Bose Condensates in Interaction with Excitations: A Two-Component Space-Dependent Model Close to Equilibrium

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Bose Condensates in Interaction with Excitations: A Two-Component Space-Dependent Model Close to

Equilibrium

Leif Arkeryd, Anne Nouri

To cite this version:

Leif Arkeryd, Anne Nouri. Bose Condensates in Interaction with Excitations: A Two-Component

Space-Dependent Model Close to Equilibrium. Journal of Statistical Physics, Springer Verlag, 2015,

160, pp.209-238. �10.1007/s10955-015-1229-6�. �hal-01261232�

Bose condensates in interaction with excitations - a two-component space-dependent model close to equilibrium.

Leif ARKERYD and Anne NOURI

Mathematical Sciences, 41296 G¨ oteborg, Sweden, [email protected]

Aix-Marseille University, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France, [email protected]

Abstract. The paper considers a model for Bose gases in the so-called ’high-temperature range’

below the temperature where Bose-Einstein condensation sets in. The model is of non-linear two- component type, consisting of a kinetic equation with periodic boundary conditions for the distri- bution function of a gas of excitations interacting with a Bose condensate, which is described by a Gross-Pitaevskii equation. Results on well-posedness and long time behaviour are proved in a Sobolev space setting close to equilibrium.

1 Preliminaries and main results.

1.1 Physics motivations.

The phenomenon of Bose-Einstein condensation occurs when a large number of particles of a Bose gas enter the same lowest accessible quantum state. Predicted by Bose and Einstein in 1924 [4] [6], it was first unambiguously produced in 1995 by E. Cornell and C. Wieman. This paper studies a Bose condensate below the transition temperature T

for condensation, and in interaction with a non- condensates component. The setting is a two-component space-dependent model well established in physics (see the monograph [10] and its references) of pair-collision interactions involving a gas of thermally excited (quasi-)particles and a condensate. The two-component model consists of a kinetic equation for the distribution function of the gas, and a Gross-Pitaevskii equation (cf [21]) for the condensate. A rather general form of the kinetic equation in the superfluid frame is (cf [22], [26])

∂

f + (∇

(E(p)) + v

) · ∇

f − ∇

(E(p) + v

· p) · ∇

f = C

(f) + C

(f, n

). (1.1) Here f is the quasi-particle phase space density, n

(resp. v

) is the mass density (resp. the velocity) of the condensate, and E(p) denotes the (Bogoliubov) quasi-particle energy. The Nordheim-Uehling- Uhlenbeck term C

for collisions between (quasi-)particles is given by

C

(f)(p) = g

~ Z

R³×R³×R³

Bδ(p + p

= p

+ p

)δ(E(p) + E(p

) = E(p

) + E(p

))

f

f

(1 + f )(1 + f

) − f f

(1 + f

)(1 + f

)

dp

dp

dp

, (1.2)

12010 Mathematics Subject Classification. 82C10, 82C22, 82C40.

2Key words; low temperature kinetics, Bose condensate, two component model, 0.7Tc.

where g =

, a is the scattering length of the interaction potential, ~ the Planck constant, m the atomic mass, B a collision kernel, and

f = f (p), f

= f (p

), f

= f (p

), f

= f (p

).

The collision term C

for collisions between (quasi-)particles and condensate is C

(f, n

)(p) = g

n

~ Z

R³×R³×R³

Aδ(p

− p

− p

)δ(E

− E

− E

)[δ(p − p

) (1.3)

−δ(p − p

) − δ(p − p

)]((1 + f

)f

f

− f

(1 + f

)(1 + f

))dp

dp

dp

, where A is a collision kernel and

f

= f(p

), E

= E(p

), 1 ≤ j ≤ 3.

The usual Gross-Pitaevskii equation for the wave function ψ (the order parameter) associated with a Bose condensate is

i ~ ∂

ψ = − ~

2m ∆

ψ + (g|ψ|

+ U

)ψ,

where U

is an external potential, i.e. a Schr¨ odinger equation complemented by a non-linear term accounting for two-body interactions.

In the present context, the Gross-Pitaevskii equation is further generalized by letting the condensate move in a self-consistent Hartree-Fock mean field 2 R

R³

f(p)dp produced by the thermally excited atoms, together with a dissipative coupling term associated with the collisions. The generalized Gross-Pitaevskii equation derived in e.g. [15], [16], [22] and [10], is of the type

i ~ ∂

ψ = −

∆

ψ +

g|ψ|

+ U

+ 2g R

R³

f dp + i

~

R

R³×R³×R³

Aδ(p

− p

− p

)δ(E

− E

− E

) ((1 + f

)f

f

− f

(1 + f

)(1 + f

))dp

dp

dp

ψ. (1.4)

The two component problem (1.1), (1.4) is extensively discussed in the physics literature (see [5], [11], [12], [13], [14], [15], [16], [17], [22], [24], [26]). It is proven in [1] that these equations as given in [22], conserve the total energy. That is not so in some of the other settings, in particular not for (1.6)-(1.8) below.

1.2 The model under study.

We restrict to the ’high temperature range’, and more particularly consider the temperature range close to 0.7T

. As discussed in [5], [15], [16], [26] and more in details in [13], then |p| >> √

2mgn

, the approximation E(p) =

+ gn

of the quasi-particle energy is commonly used, A = 1, the operator C

is negligible, and the mass of the condensate exceeds that of the excitations, i.e.

n

> R

P (p)dp. In equilibrium, the right hand side of (1.1) vanishes. Multiplying the collision term by log

and integrating in p, it follows that in equilibrium

f

1 + f

= f

1 + f

f

1 + f

, when p

= p

+ p

, |p

|

= |p

|

+ |p

|

+ 2mgn

. (1.5) Equation (1.5) implies that

is a Maxwellian, hence the phase space density f of the excitations is a Planckian, which is of the type

1

e

− 1 , α > 0, β ∈ R

, p ∈ R

.

In the equilibrium Planckian distribution function, fix the condensate as identically equal to a constant n

> 0. Set α = 1, take the x-component of β as zero, |β| = 2 √

2mgn

and write the Planckian as

e^|p−p0|2−1

with p

= −

. Changing variables p → p − p

gives P (p) := 1

e

− 1 , p ∈ R

,

as equilibrium Planckian distribution function.

The present paper studies the stability of the equilibrium (P, √

n

) of the system under small deviations, that respect the conservation laws. Although we are not deriving hydrodynamic limits, we take into account that the system is close to equilibrium and introduce a mean free path , so that C

becomes

C

. The factor g is proportional to the scattering length a, which is smaller than the mean free path . Take λ of magnitude bounded by (

)

(< 1). The functions (f (t, x, p), ψ(t, x)) are considered in the slab Ω = [0, 2π] in the x-direction with periodic boundary conditions, and taken as

(f, ψ) = (P(1 + λR), √

n

+ λΦ).

In this paper the external potential U

is assumed to be a constant that will be further discussed.

We could alternatively have left out the external potential in (1.4) but replaced ψ by e

ψ in the proofs. The atomic mass m (resp. the Planck constant ~ ) will be taken as

(resp. one) for simplicity. Contrary to the classical Boltzmann operator in velocity space, f ∈ L

(R

) does not imply C

(f ) ∈ L

( R

). This paper is restricted to distribution functions, cylindrically symmetric in p = (p

, p

) ∈ R × R

. That changes the linear moment conservation Dirac measure in the collision term to δ(p

− p

− p

). Since the collective excitations play no role within the present temperature range, the domain of integration is here taken as the set of p ∈ R

such that |p|

> 2Λ

with Λ > 2 √

gn

. Denote by ˜ χ the characteristic function of the set {(p, p

, p

, p

) ∈ R

× R

× R

× R

; |p|

, |p

|

, |p

|

, |p

|

> 2Λ

}.

The restriction |p|

> 2Λ

will be implicitly assumed below, and R

dp will stand for R

|p|²>2Λ²

dp. Set δ

= δ(p − p

) − δ(p − p

) − δ(p − p

) and δ

= δ(p

= p

+ p

, |p

|

= |p

|

+ |p

|

+ n

).

The system of equations to be satisfied by (f, ψ) is

∂

f + p

∂

f = g

√ λn

Z

R³×R³×R³

˜

χδ

δ

(f

f

− f

(1 + f

+ f

))dp

dp

dp

, (1.6)

f (0, x, p) = f

(x, p), (1.7)

and

∂

ψ−i∂

ψ =

√ λ 2

Z

R³×R³×R³

˜

χδ

(f

f

−f

(1+f

+f

))dp

dp

dp

−i(n

+ U

g +2

Z

f dp)

gψ, (1.8)

ψ(0, x) = ψ

(x). (1.9)

Here, the function n

is defined by n

= n

(t, x) := |ψ|

(t, x). The approximate energy |p|

+ gn

used in (1.5), at this range of temperature is replaced by |p|

+ gn

as an approximation of order λ.

The total initial mass is 2πM

:=

Z

Ω

|ψ

(x)|

dx + Z

Ω×R³

f

(x, p)dxdp,

which is formally conserved by the equations (1.6) and (1.8).

The initial data f

and ψ

are taken as f

:= P(1 + λR

), ψ

:= √

n

+ λΦ

, for some functions R

(x, p) and Φ

(x) with

Z

(|ψ

|

− n

+ λ Z

R³

P R

dp)dx = 0.

This is consistent with the asymptotic behavior proven in the paper, i.e. (f − P, |ψ|

− n

) tending to zero when time tends to infinity. It implies that (up to the multiplicative constant

) the initial (and conserved) total mass equals the mass of (P, n

), i.e.

M

= Z

P (p)dp + n

. (1.10)

The separate masses of condensate and excitation may, however, not be conserved. The constant U

will be taken as g(n

− 2M

). For a discussion of general modeling aspects, see also our paper [2] and its references.

1.3 The main mathematical result.

The main results of the paper concern the well-posedness and long time behaviour of the problem (1.6-9).

For an initial perturbation of an equilibrium (P, √

n

) of order (

)

and conserving the total mass, the axial momentum and the kinetic energy of the excitations, the problem is well posed and the asymptotic limit when t → +∞ of the quasi-particle phase space density and the condensate mass are P resp. 2πn

. The mass of the excitations together with the mass, the kinetic energy and the internal energy of the condensate converge exponentially to their equilibrium values when t → +∞.

Let k . k

denote the norm in L

([0, 2π]), and set k ψ k

:=k ψ k

+ k ∂

ψ k

, let k . k

denote the norm in L

1+P

([0, 2π] × R

), i.e.

k h k

:= ( Z

h

(x, p) P

1 + P dpdx)

, and let L

P(1+P)

denotes the L

-space of functions h with norm ( R

P(1+P)

dpdx)

.

The solutions of (1.6-7) will be strong solutions, i.e. such that the collision operator C

(f, n

) belongs to C

R

; L

ν⁻¹²q

P 1+P

( R

; H

(0, 2π))

, ν being the collision frequency defined in (2.5). The solutions of (1.8-9) are H

-solutions in the following sense. A function ψ ∈ C

( R

; H

(0, 2π)) is an H

-solution to (1.8-9), if for all φ ∈ C( R

; H

(0, 2π)) and all t > 0,

Z

ψ(t, x) ¯ φ(t, x)dx − Z

ψ

(x) ¯ φ(0, x)dx + i Z

0

Z

∂

ψ(s, x)∂

φ(s, x)dxds ¯

= Z

0

Z

√ λ 2

Z

R³×R³×R³

˜

χδ

(f

f

− f

(1 + f

+ f

))dp

dp

dp

− i(n

+ n

− 2M

+ 2 Z

f dp)

gψ φdxds. ¯

Theorem 1.1

There are λ

, c

and η

> 0, such that for λ < λ

and (R

, Φ

) ∈ L

1+P

( R

; H

(0, 2π)) × H

(0, 2π) with Z

R

(x, p)p

P dxdp = Z

R

(x, p)(|p|

+ gn

)P dxdp = 0, (1.11) Z

(|ψ

|

− n

+ λ Z

R³

P R

dp)dx = 0, (1.12)

and

k Φ

k

≤ η

, k R

k

+ k ∂

R

k

≤ η

, (1.13) there is a unique solution

(f, ψ) = (P (1 + λR), √

n

+ λΦ) ∈ C

( R

; L

( R

; H

(0, 2π))) × C

( R

; H

(0, 2π)) to (1.6-9) with f > 0. For all t ∈ R

, the solution satisfies,

f ∈ L

P(1+P)

([0, t] × R

; H

(0, 2π))),

k R(t, ·, ·) k

+ k ∂

R(t, ·, ·) k

≤ c

η

e

, (1.14) Z

(|∂

ψ|

+ g

2 (|ψ|

− n

)

)(t, x)dx ≤ 2λ, where ζ = c

√

λ.

Moreover, n

(t) = R

|ψ(t, x)|

dx converges exponentially of order ζ to n

, when t → +∞,

t→+∞

lim Z

(|∂

ψ|

+ g

2 (|ψ|

− n

)

)(t, x)dx (1.15)

exists, and the convergence to its limit is exponential of order ζ.

Whereas non-linear systems of the type (1.6-9) and its generalizations have been much studied in mathematical physics below T

, there are so far only few papers with their focus mainly on the non- linear mathematical questions. Starting from a similar Gross-Pitaevskii and kinetic frame, two-fluid models are derived in [1]. The space homogeneous initial value problem for this system is treated in [2] for a large data setting. A Milne problem related to the present set-up is studied in [3]. The paper [8] considers a related setting, and has its focus on linearized space homogeneous problems.

Validation aspects in the space-homogeneous case are discussed in [23]. There has also been a considerable interest recently (see ee.g. [7], [18] and references therein) in the bosonic Nordheim- Uehling-Uhlenbeck equation as a model above and around T

for blow-ups and for condensation in space-homogeneous boson gases.

A classical approach to study kinetic equations in a perturbative setting, is to use a spectral inequal- ity (resp. Fourier techniques and the k · k

norm) for controlling the non-hydrodynamic (resp.

hydrodynamic) part of a solution. An additional problem here is the coupling with the generalized Gross-Pitaevskii equation. The general approach, together with a Fourier based analysis of the gen- eralized Gross-Pitaevskii equation, provide local in time solutions to the present coupled system.

Since the condensate and the normal gas are coupled by the collision interaction, the exponential

decrease of the deviation of the kinetic distribution function from the equilibrium Planckian P , helps to control the long-term evolution of the condensate. This is an important ingredient in the passage from local to global solutions, which leads to exponential decreases of the deviation of the condensate mass from its equilibrium state n

, and of the energy (1.15) from its limit value.

Within this frame the kinetic equation (1.6) differs from earlier classical ones. The collision operator in space-homogeneous bosonic Nordheim-Uehling-Uhlenbeck papers has so far been taken isotropic, but is here, due to the space-dependent slab-context, cylindric. Mass density does not belong to the kernel of the present linearized collision operator. The scaling at infinity in its collision frequency is stronger than in the classical case.

The one-dimensional spatial frame induces simplifications of the functional analysis, mainly in the control of the condensate. The T

spatial frame, for d ≥ 2, is an open problem.

The conservation properties of the model (1.6-9), as well as some properties of the collision operator

C12

nc

and its linearized operator L around the Planckian P , are discussed in Section 2, including a spectral estimate for L. This is used in Section 3, which is devoted to a priori estimates for some linear equations related to (1.6) and (1.8). They are then employed in the proof of the main the- orem in Section 4. The proof starts with a contractive iteration scheme to obtain local solutions.

A key point in the global in time analysis is the exponential convergence to equilibrium for f when t → +∞. The analysis of ψ differs from the classical Gross-Pitaevskii case. It uses the exponential convergence to equilibrium of f to control the behaviour of the kinetic energy R

|∂

ψ|

dx and the internal energy

R

|ψ|

dx of ψ.

2 Some properties of the model and the collision operator.

The model induces total mass conservation as well as axial momentum and kinetic energy conser- vations for the excitations, as stated in the following lemma.

Lemma 2.1 It holds that d

dt Z

Ω×R³

f (t, x, p)dxdp + Z

Ω

|ψ(t, x)|

dx

= 0, (2.1)

d dt

Z

Ω×R³

p

f(t, x, p)dxdp = 0, (2.2)

d dt

Z

Ω×R³

(|p|

+ gn

)f (t, x, p)dxdp = 0. (2.3)

Proof of Lemma 2.1.

Integrate (1.6) with respect to space and momentum. Add it to (1.8) multiplied by ¯ ψ ( resp. the

conjugate of (1.8)) multiplied by ψ) integrated with respect to space. One obtains (2.1). Multi-

plying (1.6) by p

(resp. (|p|

+ gn

) and integrating it w.r.t. space and momentum leads to (2.2)

(resp. (2.3)).

Since the solutions will remain close to an equilibrium (P, √

n

), the linearized operator of C

around P is of interest. For γ := √

λ, consider the decomposition f = P (1 + γ R), ˜ ψ = √

n

+ γ Φ. ˜ It holds

|ψ|

= n

= n

+ γ √

n

( ˜ Φ + Φ) + ¯ ˜ γ

| Φ| ˜

, and the collision term can be written

Z

˜

χδ

δ

(f

f

− f

(1 + f

+ f

))dp

dp

dp

= γ

P L R ˜ + γQ( ˜ R, R) ˜ ,

where L R ˜ := 1

P Z

˜

χδ(p

= p

+ p

)δ(|p

|

= |p

|

+ |p

|

+ gn

)(δ(p − p

) − δ(p − p

) − δ(p − p

)) h − (1 + P

+ P

)P

R ˜

+ (P

− P

)P

R ˜

+ (P

− P

)P

R ˜

i

dp

dp

dp

, and

2Q( ˜ R, S) := ˜ Z

˜ χδ

δ

P

P

( ˜ R

S ˜

+ ˜ R

S ˜

) − P

R ˜

(P

S ˜

+P

S ˜

) −P

S ˜

(P

R ˜

+P

R ˜

)

dp

dp

dp

. (2.4) We recall some properties about L proved in [3].

Lemma 2.2 L is a self-adjoint operator in L

. Within the space of rotationally invariant dis- tribution functions, its kernel is the subspace spanned by (|p|

+ gn

)(1 + P ) and p

(1 + P ).

The operator L splits into K − ν, where ν(p) :=

Z

˜

χδ

(1 + P

+ P

)dp

dp

+ 2 Z

˜

χδ

(P

− P

)dp

dp

(2.5)

and

Kh(p) := 2 P (p)

Z

˜

χδ

(P

− P)P

h

dp

dp

+ Z

˜

χδ

(1 + P + P

)P

h

dp

dp

+

Z

˜

χδ

(P

− P )P

h

dp

dp

. (2.6)

Lemma 2.3 The collision frequency ν satisfies

ν

(1 + |p|)

≤ ν(p) ≤ ν

(1 + |p|)

, p = (p

, p

) ∈ R × R

, (2.7) for some positive constants ν

and ν

. The operator K is compact from L

ν_1+P^P

into L

ν⁻¹_1+P^P

.

Denote by (·, ·) the scalar product in L

, and by ˜ P the orthonormal projection on the kernel of

L. Set h

:= ˜ P h and h

:= (I − P ˜ )h.

Lemma 2.4 L satisfies the spectral inequality,

−(Lh, h) ≥ c

(νh

, h

), h ∈ L

, (2.8)

with c

> 0.

We will also need an estimate for the quadratic collision operator Q.

Lemma 2.5 For cylindrically symmetric functions (g, h) ∈ L

ν_1+P^P

× L

(resp. (g, h) ∈ L

× L

), it holds

( Z

ν

P

1 + P ( Q(g, h)

P )

dp)

≤ c Z

ν P

1 + P g

(p)dp Z P

1 + P h

(p)dp

2

,

(resp.

( Z

ν

P

1 + P ( Q(g, h)

P )

dp)

≤ c Z ν P

1 + P h

(p)dp Z P

1 + P g

(p)dp

).

Proof. Considering cylindrically symmetric functions, we will use g = g(p

, p

), h = h(p

, p

). The theorem is a consequence of the following estimates for each of the terms of

. They are of the type

Q

(g, h)(p) := 2 Z

k

(p, p

)g

dp

where k

(p, p

) := P

Z

δ(p

= p

+ p

, |p|

= |p

|

+ |p

|

+ gn

) P

h

P dp

, or

Q

(g, h)(p) := 2 Z

k

(p, p

)g

dp

where k

(p, p

) := P

h Z

δ(p

= p

+ p

, |p

|

= |p

|

+ |p|

+ gn

)dp

, or

Q

(g, h)(p) := 2 Z

k

(p, p

)g

dp

where k

(p, p

) := P

h Z

δ(p

= p

+ p

, |p|

= |p

|

+ |p

|

+ gn

)dp

, or

Q

(g, h)(p) := 2 Z

k

(p, p

)g

dp

where k

(p, p

) := P

h Z

δ(p

= p

+ p

, |p

|

= |p|

+ |p

|

+ gn

)dp

, or

Q

(g, h)(p) := 2 Z

k

(p, p

)h

dp

where k

(p, p

) := P

P Z

δ(p

= p

+ p

, |p

|

= |p|

+ |p

|

+ gn

)P

g

dp

. Let (g, h) ∈ L

ν_1+P^P

× L

. Consider first the term ( R

ν

(

)

dp)

. P is uniformly

bounded by M from above and below in the domain of integration, so in the estimates below it is

enough to use M instead of P . It holds (

Z

ν

M ( Z

k

(p, p

)g

dp

)

dp)

≤ Z

( Z

ν

M k

(p, p

)dp)

g

dp

≤ c Z

M

g

Z

ν

M

(M h)

(p

− p

, |p|

− |p

|

− gn

− |p

− p

|

)dp

dp

≤ c Z

M

g

Z

ν

( p

|p

|

+ |p

|

+ gn

)M

M

(M

h

)

)dp

dp

≤ c(

Z 1

(1 + |p

|)

M

1 2

2

g

dp

)(

Z

M

h

dp

)

≤ c(

Z

ν

M

g

dp

)

( Z

M

h

dp

)

, by the Cauchy-Schwartz inequality. For the ( R

ν

(

)

dp)

term, (

Z

ν

M ( Z

k

(p, p

)g

dp

)

dp)

≤ Z

( Z

ν

M k

(p, p

)dp)

g

dp

≤ c(

Z

M

g

dp

)(

Z

ν

M h

dp)

≤ c(

Z

ν

M

g

dp

)

( Z

M h

dp)

.

The

( R

ν

(

)

dp)

3≤i≤4

terms can be handled similarly. Finally, (

Z

ν

M ( Z

˜ k

(p, p

)h

dp

)

dp)

≤ Z

( Z

ν

M ˜ k

(p, p

)dp)

h

dp

≤ c Z

M

3 2

3

h

dp

Z

M

g

dp

≤ c(

Z

M

h

dp

)

( Z

M

g

dp

)

. This completes the proof of the lemma.

Lemma 2.6

There is a constant c > 0 such that for any cylindrically symmetric function f ∈ L

(R

),

| Z

P (Lf)(p)dp |≤ c

Z P

1 + P f

(p)dp

. Proof. Using the Cauchy-Schwartz inequality,

| Z

P (Lf )(p)dp | =|

Z √ P ν

√ P ν Lf

dp |

≤ c Z

P f

(p)dp

≤ c Z P

1 + P f

(p)dp

.

3 Rest term estimates.

Consider the decomposition f = P (1 + γ R), ˜ ψ = √

n

+ γ Φ. ˜

The equations (1.8)-(1.9) written for ˜ Φ with periodic boundary conditions when ˜ R is given, are

∂

Φ ˜ − i∂

Φ = ˜ S

Φ + ˜ S

Φ + ¯ ˜ U, Φ(0, ˜ ·) = ˜ Φ

. (3.1) Here S

and S

are the coefficients of the linear terms in ˜ Φ resp. Φ, and ¯ ˜ U contains the inhomoge- neous terms and the non-linear terms in ˜ Φ, Φ. In the following lemmas the dependence of ¯ ˜ U on ˜ Φ is not taken into account.

Lemma 3.1

Let Φ ˜

(resp. S

, S

, U ) be a given function in H

(0, 2π) (resp. L

( R

; H

(0, 2π)).

There is a unique solution Φ ˜ to (3.1) in C(R

; H

(0, 2π)). Moreover, k Φ(t, .) ˜ k

≤ (2 k Φ ˜

k

+6t

Z

k U (s, .) k

ds)e

Rt

0(kS₁(r,.)k²

H1+kS₂(r,.)k²

H1)dr

, t > 0. (3.2)

Proof of Lemma 3.1

Consider first the equations

∂

Φ ˜ − i∂

Φ = ˜ W, Φ(0, ˜ ·) = ˜ Φ

, (3.3) for a given W ∈ L

( R

; H

(0, 2π)). Writing W and ˜ Φ in Fourier series, gives

ˆ ˜

Φ

(t) + in

Φ ˆ ˜

= ˆ W

, and so

ˆ ˜

Φ

(t) = Φ ˆ ˜

(0)e

+ Z

0

W ˆ

(s)e

ds. (3.4)

Hence,

| Φ ˆ ˜

(t)| ≤ | Φ ˆ ˜

(0)| + Z

0

| W ˆ

(s)|ds, X

n∈N

(1 + n

)| Φ ˆ ˜

(t)|

≤ 2 X

n∈N

[(1 + n

)| Φ ˆ ˜

(0)|

+ t Z

0

(1 + n

)| W ˆ

(s)|

ds].

And so the function ˜ Φ defined by (3.4) belongs to L

( R

; H

(0, 2π)). Moreover, k Φ(t, .) ˜ k

≤ 2

k Φ ˜

k

+t Z

0

k W (s, .) k

ds

.

We conclude that given ˜ Φ

∈ H

(0, 2π) and W ∈ L

( R

; H

(0, 2π)), there exists a unique solution ˜ Φ ∈ L

( R

; H

(0, 2π)) to (3.3). It also follows from (3.4) that the solution is a continuous function of t ∈ R

into H

(0, 2π). For W = W ( ˜ Φ) := S

Φ + ˜ S

Φ + ¯ ˜ U it holds,

k W (s, .) k

≤ 3 k S

Φ(s, .) ˜ k

+3 k S

Φ(s, .) ¯ ˜ k

+3 k U (s, .) k

≤ 3 k S

(s, .) k

k Φ(s, .) ˜ k

+3 k S

(s, .) k

k Φ(s, .) ¯ ˜ k

+3 k U (s, .) k

. With ˜ Φ

= 0, an iterative sequence of solutions ˜ Φ

of (3.3) for j ≥ 1 with the right hand side W ( ˜ Φ

), gives

k Φ ˜

(t, .) k

≤ 2 k Φ ˜

k

+6t Z

0

(k S

(s, .) k

+ k S

(s, .) k

) k Φ ˜

(s, .) k

+ k U (s, .) k

ds, (3.5) and with δ Φ ˜

= ˜ Φ

− Φ ˜

,

k δ Φ ˜

(t, .) k

≤ 6t Z

0

(k S

(s, .) k

+ k S

(s, .) k

) k δ Φ ˜

(s, .) k

.

It follows that the sequence converges on some interval t ∈ [0, T ], and that (3.4) and (3.5) hold for the limit ˜ Φ, a unique solution of (3.1). By an iteration of the argument the existence and the continuity of ˜ Φ hold for t > 0. Using Gronwall on (3.5) for ˜ Φ gives (3.2).

The rest of this section prepares for the control of the excitation distribution function f around the equilibrium P . With

f = P (1 + γ R), ˜ f

= P (1 + γ R ˜

), the equations (1.6)-(1.7) written for ˜ R, are

∂

R ˜ + p

∂

R ˜ = gγ n

L R ˜ + γ(L

R ˜ + Q

( ˜ R, R)) ˜

, R(0, ˜ ·, ·) = ˜ R

, (3.6) where L

(resp. Q

) is a linear (resp. quadratic) operator.

The following norms are used. For 1 ≤ q ≤ ∞, k f k

= Z

R³

P 1 + P (

Z

[0,2π]

|f(x, p)|

dx)

dp

,

k f k

= Z

[0,2π]×R³

P

1 + P |f(x, p)|

+ |∂

f (x, p)|

dxdp

,

k f k

= Z

[0,T]×[0,2π]×R³

P

1 + P |f (t, x, p)|

dtdxdp

.

To study (3.6), some a priori estimates will be needed for the linear problem

∂

h + p

∂

h = gγ n

Lh + γG

, h(0, ·, ·) = h

, (3.7)

periodic in x with period 2π. Assume G ∈ L

([0, T ]; L

( R

; H

(0, 2π))) for T > 0. The function ∂

h is at least formally a solution to

∂

(∂

h) + p

∂

(∂

h) = gγ n

L(∂

h) + γ∂

G

, ∂

h(0, ·, ·) = ∂

h

, (3.8) periodic in x with period 2π. For existence of solutions to problems of the type (3.7), see [19] or alternatively, consider the Fourier transform in x of (3.7) and argue as in the proof of Lemma 3.1.

The solutions are unique and continuous as functions of t into L

( R

; H

(0, 2π)). Multiply the equation by h

, integrate on [0, T ] × [0, 2π] × R

, and use (2.8) to get

Lemma 3.2 For any η > 0,

k h(T, .) k

+γ k ν

h

k

≤ c(k h

k

+γ

k ν

G

k

+γ

η k G

k

+ γ

η k h

k

).

Lemma 3.3 Assume γ ≤ 1, Z

0

Z

P (|p|

+ gn

)G(t, x, p)dpdx = Z

0

Z

P p

G(t, x, p)dpdx = 0, t ∈ (0, T ), (3.9) and

Z

Z

P (|p|

+ gn

)h

(x, p)dpdx = Z

0

Z

P p

h

(x, p)dpdx = 0. (3.10) Then

k h

k

≤ c

γ

k h

k

+γ

(k ν

G

k

+ k G

k

) . Proof of Lemma 3.3

Let {χ

= c

p

(1 + P ), χ

= c

(|p|

+ gn

)(1 + P)} be an orthonormal basis for the kernel of L.

Consider the Fourier series in x of (3.7),

∂

h

− ikp

h

= c

+ gγ

G

, (3.11)

where c

(resp. G

) is the k-th Fourier coefficient of gγn

Lh (resp. G). Set h

= (h

, χ

) and h

= (h

, χ

). Multiply (3.11) by

χ

(resp.

χ

) and integrate in p,

∂

h

− ikκ(h

+ κ

h

) = gγ

G

, (3.12)

∂

h

− ikκ(h

+ κ

h

) = gγ

G

. (3.13) Here h

and h

, denote non-hydrodynamic moments of h, and κ := R

1+P

p

χ

χ

dp.

By adding and subtracting (3.12) resp. (3.13) with their conjugates, and dividing by 2 or 2i, we get equations for the real and imaginary parts,

∂

Rh

− ikκ(Ih

+ κ

Ih

) = gγ

RG

,

∂

Ih

− ikκ(Rh

+ κ

Rh

) = gγ

IG

,

∂

Rh

− ikκ(Ih

+ κ

Ih

) = gγ

RG

,

∂

Ih

− ikκ(Rh

+ κ

Rh

) = gγ

IG

.