On the local uniqueness of steady states for the Vlasov-Poisson system

0.22 MB·

by  Mikaela Iacobelli

On the local uniqueness of steady states for the
Vlasov-Poisson system
Mikaela Iacobelli ∗
January 3, 2017
7
1
0
2
Abstract

n
a Motivated by recent results of Lemou, Méhats, and Räphael [15] and Lemou [14]
J
concerning the quatitative stability of some suitable steady states for the Vlasov-

1 Poisson system, we investigate the local uniqueness of steady states near these one.

This researchis inspiredby analogousresults ofCouffrutandŠverákinthe contextof
]
P the 2D Euler equations [6].
A
.
h 1 Introduction
t
a
m The gravitational Vlasov-Poisson equation modelizes the evolution of a large number of
[ particles subject to their own gravity, under the assumption that both the relativistic

1 effects and the collisions between particles can be neglected. We consider the Vlasov-
v Poisson system in three dimension:
2
6
∂ f +v·∇ f −∇φ ·∇ f = 0, (t,x,v) ∈ R+×R3×R3
2 t x f v (1.1)
0 f(0,x,v) = f (x,v) ≥ 0, f dxdv = 1.
( 0 0
0
. R
1 where the Newtonian potential φ is given in terms of the density ρ :
f f
0
7 1
1 ρ (x)= f(x,v)dv, and φ (x) = − ∗ρ = K ∗ρ
f f f f
v: ZR3 4π|x|
i
X At the beginning of the last century the astrophysicist Sir J. Jeans used this system
to model stellar clusters and galaxies [13] and to study their stability properties. In this
r
a
context it appears in many textbooks on astrophysics such as [4, 10]. In the repulsive
case, this system was introduced by A. A. Vlasov around 1937 [21, 22]. Because of the
considerable importance in plasma physics and in astrophysics, there is a huge literature
on the Vlasov-Poisson system.
The global existence and uniqueness of classical solutions of the Cauchy problem for
the Vlasov-Poisson system was obtained by Iordanskii [12] in dimension 1, Ukai-Okabe
[20]inthe2-dimensional case, andindependentlybyLions-Perthame [16]andPfaffelmoser
[18] in the 3-dimensional case (see also [19]). To our knowledge, there are currently no
results about existence and uniqueness of classical solutions in dimension greater than 3.
It is important to mention that, parallel to the existence of classical solutions, there
have been a considerable amount of work on the existence of weak solutions, in particular
∗University of Cambridge, DPMMS Centre for Mathematical Sciences, Wilberforce road, Cambridge
CB3 0WB, UK.Email: [email protected]
1
under very low assumptions on the initial data. We mention in particular the classical
result by Arsen’ev [3], who proved global existence of weak solutions under the hypothesis
that f is bounded and has finite kinetic energy, and the result of Horst and Hunze [11],
0
where the authors relax the integrability assumption on f . If one wishes to relax even
0
more the integrability assumptions on the initial data then one enters into the framework
of the so called renormalized solutions introduced by Di Perna and Lions [7, 8, 9]. The
interested reader is referred to the recent papers [1, 5] for more details and references.
One of the main features of the nonlinear transport flow (1.1) is the conservation of
the total energy
1 1
H(f(t)) = |v|2f(t,x,v) dxdv− |∇φ (t,x)|2dx = H(f(0)) (1.2)
f
2 R6 2 R3
Z Z
as well as the Casimir functions: for all G∈ C1([0,∞],R+) such that G(0) = 0,
G(f(t,x,v)) dxdv = G(f (x,v)) dxdv.
0
R6 R6
Z Z
1.1 Main result
The goal of this work is to prove a local uniqueness result for steady states of (1.1). In the
recent paper [14] (see also [15, 17]), the author proves quantitative stability inequalities
for the gravitational Vlasov-Poisson system that will be crucial in the following. More
precisely, the author considers a class of steady states f¯ to the Vlasov Poisson system,
which are decreasing functions of their microscopic energy, and obtains an explicit control
of the L1 distance between f¯and any function f in terms of the energy H(f)−H(f¯) and
the L1 distance between the rearrangements f¯∗ and f∗ of f¯and f, respectively.
In the following we give some definition and we state the local functional inequality in
[14, Theorem 2]. We first recall the notion of equimeasurability and rearrangement.
Definition 1.1. Given two integrable nonnegative functions f,g :Rn → R, we say that f
and g are equimeasurable if
|{f > s}| = |{g > s}| for a.e. s> 0.
Then, the (radially decreasing) rearrangement f∗ of f is defined as the unique radially
decreasing function that is equimeasurable to f. In other words, the level sets of f∗ are
given by
{f∗ > s} = B , with r(s)> 0 s.t. |B |= |{f > s}| for a.e. s > 0.
r(s) r(s)
The following important result is proved in [14, Theorem 2(ii)].
Theorem 1.2. Consider f¯ a compactly supported steady state solution of (1.1) of the
form
|v|2
f¯(x,v) = F(e(x,v)), with e(x,v) = +φ (x), (1.3)
2 f¯
where F is a continuous function from R to R+ that satisfies the following monotonicity
property: there exists e < 0 such that F(e) = 0 for e ≥ e and F is a C1 function on
0 0
(−∞,e ) with F′ < 0 on (−∞,e ). Assume that f ∈ L1∩L∞(R6) has finite kinetic energy
0 0
and is sufficiently close to a translation of f¯in the following sense:
zi∈nRf3kφf −φf¯(·−z)kL∞ +k∇φf −∇φf¯(·−z)kL2 < R0, (1.4)
2
for some suitable constant R > 0. Then there exists a constant K > 0, depending only
0 0
on f¯, such that
inf kf −f¯(·−(x ,0))k ≤ kf∗−f¯∗k +K H(f)−H(f¯)+kf∗−f¯∗k 1/2.
0 L1 L1 0 L1
x0∈R3
(cid:2) (cid:3)
where we denote f¯(·−(x ,0)) = f¯(x−x ,v).
0 0
An immediate consequence is the following estimate, that will be the starting point of
our investigation.
Corollary1.3. Letf,f¯beasinTheorem 1.2. Assume inaddition that f isequimeasurable
to f¯. Then
inf kf −f¯(·−(x ,0))k2 ≤ K2[H(f)−H(f¯)], (1.5)
x0∈R3 0 L1 0
Let f¯bethestationary solution as above. Ourgoal is to understandif, nearbyf¯, there
exist other stationary solutions of (1.1). Because stationary solutions of (1.1) correspond
to critical points of H with respect to variations of f¯ generated by Hamiltonian flows
(see Lemma 2.3 below), it makes sense to consider a “neighborhood” of f¯ generated by
flows of smooth Hamiltonians. Noticing that f¯is supported in a ball B ⊂ R3 ×R3 for
ρ
some ρ > 0 and we shall use the flow of the functions H to move f¯, it makes sense to
consider Hamiltonians H that are all supported inside B . Hence, one should think of
2ρ
these functions H as the “tangent space” at f¯that will generate the admissible variations.
Let us introduce the following notation:
f¯H := (ΦH) f¯= f¯◦ΦH ∀s∈ R,
s s # −s
where s 7→ ΦH is the Hamiltonian flow of H, namely
s
0 Id
∂ ΦH = J∇H(ΦH), J ∈ R6×6, J =
s s s −Id 0 (1.6)
 !
 Φ0 = Id.
In other words, s7→ f¯H is the variation generated by H, and as H vary this generates a
s
“symplectic” neighborhood of f¯. Note that, since f¯H = f¯sH, to parameterize a neighbor-
s 1
hood of f¯it is enough to consider the image of the map1
H 7→ f¯H. (1.7)
1
We now give some definitions in order to clarify the hypothesis that are needed on the
Hamiltonian H. Let us start with the definition of the set Inv that represents the set of
f¯
all the Hamiltonians who acts trivially on f¯.
Definition 1.4. Inv := {H ∈ C1,1(R6) : {H,f¯} = 0}
f¯
This definition is motivated by the following simple result:
1This resembles to the exponential map in Riemannian geometry, where a neighborhood of a point
x∈M is obtained as the image of a neighborhood of 0 in T M via themap
x
v7→γ (1),
v
where s7→γ (s) is thegeodesic starting at x with velocity v.
v
3
Lemma 1.5. If H ∈ Inv then (ΦH) f¯= f¯, i.e., the Hamiltonian flow of H does not
f¯ t #
move f¯.
Proof. The function f¯H := (ΦH) f¯solves the transport equation
s s #
∂ f¯H +div(J∇Hf¯H) = 0, f¯H| = f¯.
s s s s s=0
Since also f¯ is a solution to this equation (because div(J∇Hf¯) = {H,f¯} = 0) and the
vector field J∇H is Lipschitz, f ≡ f¯by uniqueness for the above transport equation.
s
ItfollowsbythelemmaabovethatifH belongstoInv thenΦH isnotmovingf¯. Since
f¯ s
our goal is to use Hamiltonians H to parameterize a neighborhood of f¯, there is no reason
to consider H that belong to Inv(f¯), and it make sense to exclude them. Actually, for
sometechnical reasons thatwillbemoreclear later, weshallneedtoimposeaquantitative
version of the condition H 6∈ Inv . To do that, we introduce the family of sets
f¯
A := {H 6≡ 0 : k∇Hk ≤ kk{H,f¯}k }, k ≥ 1.
k L1 L1
Remark 1.6. We note that Inv = Ac. Indeed, if H ∈ Ac then k∇Hk /k ≥
f¯ k k L1
k∈N k∈N
k{H,f¯}k for all k ∈ N. Thus {H,f¯}T≡ 0, which implies thaTt H ∈ Inv . Viceversa, if
L1 ¯f
H ∈ Inv then clearly H ∈ Ac.
¯f k
k∈N
T
Because of this observation, we see that
H 6∈ Inv ⇐⇒ ∃k such that H ∈ A .
f¯ k
Motivated by this fact, in the sequel we shall fix k and consider only Hamiltonians that
belong to A . Of course this is more restrictive than assuming only H 6∈ Inv but at the
k f¯
moment it is not clear to us how to remove such an assumption.
Going further in our preliminary analysis, we observe that all translations of f¯ are
trivially stationary solutions. However, translations in v are automatically controlled by
thekineticenergyandindeedtheydonotappearin(1.5). To“kill”thespaceoftranslations
in x, we will assume that Bar (f¯H) = Bar (f¯), where
x 1 x
Bar (f) := xf(x,v)dxdv ∈ R3
x
R6
Z
denotes the “barycenter (in x)” of f. We want to emphasize that this is not a restrictive
assumption on H, since one could remove it by adding to H a Hamiltonian corresponding
to translations in the x variable in order to recenter the barycenter of f. Since this would
not add major technical difficulties to the proof but may distract the reader from the
essential points, we decided to impose this barycenter condition on f¯H.
1
As a final consideration, since our goal is prove that there are no steady states to (1.1)
in a neighborhood of f¯ generated via the map (1.7), we shall need to assume that our
Hamiltonians H are small in some suitable topology.
Our main theorem asserts that, for Hamiltonians small enough in a sufficiently strong
Sobolev norm that are quantitatively away from Inv , there cannot be a stationary point
f¯
of the form f¯H.
1
4
Theorem 1.7. Let f¯be as in (1.3), where F is a continuous function from R to R+ that
satisfies the following monotonicity property: there exists e < 0 such that F(e) = 0 for
0
e ≥ e and F is a C1 function on (−∞,e ) with F′ < 0 on (−∞,e ). Let ρ > 0 be such
0 0 0
that supp(f¯) ⊂ B . Also, assume that f¯∈ W2,q(R6) for some q > 3. Then the following
ρ
local uniqueness result for steady states holds:
Let r ≥ 22, and given ρ,ε,k > 0 consider the space of functions
Nk := {f¯H : supp(H) ⊂ B , Bar (f¯H)= Bar (f¯), H ∈A , kHk ≤ ε}.
ε 1 2ρ x 1 x k Wr,2
Then, fixed k ∈ N, there is no stationary state for (1.1) in Nk for ε small enough.
ε
1.1.1 Comments
Starting from the seminal paper of Arnold about the geometric interpretation of the Euler
equations as L2-geodesics in thespace of measurepreservingdiffeomorphisms[2], Choffrut
and Šverák recently obtained a related result for the 2D Euler equation [6]. Thebasic idea
there is that, under the evolution given by the 2D incompressible Euler equations, the
vorticity is transported by an incompressible vector field, hence the measure of all its
super-level sets is constant. This means that, given an initial vorticity ω , its evolution
0
ω(t) is in the same equimeasurability class of ω . This allows one to foliate the space
0
of vorticities into a family of leaves O (the equimeasurability class of ω ), and the
ω0 0
Euler equations preserve these leaves. In addition, thanks to the Hamiltonian structure
of the Euler equations, one can characterize stationary solutions as critical points of the
Hamiltonian energy E restricted to the orbits.
In other words, one has the following situation: the space of vorticities is foliated by
the orbits O , and the equilibria are the critical points of E restricted to the orbits. In
ω
finite dimension, the implicit function theorem would give the following: if O is smooth
ω
near a point ω¯ ∈ O , and if ω¯ is a non-degenerate critical point of E in O , then near ω¯
ω ω
the set of equilibria form a smooth manifold transversal to the foliation. In addition, the
dimension of this manifold is equation to the co-dimension of the orbits. In particular,
in a non-degenerate situation, the equilibria are locally in one-to-one correspondence with
the orbits. In [6] the authors obtain an analogue of this correspondence in the infinite
dimensional context of Euler equations. There, the authors use an infinite dimensional
version of the implicit function theorem in the space of C∞ function, via a Nash-Moser’s
interation.
With respect to their result, here we have different assumptions and results. These are
motivated by the following:
• SincetheVlasov-Poisson system(1.1)isHamiltonian, givenaninitialconditionf its
0
evolution f undertheVlasov-Poisson system will also bein thesameequimesurabil-
t
ityclass. However, whileHamiltonianmapsandmeasurepreservingmapscoincidein
2-dimension, they are very different in higher dimension (for instance, Hamiltonian
mapspreservethesymplecticstructure). Becausesolutionstothe3DVlasov-Poisson
systems describe a Hamiltonian evolution of particles in the phase-space R3 ×R3,
there is no natural reason in this context why there should be only one stationary
state in the same equimeasurability class. In particular, as already observed be-
fore, stationary solutions of (1.1) correspond to critical points of H with respect to
variations of f¯ generated by Hamiltonian flows, and not with respect to arbitrary
measure preserving variations. This is why we need to look at functions f that can
be connected to f¯via a Hamiltonian flow, namely f = f¯H for some H.
1
5
• The smallness assumption on k∇rHk is natural, and actually weaker than the one
L2
in [6], since smallness there is measured in the C∞ topology.
• As already mentioned before, the assumption on Bar (f¯H) is not fundamental: one
x 1
could easily remove it by replacing it with H − H , where H corresponds to a
0 0
translation in the phase space (multiplied by a suitable cut-off function, to make it
compactly supported). What is more essential is our assumption H ∈ A , and it is
k
unclear at the moment how to remove it. It is our plan to address this issue in a
future work.
The goal of the next section is to prove our main theorem.
2 Proof of the Theorem 1.7
2.1 Strategy of the proof
The idea of the proof is the following: first, by exploiting the results in [14], we prove that
if f¯H has the same barycenter as f¯, then
1
kf¯H −f¯k2 ≤K2[H(f¯H)−H(f¯)].
1 L1 0 1
Secondly we show that if f¯H is stationary, by a Taylor expansion of s 7→ H(f¯H) we can
1 s
prove that
H(f¯H)−H(f¯)≤ Ck∇Hk3
1 X
for some suitable norm k·k of ∇H.
X
Combining these two estimates, we get
kf¯H −f¯k ≤ Ck∇Hk3/2.
1 L1 X
We then relate the two quantities appearing in the above expression: more precisely, we
first show that
kf¯H −f¯k ≈ k{H,f¯}k = k∇H ·J∇f¯k ,
1 L1 L1 L1
and then we use our quantitative assumption on the fact that H does not belong to Inv
f¯
(namely, H ∈ A ) to say that
k
k{H,f¯}k ≈k∇Hk .
L1 L1
In this way we get
3/2
k∇Hk ≤ Ck∇Hk .
L1 X
Finally, exploiting the smallness k∇rHk ≤ ε and interpolation estimates, we are able to
L2
relate the two norms above and conclude that
k∇Hk ≤ Ck∇Hk1+δ
L2 L2
for some δ > 0, which yields a contradiction when k∇Hk is small enough.
L2
6
2.2 Lower bound
We recall the definition of the baricenter of f:
Bar (f)= xf(x,v)dxdv.
x
R6
Z
Lemma 2.1. Let f¯be as in (1.3), where F is a continuous function from R to R+ that
satisfies the following monotonicity property: there exists e < 0 such that F(e) = 0 for
0
e≥ e and F is a C1 function on (−∞,e ) with F′ < 0 on (−∞,e ). Let H ∈ C2(R6) and
0 0 0
consider the function f¯1H := (ΦH1 )#f¯for some H ∈ C1,1 with k∇HkL∞ +k∇2HkL∞ ≤ η.
Also, assume that
Bar (f¯)= Bar (f¯H). (2.1)
x x 1
Then, if η is small enough,
kf¯H −f¯k2 ≤ Kˆ [H(f¯H)−H(f¯)], (2.2)
1 L1 0 1
where Kˆ depends on the diameter of the support of f¯and f¯H.
0 1
Proof. Note that, if η is small enough, the function f = f¯H satisfies (1.4). Let (x ,0) be
1 0
the point where the minimum is achieved in (1.5). By definition,
Bar (f¯H(·−(x ,0))) = xf¯H(x−x ,v)dxdv
x 1 0 1 0
R6
Z
= (x−x )f¯H(x−x ,v)dxdv+x ,
0 1 0 0
R6
Z
= Bar (f¯H)+x .
x 1 0
hence
|x |= |Bar (f¯H)−Bar (f¯H(·−(x ,0)))| (2=.1) |Bar (f¯)−Bar (f¯H(·−(x ,0)))|
0 x 1 x 1 0 x x 1 0
≤ |x||f¯−f¯H(·−(x ,v ))|dxdz
1 0 0
R6
Z
≤ Ckf¯−f¯H(·−(x ,v ))k ≤ C[H(f¯H)−H(f¯)]1/2,
1 0 0 L1 1
where we used that f¯ and f¯H are compactly supported, so |(x,v)| is bounded on the
1
support of f¯and f¯H(·−(x ,v )). Thus,
1 0 0
kf¯−fHk ≤kf¯−f¯(·+(x ,v )))k +kf¯(·+(x ,v )))−f¯Hk
1 L1 0 0 L1 0 0 1 L1
≤|x |k∇f¯k +[H(f¯H)−H(f¯)]1/2 ≤ C[H(f¯H)−H(f¯)]1/2,
0 L1 1 1
which concludes the proof.
2.3 Upper bound
The aim of this section is to provide an estimate of the difference between the energy of f¯
and of f¯H in terms H, under the additional assumption that f¯H is a stationary solution
1 1
for (1.1). More precisely, we prove the following:
Proposition 2.2. Let f¯be a compactly supported steady state such that f¯∈L∞(R6), and
that f¯∈ W2,q(R6) for some q > 3. Let H ∈ C2(R6). Also, assume that f¯H = (ΦH) f¯is
1 1 #
a stationary solution for (1.1). Then the following estimate holds:
2
|H(f1H)−H(f¯)| ≤ Ck∇HkL∞ k∇HkL∞ +k∇2HkL∞ ,
(cid:16) (cid:17)
where C is a constant depending only on f¯.
7
As a first step towards the proof of the above result, we aim to give a characterization
of the stationary solutions of (1.1) in terms of the energy of the system H.
Lemma 2.3. Let f : R6 → R be a compactly supported function. Then f is a steady state
for (1.1) if and only if
d
H(fH)| = 0
ds s s=0
for all H ∈C2(R6), where fH := (ΦH) f.
s s #
Proof. Fix H ∈ C2(R6), and consider its flow ΦH. To simplify the notation we set ΦH =
s s
Φ . Also, it will be convenient to write Φ = (Φx,Φv) :R6 → R3×R3.
s s s s
Given a compactly supported function f, we compute the first variation of the Hamil-
tonian H around f along fH :
s
d d 1 1
H(fH) = |v|2fH(x,v) dxdv− K(x−y)fH(x,v)fH(y,w) dxdvdydw
ds s ds 2 R6 s 2 R6×6 s s
(cid:20) Z Z (cid:21)
d 1
= |Φv(x,v)|2f(x,v) dxdv
ds 2 R6 s
(cid:20) Z
1
− K(Φx(x,v)−Φx(y,v))f(x,v)f(y,w) dxdvdydw
2 R6×6 s s
Z (cid:21)
= Φv(x,v)∂ Φv(x,v)f(x,v) dxdv
s s s
R6
Z
1
− ∇ K(Φx(x,v)−Φx(y,v))·
2 R6×6 x s s
(cid:20)Z
·∂ (Φx(x,v)−Φx(y,v))f(x,v)f(y,w) dxdvdydw .
s s s
(cid:21)
Recalling that
∂ Φ = (∇ H(Φ ),−∇ H(Φ )),
s s v s x s
we have
d
H(fH) = − Φv(x,v)·∇ H(Φ (x,v))f(x,v) dxdv
ds s R6 s x s
Z
1
− ∇ K(Φx(x,v)−Φx(y,v))·
2 R6×6 x s s
(cid:20)Z
·(∇ H(Φ (x,v))−∇ H(Φ (y,v)))f(x,v)f(y,w) dxdvdydw .
v s v s
(cid:21)
Also, since K(x−y) = K(y −x), we see that ∇ K(x−y) = −∇ K(y −x), so we can
x x
rewrite the above expression as
d
H(fH) = − Φv(x,v)·∇ H(Φ (x,v))f(x,v) dxdv
ds s R6 s x s
Z
− ∇ K(Φx(x,v)−Φx(y,v))·∇ H(Φ (x,v))f(x,v)f(y,w) dxdvdydw.
x s s v s
R6×6
Z
8
Also, using that Φ preserves the Lebesgue measure and that Φ−1 = Φ , we can rewrite
s s −s
the first variation in the following way:
d
H(fH) = − v·∇ H(x,v)f(Φ (x,v)) dxdv
ds s R6 x −s
Z
− ∇ K(x−y)·∇ H(x,v)f(Φ (x,v))f(Φ (y,w)) dxdvdydw. (2.3)
x v −s −s
R6×6
Z
In particular, since Φ = Id for s= 0, we see that
s
d
H(f )| = − v·∇ H(x,v)f(x,v) dxdv
s s=0 x
ds R6
Z
− ∇ K(x−y)·∇ H(x,y))f(x,v)f(y,w) dxdvdydw. (2.4)
x v
R6×6
Z
On the other hand, for f to be a stationary solution for the system (1.1) means that
div (vf(x,v))−div (∇φ (x)f(x,v)) = 0,
x v f
or equivalently, that for all ψ ∈ C1,
− v·∇ ψ(x,v)f(x,v) dxdv+ ∇ φ (x)·∇ ψ(x,v)f(x,v) dxdv = 0. (2.5)
x x f v
R6 R3
Z Z
Since
∇ K(x−y)f(y,w)dydw = −∇φ ,
x f
Z
(2.4) proves that
d
H(fH)| = 0 ⇐⇒ (2.5) holds with ψ = H.
ds s s=0
Since C2 functions are dense in C1 for the C1 topology, this proves the result.
2.3.1 Second variation for H
As a second step, we compute the second variation for H, in line with the computation of
the first variation (2.3). Here we consider as initial condition f¯and, given a Hamiltonian
H ∈ C2, we consider f¯H := f¯◦ΦH . As before, to simplify the notation, we set Φ = ΦH.
s −s s s
Also, we define
g := ∇f¯·J∇H = {H,f¯} (2.6)
and we observe that
d
f¯(Φ ) = g(Φ ). (2.7)
−s −s
ds
Using the equations (2.3) and (2.7) the second variation is given by the following:
d2
H(f¯H) = v·∇ H(x,v)g(Φ (x,v)) dxdv
d2s s R6 x −s
(cid:20)Z (cid:21)
+ ∇ K(x−y)·∇ H(x,v)g(Φ (x,v))f¯(Φ (y,w)) dxdvdydw
x v −s −s
R6×6
(cid:20)Z (cid:21)
+ ∇ K(x−y)·∇ H(x,v)f¯(Φ (x,v))g(Φ (y,w))dxdvdydw .
x v −s −s
R6×6
(cid:20)Z (cid:21)
9
Thus, we obtain:
d2
H(f¯H) = v·∇ H(x,v)g(Φ (x,v)) dxdv (2.8)
d2s s R6 x −s
(cid:20)Z (cid:21)
+ ∇ K(x−y)·[∇ H(x,v)−∇ H(y,w)]f¯(Φ (x,v))g(Φ (y,w))dxdvdydw .
x v w −s −s
R6×6
(cid:20)Z (cid:21)
2.3.2 Proof of Proposition 2.2
Asintheprevioussection, wesetf¯H := f¯◦ΦH andΦ := ΦH. Recall that,byassumption
s −s s s
f¯H, is a stationary solution of (1.1).
1
We now study the Taylor expansion of the Hamiltonian of the gravitational Vlasov
Poisson system both in f¯ and in f¯H. Since f¯ and fH are two stationary solutions, it
1 1
follows by Lemma 2.3 applied both to f¯and to f¯H that
1
d d
H(f¯H)| = H(f¯◦Φ )| =0
ds s s=0 ds −s s=0
and
d d
H(f¯H)| = H(f¯H ◦Φ )| = 0.
ds s s=1 dτ 1 −τ τ=0
Hence, by Taylor’s formula,
1 d2
H(f¯H)= H(f¯)+ (1−s) H(f¯H)ds
1 d2s s
Z0
and
1 d2
H(f¯) = H(f¯H)+ s H(f¯H)ds.
1 d2s s
Z0
Therefore,
1 d2
H(f¯H)−H(f¯) = (1−2s) H(f¯H)ds.
1 d2s s
Z0
Since
1
(1−2s)ds = 0
Z0
we can add a constant term in the integral, and we get
1 1 d2 d2
H(f¯H)−H(f¯) = (1−2s) H(f¯H)− H(f¯H)| ds. (2.9)
1 2 Z0 d2s s d2s s s=0!
Thanks to the latter computation, in order to estimate the left hand side of (2.9), we can
estimate
d2 d2
H(f¯H)− H(f¯H)|
d2s s d2s s s=0
10