HAL Id: hal-02048776
https://hal.archives-ouvertes.fr/hal-02048776v2
Submitted on 25 Mar 2019
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de
Nonlinear instability of inhomogeneous steady states solutions to the HMF Model
Mohammed Lemou, Ana Maria Luz, Florian Méhats
To cite this version:
Mohammed Lemou, Ana Maria Luz, Florian Méhats. Nonlinear instability of inhomogeneous steady
states solutions to the HMF Model. Journal of Statistical Physics, Springer Verlag, 2020, 178 (3),
pp.645-665. �10.1007/s10955-019-02448-4�. �hal-02048776v2�
Nonlinear instability of inhomogeneous steady states solutions to the HMF Model
M. Lemou
:, A. M. Luz
‹, and F. M´ ehats
;:
Univ Rennes, CNRS, IRMAR, mohammed.lemou@univ-rennes1.fr
‹
IME, Universidade Federal Fluminense, analuz@id.uff.br
;
Univ Rennes, IRMAR, florian.mehats@univ-rennes1.fr
March 25, 2019
Abstract
In this work we prove the nonlinear instability of inhomogeneous steady states solutions to the Hamiltonian Mean Field (HMF) model. We first study the linear instability of this model under a simple criterion by adapting the techniques developed in [19]. In a second part, we extend to the inhomogeneous case some techniques developed in [14, 17, 18] and prove a nonlinear instability result under the same criterion.
1 Introduction
In this paper, we are interested in the nonlinear instability of inhomogeneous steady
states of the Hamiltonian Mean Field (HMF) system. The HMF system is a kinetic
model describing particles moving on a unit circle interacting via an infinite range
attractive cosine potential. This 1D model holds many qualitative properties of more
realistic long-range interacting systems as the Vlasov-Poisson model. The HMF model
has been the subject of many works in the physical community, for the study of non
equilibrium phase transitions [11, 26, 2, 24], of travelling clusters [6, 27] or of relaxation
processes [28, 3, 12]. The long-time validity of the N-particle approximation for the
HMF model has been investigated in [8, 9] and the Landau-damping phenomenon near
a spatially homogeneous state has been studied recently in [13]. The formal linear
stability of inhomogeneous steady states has been studied in [10, 23, 7]. In particular,
a simple criterion of linear stability has been derived in [23]. In [22], the authors of the
present paper have proved that, under the same criterion κ
0ă 1 (see below for a precise
formulation), the inhomogeneous steady states of HMF that are nonincreasing functions
of the microscopic energy are nonlinearly stable. The aim of the present paper is to
show, in a certain sense, that this criterion is sharp: we show that if κ
0ą 1, the HMF
model can develop instabilities, from both the linear and the nonlinear points of view.
In [15], Guo and Lin have derived a sufficient criterion for linear instability to 3D Vlasov-Poisson by extending an approach developped in [19] for BGK waves. Let us also mention that both works have adapted some techniques presented in [16] to prove the nonlinear instability of the 3D Vlasov-Poisson system. In the first part of this article we adapt these techniques and prove the linear instability of nonhomogeneous steady states to the HMF system. In [17], a nonlinear instability result for 1D Vlasov-Poisson equation was obtained for an initial data close to stationary homogeneous profiles that satisfy a Penrose instability criterion by using an approach developed in [14]. In [18], starting with the N- particles version of the HMF model, a nonlinear instability result is obtained for the corresponding Vlasov approximation by also considering a Penrose instability condition for stationary homogeneous profiles. In the second part of this article our aim is to prove the nonlinear instability of non-homogeneous steady states of HMF by adapting the techniques developed in [14, 17, 18].
In the HMF model, the distribution function of particles f p t, θ, v q solves the initial- valued problem
B
tf ` v B
θf ´ B
θφ
fB
vf “ 0, p t, θ, v q P R
`ˆ T ˆ R , (1.1) f p 0, θ, v q “ f
initp θ, v q ě 0,
where T is the flat torus r 0, 2π s and where the self-consistent potential φ
fassociated with a distribution function f is defined by
φ
fp θ q “ ´ ż
2π0
ρ
fp θ
1q cos p θ ´ θ
1q dθ
1, ρ
fp θ q “ ż
R
f p θ, v q dv. (1.2) Introducing the so-called magnetization vector defined by
M
f“ ż
2π0
ρ
fp θ q u p θ q dθ, with u p θ q “ p cos θ, sin θ q
T(1.3) we have
φ
fp θ q “ ´ M
f¨ u p θ q . (1.4) In this work will consider steady states of (1.1) of the form
f
0p θ, v q “ F p e
0p θ, v qq , (1.5) where F is a given nonnegative function and where the microscopic energy e
0p θ, v q is given by
e
0p θ, v q “ v
22 ` φ
0p θ q with φ
0“ φ
f0. (1.6) Without loss of generality, we assume that φ
0p θ q “ ´ m
0cos θ with m
0ą 0. Here m
0is the magnetization of the stationary state f
0defined by m
0“ ş
ρ
f0cos θdθ.
It is shown in [22] that (essentially) if F is decreasing then f
0is nonlinearly stable
by the HMF flow (1.1) provided that the criterion κ
0ă 1 is satisfied, where κ
0is given
by
κ
0“ ´ ż
2π0
ż
`8´8
F
1p e
0p θ, v qq
¨
˚ ˚
˚ ˝ ż
De0pθ,vq
p cos θ ´ cos θ
1qp e
0p θ, v q ` m
0cos θ
1q
´1{2dθ
1ż
De0pθ,vq
p e
0p θ, v q ` m
0cos θ
1q
´1{2dθ
1˛
‹ ‹
‹ ‚
2
dθdv, (1.7) with
D
e“ θ
1P T : m
0cos θ
1ą ´ e ( .
In this paper, we explore situations where this criterion is not satisfied, i.e when κ
0ą 1. Let us now state our two main results. The first one concerns the linearized HMF equation given by
B
tf “ Lf, (1.8)
where
Lf : “ ´ v B
θf ` B
θφ
0B
vf ` B
θφ
fB
vf
0. (1.9) Theorem 1.1 (Linear instability) . Let f
0P L
1p T , R q be a stationary solution of (1.1) of the form (1.5), where F is a nonnegative C
1function on R such that F
1p e
0p θ, v qq belongs to L
1p T , R q . Assume that κ
0ą 1, where κ
0is given by (1.7). Then there exists λ ą 0 and a non-zero f P L
1p T ˆ R q such that e
λtf is a nontrivial growing mode weak solution to the linearized HMF equation (1.8).
Our second result is the following nonlinear instability theorem.
Theorem 1.2 (Nonlinear instability) . Let f
0be a stationary solution of (1.1) of the form (1.5), where F is a C
8function on R , such that F p e q ą 0 for e ă e
˚, F p e q “ 0 for e ě e
˚, with e
˚ă m
0and | F
1p e q| ď C | e
˚´ e |
´αF p e q in the neighborhood of e
˚, for some α ě 1. Assume that κ
0ą 1, where κ
0is given by (1.7). Then f
0is nonlinearly unstable in L
1p T ˆ R q , namely, there exists δ
0ą 0 such that for any δ ą 0 there exists a nonnegative solution f p t q of (1.1) satisfying } f p 0 q ´ f
0}
L1ď δ and
} f p t
δq ´ f
0}
L1ě δ
0, with t
δ“ O p| log δ |q as δ Ñ 0.
Remark 1.3. Note that in these two theorems we do not assume that the profile F is a decreasing function. Besides, the set of steady states satisfying the assumptions of these theorems is not empty, as proved in the Appendix (see Lemma A.1). Note also that the instability of Theorem 1.2 is not due to the usual orbital instability. Indeed the functional space of the pertubation can be restricted to the space of even functions in p θ, v q .
The outline of the paper is as follows: Sections 2 and 3 are respectively devoted to
the proofs of Theorem 1.1 and Theorem 1.2.
2 A linear instability result: proof of Theorem 1.1
The aim of this section is to prove Theorem 1.1. This proof will be done following the framework used by Lin for the study of periodic BGK waves in [19], which was generalized to the analysis of instabilities for the 3D Vlasov-Poisson system by Guo and Lin in [15]. We divide this proof into the three Lemmas 2.1, 2.2 and 2.3, respectively proved in Subsections 2.1, Subsection 2.2 and Subsection 2.3.
A growing mode of (1.8) is a solution of the form e
λtf , where f P L
1p T , R q is an unstable eigenfunction of L, i.e. a nonzero function satisfying Lf “ λf in the sense of distributions, with λ P R
˚`
and with L defined by (1.9). Note that the equation Lf ´ λf “ 0 is invariant by translation: if f p θ, v q is an eigenfunction, then for all θ
0, f p θ ` θ
0, v q is also an eigenfunction. Since for all f P L
1we can find a θ
0P T such that ş
ρ
fp θ ` θ
0q sin θdθ “ 0, we can assume that our eigenfunction of L always satisfy ş ρ
fsin θdθ “ 0, i.e. φ
f“ ´ m cos θ with m “ ş
ρ
fcos θdθ.
Let us first define p Θ p s, θ, v q , V p s, θ, v qq as the solution of the characteristics problem
$ ’
&
’ %
dΘ p s, θ, v q
ds “ V p s, θ, v q dV p s, θ, v q
ds “ ´B
θφ
0p Θ p s, θ, v qq
(2.1)
with initial data Θ p 0, θ, v q “ θ, V p 0, θ, v q “ v. When there is no ambiguity, we denote simply Θ p s q “ Θ p s, θ, v q and V p s q “ V p s, θ, v q . Since φ
0p θ q “ ´ m
0cos θ, the solution p Θ, V q is globally defined and belongs to C
8p R ˆ T ˆ R q . Note that the energy e
0p Θ p s q , V p s qq “
Vp2sq2` φ
0p Θ p s qq does not depend on s.
We shall reduce the existence of a growing mode of (1.8) to the existence of a zero of the following function, defined for all λ P R
˚`
: G p λ q “ 1 `
ż
2π0
ż
R
F
1ˆ v
22 ´ m
0cos θ
˙
cos
2θdθdv
´ ż
2π0
ż
R
F
1ˆ v
22 ´ m
0cos θ
˙ ˆż
0´8
λe
λscos Θ p s, θ, v q ds
˙
cos θdθdv. (2.2) Lemma 2.1. Let f
0P L
1p T , R q be a stationary solution of (1.1) of the form (1.5), where F is a C
1function on R such that F
1p e
0p θ, v qq belongs to L
1p T , R q . Then the function G defined by (2.2) is well-defined and continuous on R
˚`
. Moreover, there exists a growing mode e
λtf solution to (1.8) associated with the eigenvalue λ ą 0 if and only if G p λ q “ 0.
An unstable eigenfunction f of L is defined by f p θ, v q “ ´ F
1p e
0p θ, v qq cos θ ` F
1p e
0p θ, v qq
ż
0´8
λe
λscos p Θ p s, θ, v qq ds. (2.3) Lemma 2.2. Under the assumptions of Lemma 2.1, the function G defined by (2.2) satisfies
lim
λÑ0`
G p λ q “ 1 ´ κ
0, (2.4)
where κ
0is defined by (1.7).
Lemma 2.3. Under the assumptions of Lemma 2.1, the function G defined by (2.2) satisfies
λ
lim
Ñ`8G p λ q “ 1. (2.5)
Proof of Theorem 1.1. From these three lemmas, it is clear that if κ
0ą 1, we have
λ
lim
Ñ0`G p λ q ă 0 and lim
λÑ`8
G p λ q ą 0 so by continuity of G, there exists λ ą 0 such that G p λ q “ 0. This means that there exists a growing mode to (1.8) and this proves Theorem 1.1.
2.1 First properties of the function G p λ q: proof of Lemma 2.1 In this subsection, we prove Lemma 2.1. Let λ P R
˚`
. Since, by assumption, the function F
1p e
0p θ, v qq belongs to L
1p T ˆ R q , and since
@p θ, v q ˇ ˇ ˇ ˇ
ż
0´8
λe
λscos p Θ p s qq ds ˇ ˇ ˇ ˇ ď
ż
0´8
λe
λsds “ 1, both functions
F
1p e
0p θ, v qq cos θ and F
1p e
0p θ, v qq ż
0´8
λe
λscos p Θ p s, θ, v qq ds
belong to L
1p T ˆ R q , so the function f defined by (2.3) also belongs to L
1p T ˆ R q . Hence, by integrating with respect to cos θdθdv, we deduce that G p λ q is well-defined by (2.2).
The continuity of G on R
˚`
stems from dominated convergence.
Consider now a (nonzero) growing mode e
λtf of (1.8) associated to an eigenvalue λ ą 0. Let us prove that G p λ q “ 0. From Lf “ λf and (2.1), we get, in the sense of distributions,
d ds
´
e
λsf p Θ p s q , V p s qq ¯
“ e
λsφ
1fp Θ p s qq V p s q F
1p e
0p Θ, V qq . Integrating this equation from ´ R to 0, we get, for almost all p θ, v q and all R,
f p θ, v q “ e
´λRf p Θ p´ R q , V p´ R qq ` F
1p e
0q ż
0´R
e
λsφ
1fp Θ p s qq V p s q ds,
where we recall that e
0p Θ, V q “ e
0p θ, v q . We multiply by a test function ψ p θ, v q P C
08p T ˆ R q and integrate with respect to p θ, v q ,
ż
2π0
ż
R
f p θ, v q ψ p θ, v q dθdv “ e
´λRż
2π0
ż
R
f p θ, v q ψ p Θ p R q , V p R qq dθdv
` ż
0´R
ż
2π0
ż
R
e
λsF
1p e
0q φ
1fp Θ p s qq V p s q ψ p θ, v q dsdθdv.
In the first integral of the right-hand side, we have performed the change of variable p θ, v q “ p Θ p R, θ
1, v
1q , V p R, θ
1, v
1qq . In the second integral, we remark that | φ
1fp Θ p s qq | ď } f }
L1and, the support of ψ being compact, v is bounded. Hence, by
v22´ m
0cos θ “
V2
2
´ m
0cos Θ, V p s q is bounded. Therefore, by dominated convergence (using that F
1p e
0q P L
1), as R Ñ 8 , we get
ż
2π0
ż
R
f p θ, v q ψ p θ, v q dθdv “ ż
0´8
ż
2π0
ż
R
e
λsF
1p e
0q φ
1fp Θ p s qq V p s q ψ p θ, v q dsdθdv i.e.
f p θ, v q “ F
1p e
0q ż
0´8
e
λsφ
1fp Θ p s qq V p s q ds “ F
1p e
0q ż
0´8
e
λsd
ds p φ
fp Θ p s qqq ds
“ F
1p e
0q φ
fp θ q ´ F
1p e
0q ż
0´8
λe
λsφ
fp Θ p s qq ds almost everywhere. Recall that φ
fp θ q “ ´ m cos θ, with m “ ş
2π0
ρ
fp θ q cos θdθ. Then we can rewrite this expression of f as
f p θ, v q “ ´ mF
1p e
0q cos θ ´ mF
1p e
0q ż
0´8
λe
λscos Θ p s q ds.
Integrating both sides of this equation with respect to cos θdθdv we get m “
ż
2π0
ρ
fp θ q cos θdθ “ ´ m ż
2π0
ż
R
F
1p e
0q cos
2θdθdv
` m ż
2π0
ż
R
F
1p e
0q ˆż
0´8
λe
λscos Θ p s q ds
˙
cos θdθdv, i.e. mG p λ q “ 0. It is clear that m ‰ 0, otherwise f “ 0 a.e.. Finally, we get G p λ q “ 0.
Reciprocally, assume that G p λ q “ 0 for some λ ą 0. Let f be given by (2.3). We have proved above that this function belongs to L
1p T ˆ R q . Moreover, since Θ p s, ´ θ, ´ v q “ Θ p s, θ, v q , we have ş
ρ
fsin θdθ “ 0. Multiplying (2.3) by cos θ and integrating with respect to θ and v, and using that G p λ q “ 0, we get ş
ρ
fcos θdθ “ 1, so f is not the zero function and we have φ
fp θ q “ ´ cos θ.
We now check that the function f given by (2.3) is an eigenfunction of L associated with λ. From (2.3), we get
f p Θ p t q , V p t qq “ ´ F
1p e
0q φ
fp Θ p t qq ` F
1p e
0q ż
0´8
λe
λsφ
fp Θ p s, Θ p t q , V p t qqq ds.
Note that Θ p s, Θ p t q , V p t qq “ Θ p s ` t, θ, v q . Therefore f p Θ p t q , V p t qq “ ´ F
1p e
0q φ
fp Θ p t qq ` F
1p e
0q
ż
0´8
λe
λsφ
fp Θ p s ` t qq ds
“ ´ F
1p e
0q φ
fp Θ p t qq ` F
1p e
0q e
´λtż
t´8
λe
λsφ
fp Θ p s qq ds
“ F
1p e
0q e
´λtż
t´8
e
λsφ
1fp Θ p s qq V p s q ds, where we integrated by parts. Then
e
λtf p Θ p t q , V p t qq “ F
1p e
0q ż
t´8
e
λsφ
1fp Θ p s qq V p s q ds
“ ż
t´8
e
λsφ
1fp Θ p s qq B
vf
0p Θ p s q , V p s qq ds.
Differentiating both sides with respect to t, we obtain in the sense of distributions that for all t P R ,
e
λt`
λf p Θ p t q , V p t qq ` V p t qB
θf p Θ p t q , V p t qq ´ φ
10p Θ p t qqB
vf p Θ p t q , V p t qq ˘
“ e
λtφ
1fp Θ p t qq B
vf
0p Θ p t q , V p t qq . By writing this equation at t “ 0, we get Lf “ λf: f is an unstable eigenfunction of L.
This ends the proof of Lemma 2.1.
2.2 Limiting behavior of G p λ q near λ “ 0: proof of Lemma 2.2 To study lim
λÑ0`
G p λ q we need to analyze the limit of the function g
λp θ, v q “
ż
0´8
λe
λscos Θ p s, θ, v q ds (2.6) as λ Ñ 0. We provide this result in the next lemma, where we also recall some well- known facts on the solution of the characteristics equations (2.1), which are nothing but the pendulum equations.
Lemma 2.4. Let p θ, v q P T ˆ R and e
0“
v22´ m
0cos θ. Consider the solution p Θ p s, θ, v q , V p s, θ, v qq to the characteristics equations (2.1). Then the following holds true.
(i) If e
0ą m
0then, for all s P R , we have
Θ p s ` T
e0q “ Θ p s q ` 2π, V p s ` T
e0q “ V p s q , (2.7) with
T
e0“ ż
2π0
dθ
1a 2 p e
0` m
0cos θ
1q ą 0. (2.8)
(ii) If ´ m
0ă e
0ă m
0then Θ and V are periodic with period given by T
e0“ 4
ż
θm00
a dθ
2 p e
0` m
0cos θ q “ 4
? m
0ż
π{20
b dθ
1 ´
m2m0`0e0sin
2θ ą 0, (2.9) where θ
m0“ arccos p´
me00q .
(iii) We have
λ
lim
Ñ0`g
λp θ, v q “
$ ’
’ ’
’ &
’ ’
’ ’
% 1 T
e0ż
2π0
cos θ
1a 2 p e
0` m
0cos θ
1q dθ
1if e
0ą m
0, 4
T
e0ż
θm00
cos θ
1a 2 p e
0` m
0cos θ
1q dθ
1if ´ m
0ă e
0ă m
0. (2.10) Proof. (i) Let e
0ą m
0. Without loss of generality, since Θ p s, ´ θ, ´ v q “ Θ p s, θ, v q and V p s, ´ θ, ´ v q “ V p s, θ, v q , we can only treat the case v ą 0. As we have
V p s q
22 “ e
0` m
0cos Θ p s q ě e
0´ m
0ą 0,
V p s q does not vanish and remains positive. Hence Θ p s q is the solution of the following autonomous equation
Θ 9 p s q “ V p s q “ a
2 p e
0` m
0cos Θ p s qq (2.11) and is strictly increasing with Θ p s q Ñ `8 as s Ñ `8 . Let T
e0be the unique time such that Θ p T
e0q “ θ ` 2π. By Cauchy-Lipschitz’s theorem, we have Θ p s ` T
e0q ´ 2π “ Θ p s q and (2.7) holds. Defining
P p τ q “ ż
τ0
dθ
1a 2 p e
0` m
0cos θ
1q ,
the solution of (2.11) satisfies P p Θ p s qq ´ P p θ q “ s. Therefore, we have T
e0“ P p θ ` 2π q ´ P p θ q , from which we get (2.8).
(ii) Let ´ m
0ă e
0ă m
0. In this case, Θ p s q will oscillate between the two values θ
m0“ arccos p´
me00q and ´ θ
m0with a period T
e0given by (2.9). On the half-periods where Θ is increasing, we also have (2.11). We skip the details of the proof, which is classical.
(iii) We remark that cos Θ p s ` kT
e0q “ cos Θ p s q for all s P R and k P Z . Indeed, by (i),
for e
0ą m
0we have Θ p s ` kT
e0q “ Θ p s q ` 2πk and, by (ii), for ´ m
0ă e
0ă m
0we have
Θ p s ` kT
e0q “ Θ p s q . Hence, we compute from (2.6) g
λp θ, v q “
`8
ÿ
k“0
ż
pk`1qTe0kTe0
λe
´λscos Θ p´ s q ds
“
`8
ÿ
k“0
ż
Te00
λe
´λs´kλTe0cos Θ p´ s q ds
“
˜
`8ÿ
k“0
e
´kλTe0¸ ż
Te00
λe
´λscos Θ p´ s q ds
“ λ
1 ´ e
´λTe0ż
Te00
e
´λscos Θ p´ s q ds.
Therefore, clearly, lim
λÑ0`
g
λp θ, v q “ 1 T
e0ż
Te00
cos Θ p´ s q ds “ 1 T
e0ż
Te00
cos Θ p s q ds.
If e
0ą m
0, we perform the change of variable θ
1“ Θ p s q which is strictly increasing from r 0, T
e0s to r θ, θ ` 2π s . Using (2.11), we obtain
lim
λÑ0`
g
λp θ, v q “ 1 T
e0ż
θ`2πθ
cos θ
1a 2 p e
0` m
0cos θ
1q dθ
1“ 1 T
e0ż
2π0
cos θ
1a 2 p e
0` m
0cos θ
1q dθ
1. If ´ m
0ă e
0ă m
0, we can always choose a time t
0such that Θ p t
0q “ ´ θ
m0, Θ p t
0` T
e0{ 2 q “ θ
m0, Θ p s q is strictly increasing on r t
0, t
0` T
e0{ 2 s and such that Θ p s q “ Θ p 2t
0` T
e0´ s q for s P r t
0` T
e0{ 2, t
0` T
e0s . We have
λ
lim
Ñ0`g
λp θ, v q “ 1 T
e0ż
t0`Te0{2t0
cos Θ p s q ds ` 1 T
e0ż
t0`Te0t0`Te0{2
cos Θ p s q ds
“ 1 T
e0ż
t0`Te0{2t0
cos Θ p s q ds ` 1 T
e0ż
t0`Te0t0`Te0{2
cos Θ p 2t
0` T
e0´ s q ds
“ 2 T
e0ż
t0`Te0{2t0
cos Θ p s q ds
“ 2 T
e0ż
θm0´θm0
cos θ
1a 2 p e
0` m
0cos θ
1q dθ
1“ 4 T
e0ż
θm00
cos θ
1a 2 p e
0` m
0cos θ
1q dθ
1, where, on the time interval r t
0, t
0` T
e0{ 2 s , we performed the change of variable θ
1“ Θ p s q .
Proof of Lemma 2.2. Now we come back to the definition (2.2) of G
λ, which reads G p λ q “ 1 `
ż
2π0
ż
R
F
1ˆ v
22 ´ m
0cos θ
˙
cos
2θdθdv
´ ż
2π0
ż
R
F
1ˆ v
22 ´ m
0cos θ
˙
g
λp θ, v q cos θdθdv. (2.12)
We remark that | g
λp θ, v q| ď 1 and recall that the function F
1´
v2
2
´ m
0cos θ ¯
belongs to L
1p T ˆ R q . Therefore, we can pass to the limit in the second integral by dominated convergence and deduce from Lemma 2.4 (iii) (note that the set tp θ, v q : e
0p θ, v q ď ´ m
0u is of measure zero) that
λ
lim
Ñ0`G p λ q “ 1 ` ż
2π0
ż
R
F
1p e
0q cos
2θdθdv
´ ij
e0pθ,vqąm0
F
1p e
0q
˜ 1 T
eż
2π0
cos θ
1a 2 p e
0` m
0cos θ
1q dθ
1¸
cos θdθdv
´
ij
´m0ăe0pθ,vqăm0
F
1p e
0q
˜ 4 T
eż
θm00
cos θ
1a 2 p e
0` m
0cos θ
1q dθ
1¸
cos θdθdv
“ 1 ` ż
2π0
ż
R
F
1p e
0q cos
2θdθdv
´ ż
2π0
ż
R
F
1p e
0q
¨
˚ ˚
˚ ˝ ż
De0
cos θ
1p e
0` m
0cos θ
1q
´1{2dθ
1ż
De0
p e
0` m
0cos θ
1q
´1{2dθ
1˛
‹ ‹
‹ ‚ cos θdθdv
“ 1 ` ż
2π0
ż
R
F
1p e
0q cos p θ q
2dvdθ ´ ż
2π0
ż
R
F
1p e
0q p Π
m0p cos θ qq
2dθdv, with
p Π
m0h qp e q “ ż
De
p e ` m
0cos θ q
´1{2h p θ q dθ ż
De
p e ` m
0cos θ q
´1{2dθ
, (2.13)
for all function h p θ q and
D
e“ θ
1P T : m
0cos θ
1ą ´ e ( .
Here the operator Π
m0is a variant of the operator Π given by (3.8) in [21], this operator should be understood as the “projector”onto the functions which depend only on the microscopic energy e
0p θ, v q . A projector of this type is also mentioned in the work by Guo and Lin [15].
Now we remark that straightforward calculations give
1 ´ κ
0“ 1 ` ż
2π0
ż
R
F
1p e
0q ´
cos
2θ ´ 2 cos θΠ
m0p cos θ q ` p Π
m0p cos θ qq
2¯ dθdv,
“ 1 ` ż
2πż
F
1p e
0q cos
2θdθdv ´ ij
F
1p e
0q p Π
m0p cos θ qq
2dθdv,
where
´ κ
0: “ ż
2π0
ż
R
F
1p e
0q
¨
˚ ˚
˝ ż
De
p cos θ ´ cos θ
1qp e
0` m
0cos θ
1q
´1{2dθ
1ż
De
p e
0` m
0cos θ
1q
´1{2dθ
1˛
‹ ‹
‚
2
dθdv.
This calculation uses that Π
m0is a projector. We finally get (2.4) and the proof of Lemma 2.2 is complete.
2.3 Limiting behavior of G p λ q as λ Ñ 8: proof of Lemma 2.3
In this subsection, we prove Lemma 2.3. An integration by parts in (2.6) yields g
λp θ, v q “ cos θ `
ż
0´8
e
λsV p s, θ, v q sin Θ p s, θ, v q ds.
The velocity can be bounded independently of s thanks to the conservation of the energy,
| V p s q| “ `
v
2` 2m
0cos θ ´ 2m
0cos Θ p s q ˘
1{2ď `
v
2` 4m
0˘
1{2. Thus
ˇ ˇ ˇ ˇ
ż
0´8
e
λssin Θ p s, θ, v q V p s, θ, v q ds ˇ ˇ ˇ ˇ ď `
v
2` 4m
0˘
1{2ż
0´8
e
λsds “
` v
2` 4m
0˘
1{2λ and, for all p θ, v q ,
λÑ`8
lim g
λp θ, v q “ cos θ.
Using again that | g
λp θ, v q| ď 1 and that F
1p e
0p θ, v qq belongs to L
1, we deduce directly (2.5) from (2.12) and from dominated convergence. The proof of Lemma 2.3 is complete.
3 A nonlinear instability result: proof of Theorem 1.2
We start by an analysis of the linearized HMF operator L around the inhomogeneous equilibrium state f
0, where L is given by (1.9). We write
L “ L
0` K, (3.1)
where
L
0f “ ´ v B
θf ´ E
f0B
vf, Kf “ ´ E
fB
vf
0, E
f“ ´B
θφ
f. (3.2)
3.1 Estimates on the semigroup e
tLLet us state some useful properties of the operator L
0given by (3.2). Since φ
0p θ q “
´ m
0cos θ is smooth, the characteristics equations (2.1) admit a unique solution Θ p s, θ, v q , V p s, θ, v q , which is globally defined and C
8in the variables p s, θ, v q . Moreover, this solution has bounded derivatives with respect to θ and v, locally in time. Let k P N . For any f in the Sobolev space W
k,1p T ˆ R q , the function
e
tL0f p t, θ, v q : “ f p Θ p´ t, θ, v q , V p´ t, θ, v qq , @ t ě 0, (3.3) belongs to C
0p R , W
k,1p T ˆ R qq and is clearly a solution to B
tg “ L
0g with initial data f . This means that the semigroup e
tL0generated by the operator L
0is strongly continuous on W
k,1p T ˆ R q .
Our aim is to apply the abstract results in [25] concerning perturbation theory of linear operators. Hence, we need to prove the following estimate. For all β ą 0, there exists a positive constant M
β,ksuch that
} e
tL0f }
Wk,1ď M
β,ke
tβ} f }
Wk,1@ f P W
k,1p T ˆ R q , @ t ě 0, (3.4) where M
β,kdepends on β and k. In fact this estimate will be proved for a subclass of functions f . From the assumptions of Theorem 1.2 on F , there exists e
˚ă m
0such that support of F is p´8 , e
˚s . This means that the support of f
0is contained in Ω
0, where Ω
0is the smooth open set
Ω
0“
"
p θ, v q : v
22 ´ m
0cos θ ă e
˚* . We then introduce the functional space
E
k“ !
f P W
k,1p T ˆ R q : Supp p f q Ă Ω
0) , and claim that for all f P E
k, we have e
tL0f P E
kand, for all β ą 0,
} e
tL0f }
Wk,1ď M
β,ke
tβ} f }
Wk,1, @ f P E
k, @ t ě 0, (3.5) M
β,kbeing a positive constant depending on β and k.
Assume for the moment that estimate (3.5) holds true. From the assumptions of Theorem 1.2, one deduces that B
vf
0P E
k. It is then easy to check that e
tL0K is a compact operator on E
k, for all t P R , and the map t ÞÑ e
tL0K P L p E
kq is continuous on R . Hence, K is L
0-smoothing in the sense of [25] (page 707). Assumptions of Theorem 1.1 in [25] are therefore satisfied, which implies that L generates a strongly continuous semigroup e
tL. Now, from Theorem 1.2 in [25], for all β ą 0, any point of the spectrum σ p L q lying in the half plane Re z ą β is an isolated eigenvalue with finite algebraic multiplicity. Furthermore, the set σ p L q X t Re z ą β u is finite.
The assumptions of Theorem 1.2 clearly imply those of Theorem 1.1. Hence L admits at least one eigenvalue λ P R
˚`
associated with an eigenfunction f r P L
1p T ˆ R q . We claim
that, in fact, f r P E
kwhich will be proved below. This means that the set of eigenvalues of L on E
kwith positive real part is not empty, and we therefore can choose an eigenvalue γ with positive maximal real part. Finally, we apply Theorem 1.3 in [25] and get that, for all β ą Re γ, there exists a positive constant M
β,ksuch that
} e
tLf }
Wk,1ď M
β,ke
tβ} f }
Wk,1@ f P E
k, @ t ě 0. (3.6) Proof of (3.5) and of the claim f r P E
k. Let f P E
k. From (3.3), we clearly have
} e
tL0f }
L1“ } f }
L1, @ f P L
1, @ t ě 0.
Moreover, we know from the analysis of the characteristics problem (2.1) performed in Section 2, that by conservation of the energy, for all p θ, v q P Ω
0, we have p Θ p t, θ, v q , V p t, θ, v qq P Ω
0. Thus
Supp p e
tL0f q Ă Ω
0.
Let k ě 1. By (3.3), to get an estimate of e
tL0f in W
k,1p T ˆ R q , it is sufficient to estimate Θ and V in W
k,8p T ˆ R q for p θ, v q P Ω
0. Recall that, since e
˚ă m
0, Θ and V are periodic functions with period T
e0. Moreover, (2.9) shows that T
e0is a C
8function of e
0on r´ m
0, e
˚s , which means that it is also a C
8function of p θ, v q . Note also that
?2πm0
ď T
e0ď T
e˚. Define now the following 1-periodic functions
Θ r p s, θ, v q “ Θ p sT
e0, θ, v q , V r p s, θ, v q “ V p sT
e0, θ, v q satisfying
dr Θ
ds “ T
e0V , r d V r
ds “ ´ m
0T
e0sin Θ. r Applying Gronwall Lemma, one gets
|B
rsB
θjB
vℓΘ r | ` |B
srB
jθB
vℓV r | ď C
ke
Cks, @ s ě 0, @p θ, v q P Ω
0, for r ` j ` ℓ ď k.
The period of Θ and r V r being independent of p θ, v q , the functions B
θjB
vℓΘ and r B
jθB
vℓV r are also 1-periodic and therefore
|B
rsB
θjB
ℓvΘ r | ` |B
rsB
θjB
vℓV r | ď C
k1“ C
ke
Ck, @ s ě 0, @p θ, v q P Ω
0, for r ` j ` ℓ ď k.
Coming back to Θ and V , we deduce
|B
θjB
vℓΘ | ` |B
θjB
ℓvV | ď C
kp 1 ` s
kq , @ s ě 0, @p θ, v q P Ω
0, for j ` ℓ ď k. (3.7) Using this estimate and (3.3), we finally get (3.5).
Let us finally prove that f r P E
k. By Lemma 2.1, the function f r is given by f r p θ, v q “ ´ F
1p e
0p θ, v qq cos θ ` F
1p e
0p θ, v qq
ż
0´8
λe
λscos p Θ p s, θ, v qq ds.
Hence, the support of F
1p e
0p θ, v qq being in Ω
0, the support on f r will also be contained in Ω
0. Moreover, by using (3.7), we obtain that, for some C
ką 0, we have
@ j ` ℓ ď k, @p θ, v q P T ˆ R , ˇ ˇ ˇ ˇ B
θjB
ℓvż
0´8
λe
λscos p Θ p s, θ, v qq ds ˇ ˇ ˇ ˇ ď C
k.
This is sufficient to deduce from F P C
8that f r P E
k. 3.2 An iterative scheme
In this part, we prove Theorem 1.2 by following the strategy developed by Grenier in [14], which has been also used in [17, 18] to analyse instabilities for homogeneous steady states of Vlasov-Poisson models. Let N ě 1 be an integer to be fixed later. According to the previous subsection, we can consider an eigenvalue γ of L on E
Nwith maximal real part, Re γ ą 0. Let g P E
Nbe an associated eigenfunction. With no loss of generality, we may assume that } Re g }
L1“ 1. Let
f
1p t, θ, v q “ Re `
e
γtg p θ, v q ˘
χ
δp e
0p θ, v qq , (3.8) with e
0p θ, v q “
v22` φ
0p θ q and where 0 ď χ
δp e q ď 1 is a smooth real-valued truncation function to be defined further, in order to ensure the positivity of f
0` δf
1p 0 q . Note that f
1is almost a growing mode solution to the linearized HMF model (1.8) since we have
pB
t´ L q f
1“ Re p e
γtR r
δq , where
R r
δ“ `
´ E
p1´χδqg` p 1 ´ χ
δq E
g˘
B
vf
0(3.9)
will be small. We now construct an approximate solution f
appNto the HMF model (1.1) of the form
f
appN“ f
0` ÿ
Nk“1
δ
kf
k, (3.10)
for sufficiently small δ ą 0, in which f
k(k ě 2) solves inductively the linear problem pB
t´ L q f
k`
k
ÿ
´1 j“1E
fjB
vf
k´j“ 0 (3.11) with f
kp 0 q “ 0. Then f
appNapproximately solves the HMF model (1.1) in the sense that
B
tf
appN` v B
θf
appN´ B
θφ
fNapp
B
vf
appN“ R
N` δ Re p e
γtR r q , (3.12) where the remainder term R
Nis given by
R
N“ ÿ
1ďj,ℓďN;j`ℓěN`1
δ
j`ℓE
fjB
vf
ℓ. (3.13)
Step 1. Estimate of f
k. We claim that f
kP E
N´k`1
and, for all 1 ď k ď N ,
} f
k}
WN´k`1,1ď C
ke
ktReγ. (3.14) We proceed by induction. From (3.8), this estimate is a consequence, for k “ 1, of
} gχ
δ}
WN,1ď C
1, (3.15)
which is proved below in Step 5. Let k ě 2. We have f
kp t q “ ´
ż
t0
e
Lpt´sqk
ÿ
´1 j“1E
fjp s qB
vf
k´jp s q ds.
Therefore, for Re γ ă β ă 2 Re γ, } f
k}
WN´k`1,1ď
k
ÿ
´1 j“1ż
t0
› ›
› e
Lpt´sq`
E
fjp s qB
vf
k´jp s q ˘›› ›
WN´k`1,1
ds ď M
β,N´k`1k
ÿ
´1 j“1ż
t0
e
βpt´sq› ›E
fjp s q › ›
WN´k`1,8
}B
vf
k´jp s q}
WN´k`1,1ds ď M
β,N´k`1k
ÿ
´1j“1
ż
t0
e
βpt´sq} f
jp s q}
L1} f
k´jp s q}
WN´k`2,1ds since k ´ j ď k ´ 1,
ď M
β,N´k`1˜
k´1ÿ
j“1
C
jC
k´j¸ ż
t 0e
βpt´sqe
ksReγds
ď M
β,N´k`1k Re γ ´ β
˜
k´1ÿ
j“1
C
jC
k´j¸
e
ktReγ,
where we used (3.6) and the recursive assumption. This ends the proof of (3.14).
Step 2. Estimates of f
appN´ f
0and R
N. The parameter δ and the time t will be such that
δe
tReγď min ˆ 1
2 , 1 2K
N˙
, K
N“ max
1ďkďN
C
k. (3.16) Hence, from (3.14) we obtain
} f
appN´ f
0}
W1,1ď ÿ
Nk“1
δ
kC
ke
ktReγď K
Nδe
tReγ1 ´ δe
tReγď 1 and
} R
N}
L1ď
`8
ÿ
k“N`1
δ
ke
ktReγÿ
1ďj,ℓďN;j`ℓ“k
C
jC
ℓď C r
N`
δe
tReγ˘
N`1.
Step 3. Estimate of f ´ f
appN. Let f p t q be the solution of (1.1) with initial data f
0` δ Re gχ
δand let h “ f ´ f
appN. Note that the positivity of f p t q is ensured by f
0` δ Re gχ
δě 0 and that we have
} f p 0 q ´ f
0}
L1ď δ.
The function h satisfies the following equation B
th ` v B
θh ` E
fB
vh “ ´
E
fNapp
´ E
f¯
B
vf
appN´ R
N´ δ Re p e
γtR r
δq
with h p 0 q “ 0. To get a L
1-estimate of h, we multiply this equation by sign p h q and integrate in p θ, v q . We get
d
dt } h }
L1ď › › › E
fNapp
´ E
f› › ›
L8› › B
vf
appN› ›
L1` } R
N}
L1` δe
tReγ} R r
δ}
L1ď } h }
L1› › B
vf
appN› ›
L1
` } R
N}
L1` δe
tReγ} R r
δ}
L1. From Step 2 we have › › B
vf
appN› ›
L1
ď }B
vf
0}
L1` 1, which implies that } h p t q}
L1ď
ż
t0
e
pt´sqp}Bvf0}L1`1q´
} R
Np s q}
L1` δe
sReγ} R r
δ}
L1¯ ds.
Again from Step 2, we then get } h p t q}
L1ď
ż
t0
e
pt´sqp}Bvf0}L1`1q´ C r
N`
δe
sReγ˘
N`1` δe
sReγ} R r
δ}
L1¯ ds We now fix N as follows (with the notation t ¨ u for the integer function)
N : “
Z }B
vf
0}
L1` 1 Re γ
^
` 1 ě 1 and claim that χ may be chosen such that
} R r
δ}
L1ď `
δe
sReγ˘
N, (3.17)
see Step 5 for the proof. This yields
} f ´ f
appN}
L1p t q ď C q
N`
δe
tReγ˘
N`1(3.18) with C q
N“
13 Re`CrNγ.
Step 4. End of the proof. Since Re g is not zero, we can choose a real valued function ϕ p θ, v q in L
8such that } ϕ }
L8=1 and
Re z
gą 0 with z
g“ ż
2π0
ż
R
gϕdθdv.
Denoting
z
g,δ“ ż
2π0
ż
R
gχ
δϕdθdv, we have
ij
f
1ϕdθdv “ e
tReγRe `
e
itImγz
g,δ˘ ě e
tReγRe `
e
itImγz
g˘
´ e
tReγ| z
g´ z
g,δ| ě e
tReγRe `
e
itImγz
g˘
´ e
tReγ} g p 1 ´ χ
δq}
L1We claim that
δ
lim
Ñ0}p 1 ´ χ
δq g }
L1“ 0, (3.19) which again will be proved in Step 5. In order to end the proof of Theorem 1.2, we estimate from below, using (3.18) and (3.14),
} f ´ f
0}
L1ě ij
p f ´ f
0q ϕdθdv “ ij
p f
appN´ f
0q ϕdθdv ` ij
p f ´ f
appNq ϕdθdv ě δ
ij
f
1ϕdθdv ´ ÿ
Nk“2
δ
k} f
k}
L1´ C q
N`
δe
tReγ˘
N`1ě δ ij
f
1ϕdθdv ´ ÿ
Nk“2
C
k`
δe
tReγ˘
k´ C q
N`
δe
tReγ˘
N`1ě δ ij
f
1ϕdθdv ´ 2K
N`
δe
tReγ˘
2´ C q
N`
δe
tReγ˘
N`1ě δe
tReγ´ Re `
e
itImγz
g˘
´ }p 1 ´ χ
δq g }
L1´ 2K
Nδe
tReγ´ C q
N`
δe
tReγ˘
N¯ Assume for a while that
Re `
e
itImγz
g˘
ě Re z
g2 . (3.20)
We have
} f ´ f
0}
L1ě δe
tReγRe z
g2
˜
1 ´ 2 }p 1 ´ χ
δq g }
L1Re z
g´ 4K
Nδe
tReγRe z
g´ 2 C q
N`
δe
tReγ˘
NRe z
g¸
Let δ
0ą 0 be such that 32K
Np Re z
gq
2δ
0` 2 C q
N8
Np Re z
gq
N`1δ
N0ď 1
4 and 8δ
0Re z
gď min ˆ 1
2 , 1 2K
N˙
(note that N ě 1) and consider times t such that 4δ
0Re z
gď δe
tReγď 8δ
0Re z
g. (3.21)
Owing to (3.19), we also choose δ small enough such that 2 }p 1 ´ χ
δq g }
L1Re z
gď 1 4 . We conclude from these inequalities that
} f ´ f
0}
L1ě δ
0and that (3.16) is satisfied.
To end the proof, it remains to fix the time t
δand to choose the truncation function χ
δ. Let us show that, for δ small enough, there exists a time t
δsatisfying both (3.20) and (3.21). If Im γ “ 0, then (3.20) is clearly satisfied since Re z
gą 0: a suitable t
δis then
t
δ“ 1 Re γ log
ˆ 6δ
0δ Re z
g˙ .
Assume now that Im γ ‰ 0. For δ small enough, the size of the interval of times t satisfying (3.21) becomes larger than
|Im2πγ|. This means that it is possible to find a time t
δin this interval satisfying (3.20).
Step 5. Choice of χ
δ. For all δ ą 0, we have to fix the function χ
δP C
8p R q such that (3.15), (3.17), (3.19) are satisfied and such that f
0` δf
1p 0 q ě 0. First of all, proceeding as in the proof of Lemma 2.1, we obtain that g takes the form
g p θ, v q “ ´ mF
1p e
0q cos θ ´ mF
1p e
0q ż
0´8
γe
γscos Θ p s q ds, (3.22) with m “ ş
2π0
ρ
fp θ q cos θdθ. Hence,
| g p θ, v q| ď | m | ˆ
1 ` | γ | Re γ
˙
| F
1p e
0q| . (3.23) The assumptions on F and F
1in Theorem 1.2 imply that
@ e ă e
˚, | F
1p e q| ď C p e
˚´ e q
´αF p e q (3.24) with α ě 1. Since F p e q ą 0 for e ă e
˚, the local assumption becomes global. Let χ be a C
8function such that 0 ď χ ď 1 and
$ &
%
χ p t q “ 0 for t ď 0, χ p t q ď 2t
αfor t ě 0, χ p t q “ 1 for t ě 1 and let
χ
δp e q “ χ
ˆ e
˚´ e δ
1{p2αq˙
. (3.25)
From
}p 1 ´ χ
δq g }
L1ď } g
1e˚´δ1{p2αqăe0pθ,vqăe˚}
L1and dominated convergence, we clearly have (3.19). By (3.23) and (3.24), we have, for all p θ, v q P T ˆ R ,
δ | Re g p θ, v q χ
δp e
0p θ, v qq| ď δC | e
˚´ e
0|
´αF p e
0q | e
˚´ e
0|
αδ
1{2“ Cδ
1{2f
0p θ, v q , so for δ small enough, we have f
0` δf
1p 0 q ě 0.
By differentiating (3.22) and using (3.7), we get
@ j ` ℓ ď N, |B
jθB
ℓvg p θ, v q| ď C max
kďN`1
F
pkqp e
0q ď C | e
˚´ e
0|
N´1,
where we used Taylor formulas and the fact that F P C
8with F p e q “ 0 for e ě e
˚. Besides, from (3.25), we obtain (if δ ď 1)
@ 1 ď j ` ℓ ď N, |B
jθB
ℓvχ
δp e
0p θ, v qq| ď Cδ
´N{p2αq1e˚´δ1{p2αqăe0pθ,vqăe˚. Therefore
} gχ
δ}
WN,1ď } g }
L1` Cδ
´N{p2αqż
2π0
ż
R
| e
˚´ e
0p θ, v q|
N´11e˚´δ1{p2αqăe0pθ,vqăe˚dθdv ď } g }
L1` Cδ
´N{p2αqż
e˚e˚´δ1{p2αq
p e
˚´ e q
N´1˜ 4
ż
θm00
a dθ
2 p e ` m
0cos θ q
¸ de
“ } g }
L1` Cδ
´N{p2αqż
e˚e˚´δ1{p2αq
p e
˚´ e q
N´1T
ede,
where θ
m0“ arccos p´
me0q and T
eis given by (2.9). Now we recall that for e ď e
˚we have T
eď T
e˚. This yields
} gχ
δ}
WN,1ď } g }
L1` CT
e˚δ
´N{p2αqż
e˚e˚´δ1{p2αq
p e
˚´ e q
N´1de “ } g }
L1` CT
e˚N . We have proved (3.15).
By (3.9), we have
} R r
δ}
L1ď C p}p 1 ´ χ
δq g }
L1` }p 1 ´ χ
δqB
vf
0}
L1q ď C ´
} g
1e˚´δ1{p2αqăe0pθ,vqăe˚}
L1` }B
vf
01e˚´δ1{p2αqăe0pθ,vqăe˚}
L1¯ , so by dominated convergence,
δ
lim
Ñ0} R r
δ}
L1“ 0.
We now choose δ small enough such that } R
δ}
L1ď
ˆ 4δ
0Re z
g˙
N.
From (3.21), we obtain (3.17), which ends the proof of Theorem 1.2.
A Appendix. Existence of unstable steady states
In this section, we prove that the set of steady states satisfying the assumptions of Theorems 1.1 and 1.2 is not empty. More precisely, we prove the following
Lemma A.1. Let m ą 0. There exist m ą 0, e
˚ă m and there exists a nonincreasing function F , C
8on R , such that F p e q ą 0 for e ă e
˚, F p e q “ 0 for e ě e
˚and
| F
1p e q| ď C | e
˚´ e |
´αF p e q in the neighborhood of e
˚, for some α ě 1, and such that the function f p θ, v q “ F p
v22´ m cos θ q is a steady state solution to the HMF model (1.1) and such that κ p m, F q ą 1, where κ p m, F q is given by
κ p m, F q “ ż
2π0
ż
`8´8
ˇ ˇF
1p e p θ, v qq ˇ ˇ
¨
˚ ˚
˚ ˝ ż
Depθ,vq
p cos θ ´ cos θ
1qp e p θ, v q ` m cos θ
1q
´1{2dθ
1ż
Depθ,vq
p e p θ, v q ` m cos θ
1q
´1{2dθ
1˛
‹ ‹
‹ ‚
2
dθdv,
with
e p θ, v q “ v
22 ´ m cos θ, D
e“ θ
1P T : m cos θ
1ą ´ e ( .
Proof. Let m ą 0 and F a nonincreasing C
8function on R supported in p´8 , m q , which is not identically zero on p´ m, m q . We first observe that f p θ, v q “ F p
v22´ m cos θ q is a steady state solution to the HMF model (1.1) if and only if m and F satisfy γ p m, F q “ m with
γ p m, F q : “ ż
2π0
ż
R
F ˆ v
22 ´ m cos θ
˙
cos θdθdv ą 0.
By using the linearity of γ in F we deduce that
γ mpm,Fq