for a large class of monotone models. Here
D = v · ∇x −∇xU0 · ∇v,
h(x, v) is odd in v and −∆ψ =
Dhdv. He showed that such a positivity is
equivalent to the linear stability of f0(E). In [9], Doremus, Baumann and Feix
proved the radial stability of any monotone spherical models. Their proof was
further clarified and simplified in [10], [37], [22], and more recently in [33], [21].
In particular, this implies that any monotone isotropic models are at least linearly
stable.
Unfortunately, despite its importance and a lot of research (e.g., [20], [5], [6],
[13]), to our knowledge, no rigorous and explicit instability criterion of non-monotone
models has been derived. When f ′0(E) changes sign, functional H
is indefinite
and it gives no stability information, although it seems to suggest that these mod-
els are not energy minimizers under symplectic perturbations. In this paper, we
first obtain the following instability criterion for general spherical galaxies. For
any function g with compact support within the support of f0(E), we define the
|f ′0(E)| −weighted L2
R3 ×R3
space L2|f ′0|
with the norm ‖·‖|f ′0| as
(5) ||h||2|f ′
|f ′0(E)|h2dxdv.
UNSTABLE AND STABLE GALAXY MODELS 3
Theorem 1.1. Assume that f0(E) has a compact support in x and v, and f
bounded. For φ ∈ H1, define the quadratic form
(6) (A0φ, φ) =
|∇φ|2dx + 4π
f ′0(E) (φ− Pφ)
dxdv,
where P is the projector of L2|f ′0|
kerD =
and more explicitly Pφ is given by (18) for radial functions and (26) for general
functions. If there exists φ0 ∈ H1 such that
(7) (A0φ0, φ0) < 0,
then there exists λ0 > 0 and φ ∈ H2, f (x, v) given by (14), such that eλ0t[f, φ] is
a growing mode to the Vlasov-Poisson system (1) linearized around [f0(E), Uf0 ] .
A similar instability criterion can be obtained for symmetry preserving pertur-
bations of anisotropic spherical models f0
, see Remark 2. We note that the
term Pφ in the instability criterion is highly non-local and this reflects the collective
nature of stellar instability. The proof of Theorem 1.1 is by extending an approach
developed in [25] for 1D Vlasov-Poisson, which has recently been generalized to
Vlasov-Maxwell systems ([26], [28]). There are two elements in this approach. One
is to formulate a family of dispersion operators Aλ for the potential, depending
on a positive parameter λ. The existence of a purely growing mode is reduced to
find a parameter λ0 such that the Aλ0 has a kernel. The key observation is that
these dispersion operators are self-adjoint due to the reversibility of the particle
trajectories. Then a continuation argument is applied to find the parameter λ0
corresponding to a growing mode, by comparing the spectra of Aλ for very small
and large values of λ. There are two new complications in the stellar case. First,
the essential spectrum of Aλ is [0,+∞) and thus we need to make sure that the
continuation does not end in the essential spectrum.This is achieved by using some
compactness property due to the compact support of the stellar model. Secondly,
it is more tricky to find the limit of Aλ when λ tends to zero. For that, we need an
ergodic lemma (Lemma 2.4) and use the integrable nature of the particle dynamics
in a central field to derive an expression for the projection Pφ appeared in the limit.
In the second part of the article, we further study the nonlinear (dynamical)
stability of the normalized King model:
(8) f0 = [e
E0−E − 1]+
motivated by the study of the operator A0. The famous King model describes
isothermal galaxies and the core of most globular clusters [24]. Such a model
provides a canonical form for many galaxy models widely used in astronomy. Even
though f ′0 < 0 for the King model, it is important to realize that, because of the
Hamiltonian nature of the Vlasov-Poisson system (1), linear stability fails to imply
nonlinear stability (even in the finite dimensional case). The Liapunov functional is
usually required to prove nonlinear stability. In the Casimir-energy functional (2), it
is natural to expect that the positivity of such a quadratic formH(2)f0 [g] should imply
stability for f0(E). However, there are at least two serious mathematical difficulties.
First of all, it is very challenging to use the positivity of H(2)f0 [g] to control higher
4 YAN GUO AND ZHIWU LIN
order remainder in H(f)−H(f0) to conclude stability [38]. For example, one of the
remainder terms is f3 whose L2 norm is difficult to be bounded by a power of the
stability norm. The non-smooth nature of f0(E) also causes trouble here. Second of
all, even if one can succeed in controlling the nonlinearity, the positivity of H
is only valid for certain perturbation of the form g = Dh [22]. It is not clear at all
if any arbitrary, general perturbation can be reduced to the form Dh. To overcome
these two difficulties, a direct variational approach was initiated by Wolansky [39],
then further developed systematically by Guo and Rein in [14], [15], [17], [18], [19].
Their method avoids entirely the delicate analysis of the second order variation
H(2)f0 in (3), which has led to first rigorous nonlinear stability proof for a large class
of f0(E). The high point of such a program is the nonlinear stability proof for every
polytrope [18] f0(E) = (E0 − E)k+. Their basic idea is to construct galaxy models
by solving a variational problem of minimizing the energy under some constraints
of Casimir invariants. A concentration-compactness argument is used to show the
convergence of the minimizing sequence. All the models constructed in this way
are automatically stable.
Unfortunately, despite its success, the King model can not be studied by such a
variational approach. The Casimir function for a normalized King model is
(9) Q0(f) = (1 + f) ln(1 + f)− 1− f,
which has very slow growth for f → ∞. As a result, the direct variational method
fails. Recently, Guo and Rein [21] proved nonlinear radial stability among a class
of measure-preserving perturbations
Sf0 ≡
f(t, r, vr, L) ≥ 0 :
Q(f, L) =
Q(f0, L), for Q ∈ C∞c and Q(0, L) ≡ 0.
The basic idea is to observe that for perturbations in the class Sf0 , one can write
g = f − f0 as Dh = {h,E}. Therefore, H(2)f0 [g] = H
[Dh], for which the positivity
was proved in [22] for radial perturbations. To avoid the difficulty of controlling the
remainder term by H(2)f0 [g], an indirect contradiction argument was used in [21].
As our second main result of this article, we establish nonlinear stability of King’s
model for general perturbations with spherical symmetry:
Theorem 1.2. The King’s model f0 = [e
E0−E − 1]+ is nonlinearly stable under
spherically symmetric perturbations in the following sense: given any ε > 0 there
exists ε1 > 0 such that for any compact supported initial data f(0) ∈ C1c with
spherical symmetry, if d (f (0) , f0) < ε1 then
0≤t<∞
d (f (t) , f0) < ε,
where the distance functional d (f, f0) is defined by (35).
For the proof, we extended the approach in [27] for the 1 1
D Vlasov-Maxwell
model. To prove nonlinear stability, we study the Taylor expansion ofH(f)−H(f0).
Two difficulties as mentioned before are: to prove the positivity of the quadratic
form and to control the remainder. We use two ideas introduced in [27]. The first
idea is to use any finite number of Casimir functional Qi
f, L2
as constraints.
The difference from [21] is that we do not impose Qi
f, L2
f0, L
in the
perturbation class, but expand the invariance equationQi
f (t) , L2
f0, L
UNSTABLE AND STABLE GALAXY MODELS 5
f (0) , L2
f0, L
to the first order. In this way, we get a constraint for
g = f −f0 in the form that the coefficient of its projection to ∂1Qi
f0, L
is small.
Putting these constraints together, we deduce that a finite dimensional projection of
g to the space spanned by
f0, L
is small. To control the remainder term,
we use a duality argument. Noting that it is much easier to control the potential φ,
we use a Legendre transformation to reduce the nonlinear term in g to a new one
in φ only. The key observation is that the constraints on g in the projection form
are nicely suited to the Legendre transformation and yields a non-local nonlinear
term in φ only with the projections kept. By performing a Taylor expansion of this
non-local nonlinear term in φ, the quadratic form becomes a truncated version of
(A0φ, φ) defined by (6), whose positivity can be shown to be equivalent to that of
Antonov functional. The the remainder term now is only in terms of φ and can
be easily controlled by the quadratic form. The new complication in the stellar
case is that the steady distribution f0 (E) is non-smooth and compactly supported.
Therefore, we split the perturbation g into inner and outer parts, according to the
support of f0. For the inner part, we use the above constrainted duality argument
and the outer part is estimated separately.
2. An Instability Criterion
We consider a steady distribution
f0 (x, v) = f0(E)
has a bounded support in x and v and f ′0 is bounded, where the particle energy
E = 1
|v|2 + U0(x). The steady gravitational potential U0(x) satisfies a nonlinear
Poisson equation
∆U0 = 4π
f0dv.
The linearized Vlasov-Poisson system is
(11) ∂tf + v · ∇xf −∇xU0 · ∇vf = ∇xφ · ∇vf0, ∆φ = 4π
f(t, x, v)dv.
A growing mode solution (eλtf(x, v), eλtφ(x)) to (1) with λ > 0 satisfies
(12) λf + v · ∇xf −∇xU0 · ∇vf = f ′0v · ∇xφ.
We define [X(s;x, v), V (s;x, v)] as the trajectory of
dX(s;x,v)
= V (s;x, v)
dV (s;x,v)
= −∇xU0
such that X(0;x, v) = x, and V (0;x, v) = v. Notice that the particle energy E is
constant along the trajectory. Integrating along such a trajectory for −∞ ≤ s ≤ 0,
we have
f(x, v) =
eλsf ′0(E)V (s;x, v) · ∇xφ(X(s;x, v))ds(14)
= f ′0(E)φ(x) − f ′0(E)
λeλsφ(X(s;x, v))ds.
6 YAN GUO AND ZHIWU LIN
Plugging it back into the Poisson equation, we obtain an equation for φ
−∆φ+ [4π
f ′0(E)dv]φ − 4π
f ′0(E)
λeλsφ(X(s;x, v))dsdv = 0.
We therefore define the operator Aλ as
Aλφ ≡ −∆φ+ [4π
f ′0(E)dv]φ − 4π
f ′0(E)
λeλsφ(X(s;x, v))dsdv.
Lemma 2.1. Assume that f0(E) has a bounded support in x and v and f
bounded. For any λ > 0, the operator Aλ : H
2 → L2 is self-adjoint with the
essential spectrum [0,+∞) .
Proof. We denote
Kλφ = −4π[
f ′0(E)dv]φ + 4π
f ′0(E)
λeλsφ(X(s;x, v))dsdv.
Recall that f0 (x, v) = f0(E) has a compact support ⊂ S ⊂ R3x × R3v. We may
assume S = Sx×Sv, both balls in R3. Let χ = χ (|x|) be a smooth cut-off function
for the spatial support of f0 in the physical space Sx; that is, χ ≡ 1 on the spatial
support of f0 and has compact support inside Sx. Let Mχ be the operator of
multiplication by χ. Then Kλ = KλMχ =MχKλ =MχKλMχ. Indeed,
f ′0 (x, v) = f
0 (X(s;x, v), V (s;x, v))
because of the invariance of E under the flow. So
(Kλφ) (x) = −4π[
f ′0(E)dv]φ + 4π
f ′0(E)
λeλsφ(X(s;x, v))dsdv(15)
= −4π[
f ′0(E)dv]φ + 4π
∫ ∫ 0
λeλs (f ′0(E)φ) (X(s;x, v))dsdv
= (MχKλMχφ)(x).
First we claim that
‖Kλ‖L2→L2 ≤ 8π
|f ′0(E)| dv
Indeed, the L2 norm for the first term in Kλ is easily bounded by 4π
f ′0(E)dv
For the second term, we have for any ψ ∈ L2,
4πλeλsf ′0(E)φ(X(s;x, v))dsdvψ(x)dx|
|f ′0(E)|φ2(X(s;x, v))dvdx
|f ′0(E)|ψ2(x)dvdx
|f ′0(E)|φ2(x)dvdx
|f ′0(E)|ψ2(x)dvdx
|f ′0(E)|φ2(x)dvdx
|f ′0(E)|ψ2(x)dvdx
|f ′0(E)| dv
‖φ‖2 ‖ψ‖2 .
UNSTABLE AND STABLE GALAXY MODELS 7
Moreover, we have that Kλ is symmetric Indeed, for fixed s, by making a change of
variable (y, w) → (X(s;x, v), V (s;x, v)), so that (z, v) = (X(−s; y, w), V (−s; y, w)),
we deduce that
4πf ′0(E)
λeλsφ(X(s;x, v))dsdvψ(x)dx
4πf ′0(E)φ(y)ψ(X(−s; y, w))dydwds
4πf ′0(E)
λeλsψ(X(−s; y,−w))φ(y)dydwds
4πf ′0(E)
λeλsψ(X(s;x, v))φ(x)dvdxds.
Here we have used the fact [X(s; y, w), V (s; y, w)] = [X(−s; y,−w),−V (s; y,−w)]
in the last line. Hence
(Kλφ, ψ) = (φ,Kλψ).
Since Kλ = KλMχ and Mχ is compact from H
2 into L2 space with support in Sx,
so Kλ is relatively compact with respect to −∆. Thus by Kato-Relich and Weyl’s
Theorems, Aλ : H
2 → L2 is self-adjoint and σess(Aλ) = σess(−∆). �
Lemma 2.2. Assume that f ′0(E) has a bounded support in x and v and f
bounded. Let
k(λ) = inf
φ∈D(Aλ),||φ||2=1
(φ,Aλφ),
then k(λ) is a continuous function of λ when λ > 0. Moreover, there exists 0 <
Λ <∞ such that for λ > Λ
(17) k(λ) ≥ 0.
Proof. Fix λ0 > 0, φ ∈ D(Aλ), and ||φ||2 = 1. Then
k(λ0) ≤ (φ,Aλ0φ)
≤ (φ,Aλφ) + |(φ,Aλ0φ)− (φ,Aλφ)|
≤ (φ,Aλφ) + 4π
|f ′0(E)|
[λeλs − λ0eλ0s]φ(X(s;x, v))φ(x)dsdvdx
≤ (φ,Aλφ) + 4π
|f ′0(E)|
[λ̃|s|eλ̃s + eλ̃s]dλ̃φ(X(s;x, v))φ(x)dsdvdx
≤ (φ,Aλφ) + C
[λ̃|s|eλ̃s + eλ̃s]dλ̃ds
≤ (φ,Aλφ) + C| lnλ− lnλ0|.
We therefore deduce that by taking the infimum over all φ,
k(λ0) ≤ k(λ) + C| lnλ− lnλ0|.
Same argument also yields k(λ) ≤ k(λ0) + C| lnλ − lnλ0|.Thus |k(λ0)− k(λ)| ≤
C| lnλ− lnλ0| and k(λ) is continuous for λ > 0.
To prove (17), by (14), we recall from Sobolev’s inequality in R3
8 YAN GUO AND ZHIWU LIN
|(Kλφ, ψ)| =
4πf ′0(E)e
λs∇φ(X(s;x, v))V (s)dsdvψ(x)dx
|ψ|2|f ′0(E)|dvdx
|∇φ(X (s))|2|f ′0(E)||V (s) |2dxdv]1/2ds
|ψ|2|f ′0(E)|dvdx
)1/2 ∫ ∫
v2|∇φ(x)|2|f ′0(E)|dxdv]1/2ds
||ψ||6||∇φ||2 ≤
||∇ψ||2||∇φ||2,
since f0 has compact support. Therefore,
(Aλφ, φ) = ||∇φ||2 − (Kλφ, φ) ≥ (1 −
)||∇φ||2 ≥ 0
for λ large. �
We now compute limλ→0+Aλ. We first consider the case when the test function
φ is spherically symmetric.
Lemma 2.3. For spherically symmetric function φ(x) = φ (|x|) , we have
(Aλφ, φ) = (A0φ, φ) ≡
|∇φ|2dx+ 4π
f ′0(E)dvφ
− 32π3
minU0
f ′0(E)
∫ r2(E,L)
r1(E,L)
2(E−U0−L2/2r2)
∫ r2(E,L)
r1(E,L)
2(E−U0−L2/2r2)
|∇φ|2 + 32π3
f ′0(E)
∫ r2(E,L)
r1(E,L)
(φ− φ̄)2 drdEdL√
2(E − U0 − L2/2r2)
Proof. Given the steady state f0(E), U0(|x|) and any radial function φ (|x|) . To
find the limit of
(Aλφ, φ) =
|∇φ|2dx+ 4π
f ′0(E)dvφ
2dx(19)
f ′0(E)
λeλsφ(X(s;x, v))ds
φ (x) dxdv,
we study the following
(20) lim
λeλsφ(X(s;x, v))ds.
Note that we only need to study (20) for points (x, v) with E = 1
|v|2+U0| (x|) < E0
and L = |x× v| > 0, because in the third integral of (19) f ′0(E) has support in
UNSTABLE AND STABLE GALAXY MODELS 9
{E < E0} and the set {L = 0} has a zero measure. We recall the linearized Vlasov-
Poisson system in the r, vr, L coordinates takes the form
∂tf + vr∂rf +
− ∂rU0
∂vrf = ∂rUf∂vrf0,
∂rrUf +
∂rUf = 4π
For the corresponding linearized system, for points (x, v) with E < E0 and L > 0,
the trajectory of (X(s;x, v), V (s;x, v)) in the coordinate (r, E, L) is a periodic
motion described by the ODE (see [8])
dr(s)
= vr(s),
dvr(s)
= −U ′0(r) +
with the period
T (E,L) = 2
∫ r2(E,L)
r1(E,L)
2(E − U0 − L2/2r2)
where 0 < r1(E,L) ≤ r2(E,L) < +∞ are zeros of E − U0 − L2/2r2.So by Lin’s
lemma in [[25]],
λeλsφ(X(s;x, v))ds =
φ(X(s;x, v))ds.
Since φ(X(s;x, v) = φ(r(s)), a change of variable from s→ r(s) leads to
φ(X(s;x, v))ds = 2
φ(r)dr
2(E − U0 − L2/2r2)
For any function g(r, E, L), we define its trajectory average as
ḡ(E,L) ≡
∫ r2(E,L)
r1(E,L)
g(r,E,L)dr√
2(E−U0−L2/2r2)
∫ r2(E,L)
r1(E,L)
2(E−U0−L2/2r2)
λeλsφ(X(s;x, v))ds = 2
φ(r)dr
2(E − U0 − L2/2r2)
/T (E,L) = φ̄ (E,L)
10 YAN GUO AND ZHIWU LIN
and the integrand in third term of (19) converges pointwise to f ′0(E)φ̄φ. Thus by
the dominated convergence theorem, we have
(Aλφ, φ) =
|∇φ|2dx+ 4π
f ′0(E)φ
2dxdv − 4π
f ′0(E)φ̄φ dxdv
|∇φ|2dx+ 4π
f ′0(E)φ
2dxdv
− 32π3
minU0
f ′0(E)
∫ r2(E,L)
r1(E,L)
φ̄ (E,L)φ (r)
drdEdL
2(E − U0 − L/2r2)
|∇φ|2dx+ 4π
f ′0(E)φ
2dxdv
− 32π3
minU0
f ′0(E)
∫ r2(E,L)
r1(E,L)
2(E−U0−L/2r2)
∫ r2(E,L)
r1(E,L)
2(E−U0−L/2r2)
|∇φ|2 + 32π3
f ′0(E)
∫ r2(E,L)
r1(E,L)
(φ− φ̄)2
drdEdL
2(E − U0 − L/2r2)
This finishes the proof of the lemma. �
To compute limλ→0+(Aλφ, φ) for more general test function φ, we use the fol-
lowing ergodic lemma which is a direct generalization of the result in [26].
Lemma 2.4. Consider the solution (P (s; p, q) , Q (s; p, q)) to be the solution of a
Hamiltonian system
Ṗ = ∂qH (P,Q)
Q̇ = −∂pH (P,Q)
with (P (0) , Q (0)) = (p, q) ∈ Rn ×Rn. Denote
Qλm =
λeλsm (P (s) , Q (s)) ds.
Then for anym (p, q) ∈ L2 (Rn ×Rn), we have Qλm→ Pm strongly in L2 (Rn ×Rn).
Here P is the projection operator of L2 (Rn ×Rn) to the kernel of the transport
operator D = ∂qH∂p − ∂pH∂q and Pm is the phase space average of m in the set
traced by the trajectory.
Proof. Denote U (s) : L2 (Rn ×Rn) → L2 (Rn ×Rn) to be the unitary semigroup
U (s)m = m (P (s) , Q (s)). By Stone Theorem ([40]), U (s) is generated by iR = D,
where R = −iD is self-adjoint and
U (s) =
eiαsdMα
where
Mα;α ∈ R1
is spectral measure of R. So
λeλsm(P (s), Q(s))ds =
eiαsdMαm ds =
λ+ iα
dMαm.
UNSTABLE AND STABLE GALAXY MODELS 11
On the other hand, the projection is P = M{0} =
ξdMα where ξ(α) = 0 for
α 6= 0 and ξ(0) = 1. Therefore
λeλsm(P (s), Q(s))ds − Pm
λ+ iα
− ξ(α)
d‖Mαm‖2L2
by orthogonality of the spectral projections. By the dominated convergence theo-
rem this expression tends to 0 as λ→ 0+, as we wished to prove. The explaination
of Pm as the phase space average of m is in our remark below. �
Remark 1. Since
λeλsds = 1, the function
(x, v) =
λeλsm (P (s), Q(s)) ds
is a weighted time average of the observable m along the particle trajectory. By the
same proof of Lemma 2.4, we have
(22) lim
m (P (s), Q(s)) ds = Pm.
But from the standard ergodic theory ([3]) of Hamiltonian systems, the limit of the
above time average in (22) equals the phase space average of m in the set traced
by the trajectory. Thus Pm has the meaning of the phase space average of m and
Lemma 2.4 states that the limit of the weighted time average (21) yields the same
phase space average. In particular, if the particle motion is ergodic in the invariant
set SI determined by the invariants E1, · · · , Ik, and if dσI denotes the induced
measure of Rn ×Rn on SI , then
(23) Pm =
σI (SI)
m (p, q) dσI (p, q) .
For integral systems, using action angle variables (J1, · · · , Jn;ϕ1, · · · , ϕn) we have
(Pm) (J1, · · · , Jn) = (2π)−n
· · ·
m (J1, · · · , Jn, ϕ1, · · · , ϕn) dϕ1, · · · dϕn
for the generic case with independent frequencies (see [4]).
Recall the weighted L2 space L2|f ′0|
in (5). Then U (s) : L2|f ′0|
→ L2|f ′0|
defined by
U (s)m = m (X(s;x, v), V (s;x, v)) is an unitary group, where (X(s;x, v), V (s;x, v))
is the particle trajectory (13). The generator of U (s) is D = v ·∂x−∇xU0 ·∇v and
R = −iD is self-adjoint by Stone Theorem. By the same proof, Lemma 2.4 is still
valid in L2|f ′0|
. In particular, for any φ (x) ∈ L2
we have
λeλsφ(X(s;x, v))ds → Pφ
in L2|f ′0|
, where P is the projector of L2|f ′0|
to kerD.
Now we derive an explicit formula for the above limit Pφ. Note that as in the
proof of lemma 2.3, we only need to derive the formula of Pφ for points (x, v) with
E < E0 and L > 0. Since U0 (x) = U0 (r), the particle motion (13) in such a center
field is integrable and has been well studied (see e.g. [8], [4]). For particles with
12 YAN GUO AND ZHIWU LIN
energy E < E0 < 0, L > 0 and momentum ~L = x× v, the particle orbit is a rosette
in the annulus
AE,L = {r1(E,L) ≤ r ≤ r2(E,L)} =
E − U0 − L2/2r2 ≥ 0
lying on the orbital plane perpendicular to ~L. So we can consider the particle
motion to be planar. For such case, the action-angle variables are as follows (see
e.g. [30]): the actions variables are
T (E,L)
, Jθ = L,
where
T (E,L) = 2
∫ r2(E,L)
r1(E,L)
2(E − U0 − L2/2r2)
is the radial period, the angle variable ϕr is determined by
dϕr =
T (E,L)
2(E − U0 − L2/2r2)
and ϕθ = θ −∆θ where
d (∆θ) =
Lr−2 − Ωθ
2(E − U0 − L2/2r2)
Ωθ (E,L) =
T (E,L)
∫ r2(E,L)
r1(E,L)
2(E − U0 − L2/2r2)
is the average angular velocity. For any function φ (x) ∈ H2
, we denote
φ~L (r, θ) to be the restriction of φ in the orbital plane perpendicular to
~L. Then by
(24), for the generic case when the radial and angular frequencies are independent,
we have
E, ~L
= (2π)
φ~Ldϕθdϕr(26)
πT (E,L)
∫ r2(E,L)
r1(E,L)
φ~L (r, θ) dθdr
2(E − U0 − L2/2r2)
In particular, for a spherically symmetric function φ = φ (r), we recover
(27) (Pφ) (E,L) = 2
T (E,L)
∫ r2(E,L)
r1(E,L)
φ(r)dr
2(E − U0 − L2/2r2)
We thus conclude the following
Lemma 2.5. Assume that f0(E) has a bounded support in x and v and f
bounded. For any φ ∈ H1
, we have
(Aλφ, φ) = (A0φ, φ)
|∇φ|2dx+ 4π
f ′0(E)dvφ
2dx− 4π
f ′0(E) (Pφ)
|∇φ|2dx+ 4π
f ′0(E) (φ− Pφ)
UNSTABLE AND STABLE GALAXY MODELS 13
where P is the projector of L2|f ′0|
to kerD and more explicitly Pφ is given by (26).
The limiting operator A0 is
(29) A0φ = −∆φ+ [4π
f ′0(E)dv]φ − 4π
f ′0(E)Pφdv.
Now we give the proof of the instability criterion.
Proof of Theorem 1.1. We define
λ∗ = sup
k(λ)<0
By Lemmas 2.1 and 2.5, we deduce that
−∞ < λ∗ ≤ Λ <∞.
Therefore, by the continuity of k(λ), we have
k(λ∗) = 0.
Hence, there exists an increasing sequence of λn < λn+1 < λ∗ so that λn → λ∗,
kn ≡ k(λn) < 0, and
kn → k(λ∗) = 0.
Therefore, kn are negative eigenvalues. By Lemma 2.2, we get a sequence φn ∈ H2
such that
(30) Aλnφn = knφn
with kn < 0, kn → 0 and λn → λ0 > 0, as n → ∞. Recall χ the cutoff function
of the support of f0(E) such that χ ≡ 1 for f0(E) > 0. We claim that χφn is a
nonzero function for any n. Suppose otherwise, χφn ≡ 0, then from the equation
(30) we have (−∆− kn)φn = 0 which implies that φn = 0, a contradiction.Thus
we can normalize φn by ‖χφn‖2 = 1. Taking inner product of (30) with φn and
integrating by parts, we have
‖▽φn‖22 ≤ −4π
f ′0(E)φ
n dvdx+
4πf ′0(E)
λnsφn(X(s;x, v))dsφn (x) dx
= −4π
f ′0(E) (χφn)
4πf ′0(E)
λns (χφn) (X(s;x, v))ds (χφn) (x) dx
f ′0(E)dv
‖χφn‖22 .
Here in the second equality above, we use the fact χ = 1 on the support of
f ′0(E) (f0(E)) and that (χφn) (X(s;x, v)) = φn(X(s;x, v)χ due to the invariance
of the support under the trajectory flow, as in (15). In the last inequality, we use
the same estimate as in (16). Thus,
||φn||L6 ≤ C sup
‖▽φn‖2 < C
for some constant C′ independent of n. Then there exists φ ∈ L6 and ∇φ ∈ L2
such that
φn → φ weakly in L6, and ∇φn → ∇φ weakly in L2.
14 YAN GUO AND ZHIWU LIN
This implies that χφn → χφ strongly in L2. Therefore ‖χφ‖2 = 1 and thus φ 6= 0.
It is easy to show that φ is a weak solution of Aλ0φ = 0 or
(31) −∆φ = −[4π
f ′0(E)dv]φ + 4πf
λ0sφ(X(s;x, v))dsdv = ρ.
We have that
ρdx = −4π
f ′0(E)φ (x) dxdv +
4πf ′0(E)φ(X(s;x, v))dxdvds
= −4π
f ′0(E)φ (x) dxdv +
4πf ′0(E)φ(x)dxdvds = 0
and by (31) ρ has compact support in Sx, the x−support of f0(E). Therefore from
the formula φ (x) =
|x−y|
dy, we have
φ (x) =
ρ (y)
|x− y|
ρ (y)
|x− y|
ρ (y)
dy = O
|x|−2
for x large, and thus φ ∈ L2. By elliptic regularity, φ ∈ H2. We define f (x, v) by
(14), then f ∈ L∞ with the compact support in S. Now we show that eλ0t[f, φ] is a
weak solution to the linearized Vlasov-Poisson system. Since φ satisfies the Poisson
equation (31), we only need to show that f satisfies the linearized Vlasov equation
(12) weakly. For that, we take any g ∈ C1c
3 × R3
, and
R3×R3
(Dg) fdxdv
R3×R3
(Dg) (f ′0(E)φ(x)) dxdv −
R3×R3
(Dg) f ′0(E)
λ0sφ(X(s;x, v))dsdxdv
= I + II.
Since D is skew-adjoint, the first term is
I = −
R3×R3
gD (f ′0(E)φ) dxdv = −
R3×R3
f ′0(E)gDφdxdv.
UNSTABLE AND STABLE GALAXY MODELS 15
For the second term,
II = −
R3×R3
f ′0(E) Dg(x, v) φ (X(s;x, v)) dxdvds
R3×R3
f ′0(E) (Dg) (X(−s), V (−s))φ (x) dxdvds
R3×R3
f ′0(E)
g (X(−s), V (−s))
ds φ (x) dxdv
R3×R3
f ′0(E)
λ0g (x, v)−
λ0sg (X(−s), V (−s)) ds
φ (x) dxdv
R3×R3
f ′0(E)λ0φ (x)− f ′0(E)
λ0sφ (X(s), V (s)) ds
g (x, v) dxdv
R3×R3
f ′0(E)φ (x)− f ′0(E)
λ0sφ (X(s), V (s)) ds
g dxdv
= .λ0
R3×R3
fgdxdv.
Thus we have
R3×R3
(Dg) fdxdv =
R3×R3
(λ0f − f ′0(E)Dφ) gdxdv
which implies that f is a weak solution to the linearized Vlasov equation
λ0f +Df = f
0 (E) v · ∇xφ.
Remark 2. Consider an anisotropic spherical galaxy with f0 (x, v) = f0
For a radial symmetric growing mode eλt (φ, f) with φ = φ (|x|) and f = f
|x| , E, L2
The linearized Vlasov equation (11) becomes
λf + v · ∇xf −∇xU0 · ∇vf
= ∇xφ · ∇vf0 = ∇xφ ·
|x× v|2
= φ′ (|x|) x
v + 2
[(x× v)× x]
v · ∇xφ,
which is of the same form as in the isotropic case (20). So by the same proof of The-
orem 1.1, we also get an instability criterion for radial perturbations of anisotropic
galaxy, in terms of the quadratic form (18) with f ′0(E) being replaced by
3. Nonlinear Stability of the King’s Model
In the second half of the article, we investigate the nonlinear stability of the King
model (8). We first establish:
Lemma 3.1. Consider spherical models f0 = f0 (E) with f
0 < 0. The operator
A0 : H
r → L2r
A0φ = −∆φ+ [4π
f ′0dv]φ − 4π
f ′0Pφdv
16 YAN GUO AND ZHIWU LIN
is positive, where H2r and L
r are spherically symmetric subspaces of H
2 and L2,
and the projection Pφ is defined by (27). Moreover, for φ ∈ H2r we have
(32) (A0φ, φ) ≥ ε
|∇φ|22 + |φ|
for some constant ε > 0.
Proof. Define k0 = inf (A0φ, φ) / (φ, φ) .We want to show that k0 > 0. First, by
using the compact embedding of H2r →֒ L2r it is easy to show that the minimum
can be obtained and k0 is the lowest eigenvalue. Let A0φ0 = k0φ0 with φ0 ∈ H2r
and ‖φ0‖2 = 1. The fact that k0 ≥ 0 follows immediately from Theorem 1.1 and the
nonexistence of radial modes ([9], [22]) for monotone spherical models. The proof
of k0 > 0 is more delicate. For that, we relate the quadratic form (A0φ, φ) to the
Antonov functional (4). We define D = v ·∂x−∇xU0 ·∇v to be the generator of the
unitary group U (s):L
|f ′0|
→ L2,r|f ′0|
defined by U (s)m = m (X(s;x, v), V (s;x, v)) .
Here L
|f ′0|
is the spherically symmetric subspace of L2|f ′0|
, which is preserved under
the flow mapping U (s). By the definition of Pφ, we have φ0 − Pφ0 ⊥ kerD. By
Stone theorem iD is self-adjoint and in particular D is closed. Therefore by the
closed range theorem ([40]), we have (kerD)
= R (D) , where R (D) is the range
of D. So there exists h ∈ L2,r|f ′0|
such that Dh = φ0−Pφ0. Moreover, since φ0−Pφ0
is even in v and the operator D reverses the parity in v, the function h is odd in v.
Define f− = f ′0h. We have
k0 = (A0φ0, φ0) =
|∇φ0|2 dx + 4π
f ′0 (φ0 − Pφ0)
|∇φ0|2 dx− 8π
|f ′0| (φ0 − Pφ0)φ0dxdv
|f ′0| (φ0 − Pφ0)
|Df−|2
|f ′0|
dxdv + 2
Df−dvdx +
|∇φ0|2 dx
|Df−|2
|f ′0|
dxdv +
|∇φ0|2 dx
|Df−|2
|f ′0|
dxdv +
|∇φ0|2 − 2∇φ0 · ∇φ−
|Df−|2
|f ′0|
dxdv − 1
where ∆φ− = 4π
Df−dv.Notice that the last expression above is the Antonov
functional 4πH (f−, f−). Since f− is spherical symmetric and odd in v,we have
H (f−, f−) > 0 by the proof in [22] which was further clarified in [33] and [21].
Therefore we get k0 > 0 as desired and (A0φ, φ) ≥ k0 |φ|22.
UNSTABLE AND STABLE GALAXY MODELS 17
To get the estimate (32), we rewrite
(A0φ, φ) = ε
|∇φ|2 dx + 4π
f ′0 (φ− Pφ)
+ (1− ε) (A0φ, φ)
|∇φ|2 dx− 4πε ‖φ− Pφ‖2L2
+ (1− ε) k0 |φ|22
|∇φ|2 dx− 8πε ‖φ‖2L2
+ (1− ε) k0 |φ|22 (since ‖P‖L2
|∇φ|2 dx+ ((1− ε) k0 − Cε) |φ|22 ≥ ε
|∇φ|2 dx+ |φ|22
if ε is small enough. �
Next, we will approximate the kerD by a finite dimensional approximation. Let
{ξi(E,L) = αi(E)βi(L)}∞i=1 be a smooth orthogonal basis for the subspace kerD =
{g(E,L)} ⊂ L2,r|f ′0|
.Define the finite-dimensional projection operator PN : L2,r|f ′0|
|f ′0|
(33) PNh ≡
(h, ξi)|f ′0|ξi
and the operator AN : H2r → L2r by
ANφ = −∆φ+ [4π
f ′0dv]φ− 4π
f ′0PNφdv.
Lemma 3.2. There exists K, δ0 > 0 such that when N > K we have
ANφ, φ
≥ δ0 |∇φ|22
for any φ ∈ H2r .
Proof. First we have AN → A0 strongly in L2. In deed, for any φ ∈ H2r ,
∥ANφ−A0φ
4πf ′0 (PNφ− Pφ) dv
≤ C ‖PNφ− Pφ‖L2
as N → ∞.We claim that for N sufficiently large, the lowest eigenvalue of AN
is at least k0/2 where k0 > 0 is the lowest eigenvalue of A0. Suppose otherwise,
then there exists a sequence {λn} and {φn} ⊂ H2r with λn < k0/2, ‖φn‖2 =
1 and Anφn = λnφn. This implies that ∆φn is uniformly bounded in L
2, by
elliptic estimate we have ‖φn‖H2 ≤ C for some constant C independent of n.
Therefore there exists φ0 ∈ H2r such that φn → φ0 weakly in H2r . By the compact
embedding of H2r →֒ L2r, we have φn → φ0 strongly in L2r and ‖φ0‖2 = 1. The
strong convergence of Anφ0 → A0φ0 implies that
Anφn → A0φ0
weakly in L2. Let λn → λ0 ≤ k0/2, then we have A0φ0 = λ0φ0, a contradiction.
Therefore we have
ANφ, φ
≥ k0/2 |φ|22 for φ ∈ H2r , when N is large enough. The
estimate (34) is by the same proof of (32) in Lemma 3.1. �
18 YAN GUO AND ZHIWU LIN
Recalling (8) with f0 = [e
E0−E−1]+ and Q0(f) = (f+1) ln(f+1)−f, we further
define functionals (related to the finite dimensional approximation of kerD) as
Ai(f) ≡
αi(− ln(s+ 1) + E0)ds,
Qi(f, L) ≡ Ai(f)βi(L), for 1 ≤ i ≤ N.
for 1 ≤ i ≤ N. Clearly,
∂1Qi(f0, L) = αi(− ln(f0 + 1) + E0)βi(L) = αi(E)βi(L) = ξi(E,L),
where {ξi(E,L)}Ni=1 are used to define PN in Lemma 3.2. Define the Casimir
functional (E0 < 0 )
I(f) =
[Q0(f) +
|v|2f − E0f ]dxdv −
|∇φ|2dx
which is invariant of the nonlinear Vlasov-Poisson system. We introduce additional
N invariants
Ji(f, L) ≡
Qi(f, L)dxdv.
for 1 ≤ i ≤ N . We define Ω to be the support of f0(E). We first consider
I(f)− I(f0) =
[Q0(f)−Q0(f0) +
|v|2(f − f0)− E0(f − f0)]dxdv
∇U0 · ∇(U − U0)−
|∇(U − U0)|2dx
[Q0(f)−Q0(f0) + (E − E0)(f − f0)]dxdv −
|∇(U − U0)|2dx.
We define
g = f − f0, φ = U − U0
gin ≡ (f − f0)1Ω, gout ≡ (f − f0)1Ωc , ∆φin ≡
gin, ∆φout ≡
gout .
And we define the distance function for nonlinear stability as
d(f, f0) ≡
[Q0(gin + f0)−Q0(f0) + (E − E0)gin]dxdv
|∇φin|2dx
Q0(gout)dxdv +
(E − E0)goutdxdv
= din +
|∇φin|2dx + dout,
for which each term is non-negative. We therefore split:
I(f)− I(f0)
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv −
|∇φin|2dx
Q0(gout)dxdv +
(E − E0)goutdxdv −
|∇φout|2dx−
∇φout · ∇φindx
= Iin + Iout .
UNSTABLE AND STABLE GALAXY MODELS 19
In the estimates below, we use C,C′, C′′ to denote general constants depending
only on f0 and quantities like ‖f (t)‖Lp (p ∈ [1,+∞]) which equals ‖f (0)‖Lp and
therefore always under control. We first estimate ‖∇φout‖22 to be of higher order of
d, which also implies that
∇φout · ∇φindx is of higher order of d.
Lemma 3.3. For ε > 0 sufficiently small, we have
|∇φout|2dx ≤ C
εd(f, f0) +
[d(f, f0)]
Proof. In fact, since
|∇φout|2dx ≤ C||
gout dv||2L6/5
≤ C||
gout 1E0≤E≤E0+εdv||2L6/5 + C||
gout 1E>E0+εdv||2L6/5 .
The first term is bounded by
g2out dv]
1E0≤E≤E0+εdv]
3/5dx
g2out dvdx]×
1E0≤E≤E0+εdv]
3/2dx
≤ Cε[
g2out dvdx] ≤ Cε[
g2out dvdx]
≤ Cεd(f, f0).
In the above estimates, we use that
Q0(gout)dvdx ≥ c
g2out dvdx and
1E0≤E≤E0+εdv ≤ Cε,
which can be checked by an explicit computation when ε > 0 is sufficiently small
such that E0 + ε ≤ 0.
On the other hand, by the standard estimates (see [12, P. 120-121])
gout 1E>E0+εdv||2L6/5
gout 1E>E0+εdxdv
|v|2gout 1E>E0+εdxdv
(E − E0)gout 1E>E0+εdxdv
(E − E0)gout 1E>E0+εdxdv + 2 sup |U0|
gout 1E>E0+εdxdv
2 sup |U0|
d5/3.
20 YAN GUO AND ZHIWU LIN
By Lemma 3.3, we have
∇φout · ∇φindx
≤ ‖∇φout‖2 ‖∇φin‖2
ε1/3d(f, f0) +
[d(f, f0)]
and therefore for ε sufficiently small,
(36) Iout ≥ dout − C
ε1/3d(f, f0) +
[d(f, f0)]
4/3 +
[d(f, f0)]
To estimate Iin, we split it into three parts:
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin + φingin]dxdv +
|∇φin|2dx
(1− τ)
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin + (I − PN )φingin]dxdv +
|∇φin|2dx
+ (1− τ)
PNφingindxdv
= I1in + I
in + I
where ∆φin = 4π
gin dv. We estimate each term in the following lemmas.
Lemma 3.4.
(38) I1
din − Cτ
|∇φin|2dx.
Proof. In fact, since the integration region Ω is finite, we have
I1in =τ
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin + φingin]dxdv +
|∇φin|2dx
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv − Cτ ||φin||L6 ||gin||L6/5
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv − C′τ ||∇φin||L2 ||gin||2
din − C′′τ ||∇φin||22,
since
din =
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv ≥ C||gin||22.
To estimate I2in, we need the following pointwise duality lemma from elementary
calculus.
Lemma 3.5. For any c, and any h, we have
gc,f0 (h) = Q0(h+ f0)−Q0(f0)−Q′0(f0)h− ch ≥ (f0 + 1)(1 + c− ec).
Proof. Direct computation yields that the minimizer fc of gc,f0 (h) satisfies the
Euler-Lagrange equation
ln (fc + f0 + 1)− ln (f0 + 1)− c = 0,
UNSTABLE AND STABLE GALAXY MODELS 21
fc = (f0 + 1) (e
c − 1) .
Thus by using the Euler-Lagrange equation, we deduce
min gc,f0 (h) = gc,d (fc)
= (fc + f0 + 1) ln(1 + fc + f0)
− (f0 + 1) ln(1 + f0)− [1 + ln(f0 + 1)]fc − cfc
= (fc + f0 + 1)[ln(1 + fc + f0)− ln(f0 + 1)− c]
+ fc ln(1 + f0) + c(f0 + 1)− [1 + ln(f0 + 1)]fc
= (f0 + 1)(1 + c− ec).
Lemma 3.6.
(39) I2
(1− τ) δ0
|∇φin|2dx− CeC
Proof. Recall (37). By using Lemma 3.5 for c = − (φin − PNφin) and using the
Taylor expansion, we have
I2in = (1 − τ)
[Q0(f0 + gin)−Q0(f0) + (E − E0)gin + (φin − PNφin) fin]dxdv
(1− τ)
|∇φin|2dx
(1− τ)
|∇φin|2dx+ (1− τ)
(f0 + 1)1Ω(1 + φin − PNφin − eφin−PNφin)dxdv
≥ 1− τ
|∇φin|2dx− 4π
|f ′0 (E)| (φin − PNφin)
− Ce|φin−PNφin|∞
|f ′0 (E)| |φin − PNφin|
dxdv (Note (f0(E) + 1)1Ω = |f ′0(E)|)
≥ (1− τ) δ0
|∇φin|2dx− Ce|φin−PNφin|∞
|f ′0 (E)| |φin − PNφin|
dxdv.
In the last line, we have used Lemma 3.2. To estimate the last term above and
conclude our lemma, it suffices to show
|φin − PNφin|∞ ≤ CNd
This follows from the facts that for the fixed N smooth functions ξi, we have
|PNφin|∞ =
(φin, ξi)|f ′0|ξi
≤ CN |φin|∞ ,
and since φ is spherically symmetric,
|φin| (r) =
u2ρin (u) du+
uρin (u)du
R |ρin|2 ≤ C
′′ ‖gin‖2 ≤ CNd
where ρin =
gindv and R is the support radius of ρin. �
22 YAN GUO AND ZHIWU LIN
We now estimate the term
PNφinfindxdv, for which we use the additional
invariants.
Lemma 3.7. For any ε > 0, we have
∣ ≤ C(d1/2(0) + ε1/2d1/2 + 1
d)d1/2.
Proof. By the definition of I3in in (37), it suffices to estimate (gin, ξi). We expand
Ji(f, L)− Ji(f0, L)
= Ji(f0 + gin, L)− Ji(f0, L) + Ji(gout, L)
= (gin , ξi) +O(d) + Ji(gout, L).
Notice that
|Ji(gout, L)| ≤ C||gout||L1 ≤ C||1{E0≤E≤E0+ε}gout||L1 + C||1{E≥E0+ε}gout||L1
≤ ε1/2||gout||L2 +
||1{E≥E0+ε}(E − E0)gout||L1 ≤ C[ε
1/2d1/2 +
It thus follows that
|(gin , ξi)| ≤ |Ji(f(0), L)− Ji(f0, L)|+ C[ε1/2d1/2 +
≤ C[d1/2(0) + ε1/2d1/2 + 1
Therefore
∣I3in
∣ = (1− τ)
PNφingin dxdv
(φin, ξi)|f ′0|ξi
gin dxdv
(φin, ξi)|f ′0|
|(ξi, gin)| ≤ C′
|φin|∞ |(ξi, gin)|
≤ Cd1/2[d1/2(0) + ε1/2d1/2 +
Now we prove the nonlinear stability of King model.
Proof of Theorem 1.2. The global existence of classical solutions of 3D Vlasov-
Poisson system was shown in [34] for compactly supported initial data f (0) ∈ C1c .
Let the unique global solution be (f (t) , φ (t)). Let d (t) = d(f (t) , f0). Combining
estimates (36), (38), (39) and (40), we have
I(f (0))− I(f0) = I(f (t))− I(f0)
≥ dout +
din +
(1− τ) δ0
|∇φin|2dx
ε1/3d (t) +
d (t)
d (t)
− CeC
′d(t)
d (t)
− Cd (t)1/2 [d1/2(0) + ε1/2d (t)1/2 +
d (t)].
UNSTABLE AND STABLE GALAXY MODELS 23
Thus by choosing ε and τ sufficiently small, there exists δ′ > 0 such that
I(f (0))− I(f0) ≥ δ′d(t)− C
d (t)
+ d (t)
+ d (t)
− CeC
′d(t)
d (t)
− Cd (t)1/2 d1/2(0).
It is easy to show that I(f (0)) − I(f0) ≤ C′′d (0). Define the functions y1 (x) =
δ′x2 − CeC′xx3 − C
x8/3 + x10/3 + x3
and y2 (x) = Cd (0)
x + C′′d (0). Then
above estimates implies that y1
d (t)
d (t)
. The function y1 is in-
creasing in (0, x0) where x0 is the first maximum point. So if d (0) is sufficiently
small, the line y = y2 (x) intersects the curve y = y1 (x) at points x1, x2, · · · ,
with x1 (d (0)) < x0 < x2 (d (0)) < · · · . Thus the inequality y1 (x) ≤ y2 (x) is
valid in disjoint intervals [0, x1 (d (0))] and [x2 (d (0)) , x3 (d (0))], · · · . Because d (t)
is continuous, we have that d (t)
< x1 (d (0)) for all t < ∞, provided we choose
d (0)
< x0. Since x1 (d (0)) → 0 as d (0) → 0, we deduce the nonlinear stability
in terms of the distance functional d (t)
Acknowledgements
This research is supported partly by NSF grants DMS-0603815 and DMS-0505460.
We thank the referees for comments and corrections.
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UNSTABLE AND STABLE GALAXY MODELS 25
Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown
University, Providence, RI 02912, USA
E-mail address: guoy@cfm.brown.edu
Mathematics Department, University of Missouri, Columbia, MO 65211 USA
E-mail address: lin@math.missouri.edu
	1. Introduction
	2. An Instability Criterion
	3. Nonlinear Stability of the King's Model
	References
