pp. X–XX EXPONENTIAL STABILITY OF THE SOLUTIONS TO THE BOLTZMANN EQUATION FOR THE BENARD PROBLEM

We complete the result in [2] by showing the exponential decay of the perturbation of the laminar solution below the critical Rayleigh number and of the convective solutions above the critical Rayleigh number, in the kinetic framework.


1.
Introduction. The arising of convective motions in a fluid between two thermal walls under the action of the gravity field g, when the bottom wall is hotter than the top wall, is one of the classical examples of bifurcation of a stationary solution in Fluid-Dynamics and is known as the "Benard problem". The bifurcation is driven by a parameter Ra, the Rayleigh number which is proportional to the product of the gravity and the temperature difference. It consists in the fact that, when the Rayleigh number Ra is below a critical value Ra c , the incompressible Navier-Stokes-Fourier system (INSF) in an external gravity has only the conductive solution, characterized by vanishing velocity field and a linear temperature profile. Instead, when Ra crosses the threshold Ra c convective solutions appear with non vanishing velocity field. With the increase of the Rayleigh number, a large variety of complex phenomena occur. Here we wish to restrict our attention to a small right neighborhood of Ra c , where only the first bifurcation occurs and the laminar solution bifurcates: above Ra c both the laminar and the two convective motions, corresponding to clockwise and anti-clockwise rotation, are stationary solutions, but only the last two are stable.
The analysis of the linear and non linear stability of the stationary solutions to the Benard problem, at the level of Fluid-Dynamics, has been performed in a vast literature ( [4,6,8,9,10,13,11]). The same problem, in the framework of the Boltzmann equation, has been addressed in [1,2], where the stationary solutions to the Boltzmann equation (1.1), both below and above the critical Rayleigh number, have been constructed and their asymptotic stability has been proved, without computing the rate of decay of the perturbation.
The aim of this paper is to complete the result in [2] by proving exponential decay rate of the perturbation. Unfortunately, the key spectral inequality we used in [2] is incorrect. Therefore, we begin with fixing this error by giving the correct inequality, then we modify consequently the proofs given in [2]. This requires a slight change of perspective. As already mentioned, the Rayleigh number is proportional to the product of the gravity times the temperature difference. Therefore, in order to achieve a supercritical Rayleigh number, either we consider a sufficiently small gravity and a corresponding temperature difference, or we fix a sufficiently small temperature difference and deal with a corresponding gravity. The former point of view is the one used in [2]. In this paper, due to the extra terms deriving from the corrected spectral inequality, we adopt, at least in two dimensions, the latter point of view, which requires minor modifications in several lemmas.
To be more specific, we state the main problem. We follow as closely as possible the notation of [2] to which we will also refer for many details which are just a repetition of the arguments given there.
We look for the solution to the initial-boundary-value problem for the Boltzmann equation with diffuse reflection at the boundary modelling two thermal walls the bottom one at temperature T − = 1 and the top one at themperature T + = 1−2πελ: |w z |F (t, x, ∓π, w)dw, t > 0, v z ≷ 0, x ∈ [−π, π), where µ = h d is the aspect ratio of the convective cell, ∇ µ = (µ∂ x , ∂ z ) and v · ∇ µ = µv x ∂ x + v z ∂ z . Indeed, we have rescaled the variables z to make the width of the slab 2π and the variable x so that all the functions are periodic in x with fixed period 2π. Moreover, The parameter ε = 0 d is the ratio between the mean free path and the width of the slab, T + and T − > T + are the temperatures on the top and bottom plates, G = 1 ε dg 2T− is the rescaled gravity field, λ = 1 ε T−−T+ 2πT− measures the rescaled temperature gradient. Moreover, , B is the differential cross section 2B(ω, V ) = |V · ω| corresponding to hard spheres, and v, v * and v , v * are pre-collisional and postcollisional velocities or conversely. Note that the boundary conditions are chosen so that the impermeability condition is formally satisfied at the boundaries.

EXPONENTIAL STABILITY FOR THE BENARD PROBLEM 675
A comment is in order about the assumptions on collision cross section and boundary conditions: the method presented here can be probably extended to collision cross sections corresponding to hard potentials with Grad angular cutoff. This would require extra technical efforts and we prefered to restrict ourselves to the simplest case. It does not seem possible to include soft potentials with cutoff and non cutoff potentials in this treatment. About boundary conditions we remark that the Benard setup requires thermal walls that could also be modeled by a combination of diffuse reflection and elastic or reverse reflection. Unfortunately the boundary terms due to elastic or reverse reflection are too singular to be treated with our methods, hence we have to confine our analysis to the purely diffusive boundary conditions. Purely elastic reflection or reverse reflection are not considered because they do not model thermal walls.
We note that above definitions of the parameters correspond to the choice ε = 2Kn 6 5π where Kn is the Knudsen number. We have also set the Mach number Ma = ε 6 5 . With such a choice of the parameters, the Rayleigh number is given by (see for example [14]) independent of ε. As mentioned before (see e.g. [4]), there is a critical value of Ra, denoted by Ra c , such that the laminar solution to the hydrodynamic equations becomes linearly unstable. In the rest of this paper λ > 0 will be a fixed value, smaller than a suitable λ 0 that will be specified later, and G will be the control parameter of the bifurcation, which will occur when G crosses the threshold G c such that 32λG c = Ra c . Moreover, we will use the notation δ = (G − G c )G −1 c , and our analysis will hold either for 0 ≤ G ≤ G c or for δ > 0 sufficiently small. We stress that the smallness of the parameters λ and δ is independent of ε, so that the results we obtain are valid also in the limit ε → 0, the hydrodynamic limit, with λ and δ small but fixed.
We now recall the Fluid-Dynamics results for the Benard problem relevant to our purposes. We refer to [6,7,8] for more details. The laminar solution to the INSF system is characterized by the temperature field T l = −λ z+π 2π and u l = 0. We write the INSF system for the deviations from the laminar solution. They are: with u a vector in R 2 whose components are u x and u z respectively, u · ∇ µ = µu x ∂ x + u z ∂ z , ∆ µ = µ 2 ∂ xx + ∂ zz , e z the unit vector in the positive z direction. p is the pressure of the incompressible fluid, θ is the deviation from the linear temperature profile,η is the kinematic viscosity andk is the heat conductivity multiplied by a factor 2 5 . The INSF system (1.4) has to be solved with homogeneous boundary data: and periodic boundary conditions in the variable x. The couple h = (u, θ) denotes the solution to the problem (1.4), (1.5). For G ≤ G c , the laminar solution, h = 0 is the only steady solution and it is stable up to the critical Rayleigh number. Moreover, there is δ 1 > 0 such that, if G ∈ (G c , G c (1 + δ)) for δ < δ 1 , then there are two periodic roll solutions, h s , with period which fixes the aspect ratio µ, rotating clockwise and anti-clockwise respectively, such that where h con are the eigenvectors corresponding to the least eigenvalues of the linearization of the problem (1.4),(1.5) around the laminar solution h = 0. The remainder h R is in a suitable Sobolev space (H k (Ω)) 3 with its Sobolev norm bounded uniformly in δ: namely, there is a constant C such that, for any δ < δ 1 , Furthermore, there are n 0 and ζ 1 such that if h 0 ∈ (H k (Ω)) 3 for k sufficiently large and has H k -norm smaller than n 0 , then the time dependent solution to the problem (1.4), (1.5) is such that for any k < k (see Proposition 3.1).
A stationary solution to the problem (1.1) is constructed by means of a truncated expansion in ε with remainder, so that we have the representation (1.9) The first term of the expansion is the standard Maxwellian M the first order correction is given by where, for G ≤ G c we have u s = 0, T s = T l and ρ s = −(λ + G)z is computed by using the Boussinesq condition When G > G c and δ < δ 1 , u s , T s and ρ s are computed in terms of h s . The higher order terms will be described later. Now we are in position to state the main theorem. In the statement we use the norm · 2,2 which represents the L 2 -norm on the phase space Ω × R 3 with weight M −1 , and the norm · 2,2,2 , the L 2 -norm on Ω × R 3 × R + with weight M −1 , including also integration of all the positive times. Theorem 1.1. There are λ 0 > 0, δ 0 > 0, ε > 0 such that, if 0 ≤ λ < λ 0 and G ∈ (0, G c (1 + δ)) with δ < δ 0 , then there is a positive, locally unique, stationary solution F s to the Boltzmann equation such that for any ε < ε 0 , (1.13) Furthermore, if the initial perturbation Φ ε 0 to the stationary solution is such that (1.14) Section 2 is devoted to the construction of the stationary solution. In Section 3 we show the exponential decay of the perturbation.
2. Stationary solution. As discussed before, in this paper we want to show that a small perturbation of the stationary solution F s to the problem (1.1) decays exponentially fast, as t → +∞. However, since the paper [2] contains an inconsistency in the construction of such a stationary solution, we need to review part of the proof of the main existence result for the stationary solutions.
We recall the notation adopted for the norms: the norm in the bulk is defined, for any 1 ≤ q ≤ +∞ as The space of measurable functions on Ω × R 3 with the above norm finite is denoted byL q .
The boundary norm is defined as The stationary solution F s corresponding to the laminar and convective solutions to the INSF system will be constructed as follows: set s , for j > 1 are constructed by means of a bulk-boundary layer expansion already discussed in [5,1,2]. Here we summarize the relevant properties of the Φ (n) s 's in the following theorem taken from [2]: Proposition 2.1. The functions Φ (n) s , n = 1, . . . , 5 and ψ n,ε can be determined so as to satisfy the boundary conditions and the normalization condition R 3 ×[−π,π] 2 dvdxdzΦ (n) = 0, so that the asymptotic expansion in ε for the stationary problem (1.1), truncated to the order 5 is given by .
If G ≤ G c then the functions Φ (n) 's, corresponding to the laminar solution satisfy the conditions for a suitable constant C. Moreover if G ≥ G c and δ < δ 1 , then the Φ (j) 's differ from those of the laminar solution by O(δ) and the inequalities (2.1) are replaced by The functions ψ n,ε are such that ψ n,ε q,2,∼ , q = 2, ∞ are exponentially small as ε → 0, and R 3 dvv z M (v)ψ n,ε = 0. The space where the remainder will be constructed is the following: Here, Df denotes first order derivatives of f and γ ± f are the ingoing (resp. outgoing) trace operators defined as the restrictions of f to the ingoing (resp. outgoing) boundary, . Before stating the main theorem of this section we recall the properties of the linearized Boltzmann operator L, The operator L has a non trivial null space. An orthonormal basis in the null space is given by the functions ψ 0 = 1, . The orthogonal projection on the null space of L is denoted by P . For the operator L the decomposition L = −νI + K holds, where I is the identity, ν is a positive function of |v| which, for hard sphere is such that ν ∼ (1 + |v|) and K is a compact operator on H. Finally L is symmetric on H and the quadratic form associated to L is negative semi-definite in the sense that there is a positive constant C such that

Now we state the main theorem of this section:
Theorem 2.1. There are positive ε 0 , δ 0 and λ 0 such that given λ < λ 0 , δ < δ 0 , for any ε < ε 0 there exists a stationary solution to (1.1) in the form s 's, bounded in · q,2 , q = 2, ∞, and R 3 dvM (v)A = 0. Moreover, the remainder satisfies the impermeability conditions The construction of the stationary solution is obtained by an iteration scheme where, in the equation for the iterate R n+1 , the non linear term is computed in terms of R n . Therefore, the main step of the analysis is the study of the equation As in [2], we introduce the operator L J for fixed x, z as follows: for any f in the domain of L, In the rest of this section we will use the function With this choice of the function q, we have for some constant C. In next section there will be a different choice of q, and the above estimate will be consequently modified.
The operator L J also has a non trivial null space Kern(L J ), which is spanned by the vectorsψ j , j = 0, . . . , 4 as proved in [2]. The vectorsψ j 's differ from the ψ j 's for terms of order ε:ψ (2.7) The operator P J denotes the orthogonal projector on Kern(L J ). We underline that L J is not symmetric, so we will also consider the adjoint of L J , denoted L * J . The null space of L * J coincides with the null space of L, Kern(L). The difference P J − P is estimated as follows: for some constant C.
The following proposition replaces Proposition 2.1 in [2]: There is ε 0 > 0 such that for ε < ε 0 there are positive constants c 1 and c 2 such that The first part is bounded from below as in Proposition 2.1 of [2] by Thus we obtain (2.9) by choosing η = 3C c εq ∞ . Consequently c 1 = c 2 and c 2 = The first consequence of the extra term appearing in Proposition 2.2 is in the Green inequality (2.15) of [2] which is modified as follows: with the prescribed inhomogeneous term g such that Mgdv = 0 and prescribed incoming data Then, for any η > 0, A similar inequality holds when L J is replaced by L * J . The proof is the same as in [2], taking into account the modified spectral gap inequality for L J .
The Fourier transform with respect to the variable x, F x f (sometimes just Ff for brevity) is defined as follows: for any ξ ∈ Z, (2.14) andf In the rest of this paper, constants which, independently of the parameter ε, can be made sufficiently small for the purposes of the proofs, will generically be denoted η.
The statement of Lemma 2.1 in [2] holds provided that q ∞ is sufficiently small: periodic in x of period 2π, and with zero ingoing boundary values at z = −π, π. Then, if λ + δ + is sufficiently small, it results: The statement of Lemma 2.1 is still true, if we replace the operator L * J with the operator L J and the operator P with P J .
Proof. Lemma 2.1 is proved as in [2]. Equation (2.21) in [2] provides a bound for P ϕ 2 2,2 in terms of ε −2 ν 1 2 (I − P )ϕ 2 2,2 . This, by the Green inequality, gives a term q 2 ∞ P ϕ 2 2,2 in the right hand side, which can be absorbed in the left hand side provided that q 2 ∞ is sufficiently small. This is true, by (2.6), provided that λ + δ + is sufficiently small.
s , R) and decompose H in accordance with the operator L J . Set H 1 ( · ) = H( · ) − J(q, P · ) = J(q, f) − J(q, P f). We notice that H 1 (·) is of order zero in ε and only depends on the non-hydrodynamic projection (I − P ).
At the stage n + 1 of the iterative procedure we need to compute the remainder R n+1 , still for brevity denoted by R. We decompose it into two parts R 1 and R 2 , solutions of two different equations. The part R 1 is periodic in x and solves the boundary value problem where the incoming data are prescribed and the inhomogeneous term g includes A and the non linear term computed at the previous step. The part R 2 is discussed later. An existence proof for this problem can be obtained by the method of [12] . The nonhydrodynamic part of R 1 is estimated along the same lines of the proof of Lemma 2.1: for small η > 0, by using the inequality The duality technique used in [2] can still be applied to estimate P J R 1 . The term H 1 (R 1 ) is treated as a perturbation, after dealing in next lemma with the system without it.
By using Lemma 2.1, one is then left with a < P ϕ >-term which is the projection of < ϕ >. Now < ϕ > is the average over the variable x, and thus satisfies a one dimensional equation similar to eq. (3.5) in [1]. By using the argument of Lemma 3.4 in [1], we obtain where < · > x denotes the average on x.
Remark. We note that in [2], instead of Lemma 3.4 in [1], we used the arguments of Lemma 3.5 in the same paper, which require G small but permit any value of λ.
In the present setup the use of Lemma 3.4 in [1] allows us to use G ∈ [0, G c (1 + δ)] provided that λ is sufficiently small. This is the only point where the condition G small was used in [2]. The final estimates for R 1 then follow as in [2]: If R 1 is a solution to the system (2.18), then, under the same conditions on the parameters as before, Now we discuss R 2 . It is solution to the following boundary value problem:

EXPONENTIAL STABILITY FOR THE BENARD PROBLEM 683
In order to estimate R 2 , one can use the arguments given in [2], Lemmas 2.4, 2.5. Indeed the only modifications arise from the extra term in the Green inequality and they are managed by using the smallness of q given in (2.6). One thus gets the final estimates for R 2 given in the following The linear estimates of Lemmas 2.2 and 2.4 are sufficient to prove the existence of the solution to the equation for the remainder. This is Theorem 2.2 in [2], which we restate here: 3. Initial boundary value problem. We now study the initial boundary value problem (1.1) for an initial datum F 0 suitably close to the stationary solution. Indeed, we introduce the perturbation Φ = M −1 (F − F s ). The equation for the perturbation Φ is: The initial conditions for M −1 (F (0, x, z, v) − F s (x, z, v)) = Φ ε 0 (x, z, v) are given with the initial datum Φ 0 specified as follows: where Φ (n) (0, x, z, v) is the n-th term of the expansion introduced in the next paragraph, computed at time t = 0, and the ε-dependent contribution p 5 is arbitrary but for having total mass dvdxdzM (v)p 5 (x, z, v) = 0 and for some constant c.
We write also the time dependent solution in terms of a truncated expansion in ε, The first term of the expansion in ε is where the fields (u(t, x, z), θ(t, x, z)) are solutions of the hydrodynamic equations for the perturbation, while ρ(t, x, z) is determined by the Boussinesq condition (1.12). The hydrodynamic initial data are chosen as follows: let (u 0 , θ 0 ) be an initial perturbation of the convective solution (u s , θ s ) sufficiently small to ensure that the solution to (1.4), denoted here (ũ(t, x, z),θ(t, x, z)) = (u s (x, z) + u(t, x, z), θ s (x, z) + θ(t, x, z)), exists globally in time and converges exponentially to (u s , θ s ) as t → +∞, as stated in (1.8).
The construction of the time dependent solution is based, as the stationary solution, on an expansion which starts with the solution to the hydrodynamic equations. We need the following proposition on the stability of the hydrodynamic solution, whose proof is referred to the literature [6,7,8,9,10,11,13]: Proposition 3.1. For δ < δ 1 , let (u, θ) be the periodic solution of the following equation for the perturbation If (u 0 , θ 0 ) ∈ (H k ) 3 , for k sufficiently large, and u 0 H k + θ 0 H k < n 0 , for n 0 small enough, then there is ζ 1 > 0 such that (u, θ)(x, z, t) is in (H k ) 3 for any t > 0 and lim t→∞ (e ζ1t u, e ζ1t θ) = 0 in (H k ) 3 , for any k < k.
The terms of the expansion Φ (n) , n = 1, . . . , 5 are constructed by means of an Hilbert type expansion in the bulk, corrected by a boundary layer expansion designed to restore the correct boundary conditions. For the construction of the expansion we refer to [5]. To state next proposition, we need the norms

EXPONENTIAL STABILITY FOR THE BENARD PROBLEM 685
Moreover f 2,2,2 is the norm f 2t,2,2 with t = +∞. We also use the boundary norms and, as before, · 2,2,∼ corresponds to t = +∞. The estimates we need on the terms of the expansion Φ (n) are summarized in the following proposition, whose proof can be readily obtained along the lines of [5,1]: Proposition 3.2. Assume that at time zero, for some suitably large k, (3.5) Then for δ < δ 1 and for n 0 of Proposition 3.1, it is possible to determine the functions Φ (n) , n = 1, . . . , 5 and the boundary functions ψ n,ε , n = 2, . . . , 5 in the asymptotic expansion so that the following boundary conditions are satisfied: The Φ (n) satisfy the zero mass condition Moreover, there are constants C and C 1 such that for n = 1, . . . , 5, and, for n = 2 . . . , 5, for any 0 ≤ ζ < ζ 1 , with ζ 1 the decay rate of the hydrodynamic equation given in Proposition 3.1.
The remainder Y satisfies the following initial boundary value problem: where ψ(t, x, ±π, v) = 5 n=1 ε n ψ n,ε (t, x, ±π, v). We have set where we recall that Φ ε s is the full stationary solution constructed in Section 2, and Φ (n) are the terms of the time dependent expansion.
The inhomogeneous term A is such that The expression for A is given in [5]. We omit it because we only use the following estimate for A, Proposition 3.3. There are C > 0 and C 1 > 0 such that for any 0 ≤ ζ < ζ 1 ,
The main result of this section is the stability result: Theorem 3.1. There are λ 0 > 0, δ 0 > 0, ε 0 > 0 (possibly smaller than those introduced in Section 2), n 0 and ζ > 0 such that, if λ < λ 0 , δ < δ 0 , and p 5 satisfies (3.3), then the solution to the initial boundary value problem (3.1) exists and has the following decay property: there is a constant C independent of ε such that e 1 2 ζt Φ ε 2,2,2 < Cε In order to prove Theorem 3.1 the strategy we have discussed before consists in writing Φ ε , as Φ ε =Φ + εY . Since the terms of the expansion are estimated by means of Proposition 3.2, we only need to estimate the remainder term Y . Again by Proposition 3.2,Φ decays to zero exponentially in t. Therefore, to prove (3.10), we need to show that also the remainder Y decays exponentially. For this purpose, let us fix a positive ζ < ζ 1 and put R = e ζt Y . Then, R is solution of HereĀ = e ζt A andψ(t, x, ±π, v) = e ζt ψ(t, x, ±π, v). The estimates of Proposition 3.3 imply that for any ζ < ζ 1 and We follow closely the approach in [2]. Therefore we will just recall the main theorems proved there, which are valid also in the present situation, and give explicitly the proofs when modifications are needed.
We decompose the operator H as in Section 2: Then we define L J = L + εJ(q, P Y ). Note that Proposition 2.2 holds for the newly defined L J and that, under the assumptions of Theorem 3.1, inequality (2.6) is replaced by (3.14) We notice that H 1 (R) is of order zero in ε, and only depends on the nonhydrodynamic part (I − P )R. To solve the equation for R we shall use an iteration procedure based on the decomposition of R in the sum R 1 + R 2 , where R 1 and R 2 are solutions of two different problems. R 1 solves a problem with prescribed incoming data and prescribed inhomogeneous term, while R 2 solves a problem with diffusive boundary conditions plus prescribed incoming data (depending on R 1 ), zero initial condition and no inhomogeneous term. We recall that R satisfies the vanishing mass condition dxdzdvM R(x, z, v, t) = 0, t ∈ R + , so that we have also dxdzdvM R 1 (x, z, v, t) = − dxdzdvM R 2 (x, z, v, t). The equations for R 1 and R 2 are Note that we have multiplied by ε the equations for R 1 and R 2 because, in some arguments we will use the "microscopic time"τ = ε −1 t and such a rescaling corresponds just to replace ε∂ t with ∂τ in above equations. In the problems (3.15) and (3.16) the unknowns R 1 and R 2 are sought for as periodic functions in x ∈ [−π, π), and g is some given function, periodic on the same interval, such that Mg(·, x, z, v)dxdzdv ≡ 0. The existence of the solution, is obtained as in [1]. We start by giving a priori estimates obtained by Green's formula, for the nonhydrodynamic part of R 1 and the outgoing flux γ − R 1 ; multiply (3.15) by 2R 1 Mκ, (where as in [2], κ = e εG(z+π) ) integrate with respect to the variables (τ , x, z, v) over [0, T ] × [0, 2π] 2 × R 3 , integrate by parts and use the spectral inequality for L J and the bounds 1 ≤ κ(z) ≤ e 2εGπ and (3.14), to obtain, for every η 1 > 0, 2T ,2,∼ , whereη = η1 2 + ε(ζ + λ + δ + + n 0 ). We consider now the so called dual problem, namely we seek for the space-periodic solutions to a linear problem in the rescaled time variableτ = ε −1 t. This problem is discussed in the following lemma, where we use the notation introduced in Section 2, but with the function q also time dependent. Next lemma follows as in [2], Lemma 4.1, by taking into account the modified spectral inequality (2.9) and the consequent Green inequality.
We write the analog of (3.17) for the dual problem: for any solution ϕ to (3.18) below, with vanishing initial and incoming data. Inequality (3.22) below follows as in [2], page 47.

18)
with vanishing initial and incoming data. Setφ = ϕ− < ϕ >= ϕ − (2π) −2 ϕdxdz. If the parameters λ, , δ and n 0 satisfy the assumptions of Theorem 3.1, then there exists η small such that, An a priori bound for P J R 1 is obtained in the following lemma based on dual techniques involving the simultaneous considerations of the problems (3.18) and (3.15). Consider first the problem (3.15) without the term H 1 (R 1 ).
Proof of Lemma 3.2. In the variables (τ , x, z, v), the function R 1 is 2π-periodic in x and solution to (3.17) with the term H 1 (R) missing. Let ϕ be a 2π-periodic function in x, solution to (3.18) with zero initial values and ingoing boundary values at z = −π, π. We multiply the equation for ϕ by κM R 1 and the one for R 1 by κM ϕ, then sum them and integrate on the variablesτ ∈ [0, T ], x ∈ [−π, π), z ∈ (−π, π) and v ∈ R 3 . Then we use the periodicity in x to cancel the terms ∂ x and take an integration by parts on the variable z. Using the equilibrium condition v · ∇ µ (κM ) + εG∂ vz (κM ) = 0, we obtain: We use the above equation to get an estimate for the term before the last in the r.h.s: hP J R 1 = h 2 . All the terms are estimated as in [2] but we give the explicit computation here for sake of completeness. We need to track the ζ-term and take care of the terms due to the modified spectral inequality. The last term is bounded as Therefore, for any arbitrary choice of K i , i = 0, 1, . . . , 4 we get, for ζ small, All the ϕ-terms computed at time T on the l.h.s can be estimated using (3.19)-(3.22) in Lemma 3.1. Using the Green inequality (3.17) to bound the R 1 -terms, we obtain, forT → ∞, The term < P ϕ > is bounded by using (3.22) in Lemma 3.1 as We recall thatη = η 1 + ε(ζ + λ + δ + + n 0 ). So choosing ε small, then K 1 , K 0 and K 3 (resp. K 2 ) of order ε −1 (resp. ε −2 ) times a big constant, K 4 big and η 1 , η 2 of order ε times a small constant and ζ + λ + δ + + n 0 sufficiently small, leads to The final estimates for R 1 are summarized in Lemma 3.3. The solution R 1 to (3.15) satisfies withη = η + ε(ζ + λ + δ + + n 0 ), for any η > 0. Moreover, it follows from Lemma 3.2 that Choosing η = √ ε leads to the first two inequalities of Lemma 3.3. The last inequality of Lemma 3.3 is obtained as in [1], by studying the solution along the characteristics. Adding the term ε −1 H 1 (R 1 ) does not change these results.
The remaining part R 2 of R satisfies the problem (3.16). Its analysis is more involved and will use a careful study of the Fourier transform of R 2 . The existence for the problem (3.16) can be adapted from the corresponding study in [12], if one includes into that approach the spectral estimate for L J , and the characteristics due to the force term.
In (3.16) the given indata part is By Green's formula for (3.16), and noting that H 1 (R 2 ) only depends on (I − P )R 2 , we get 2t,2,2 ≤ γ + R 2 2 2t,2,∼ +εδP J R 2 2 2t,2,2 , (3.23) withδ = ζ + λ + δ + ε + n 0 . By arguing along the same lines of [2], pag. 142-143, we can estimate the outgoing flux part of R 2 appearing in the r.h.s. of (3.23) and thus obtain The hydrodynamic estimates for R 2 are obtained in two steps: first we consider a 1-d (x-independent) case, with an inhomogeneous term g 1 which will take into account the x-dependence in later proofs, By the relation betweeng 1 and g 1 and the Green inequality, we also have Now we have to examine the 2-dimensional case. This is treated in [2] in Lemma 4.5. The inclusion of the term εζR 2 can be handled as before. We sketch the approach and give the details for the R 20 -moment. Lemma 3.5. Let R 2 be solution to (3.16). Then there is c > 0 such that Proof. The equation r denoting the difference between the ingoing and outgoing boundary values, For any function φ(v) we denote R 2φ = dvM R 2 φ(v). In particular, we denote R 20 = dvM R 2 and R 24 = dvM R 2 v 2 . All the functions below depend on t but we omit such a dependence.
First, we consider the case ξ x = 0. We apply Lemma 3.4 toR 2 (0, ξ z , v) = dxR 2 (x, z, v). By integrating (3.16) over x and taking into account the periodic conditions in the direction x, we get the 1-dimensional equation (3.25), where the term g 1 comes from the x-dependent terms in the expansion appearing in L J . Since the limiting solution is close to the laminar 1-dimensional solution up to order δ, g 1 is of order δ and is linear in R 2 . Thus, by Lemma 3.4 we get a bound for the Fourier components P JR2 (0, ξ z ), for δ small.
Then we need to estimate P JR2 (ξ x , ξ z ) for ξ x = 0. Arguments similar to those used in the proof of Lemma 4.1 in [2] (Lemma 3.1 here) imply that large values of ξ can be dealt with by taking advantage of the factor |ξ| −2 and the estimates for r due to the inequality (3.24). Therefore we need only to consider finitely many (ξ x , ξ z ) with ξ x = 0.
The strategy used in [2] and repeated here is to get estimates of all the hydrodynamic moments R 2vx , R 2vy , R 2vz , R 2v 2 in terms of R 20 which is estimated at the end.
The first moment considered is the v x -moment for ξ z = 0. Multiplying (3.26) by M and integrating over the velocity we get an equation forR 20 . Multiplying the conjugate of (3.26) by v x M and integrating over the velocity we get an equation forR * 2vx (ξ x , 0). Then, we multiply the first byR * 2vx (ξ x , 0) and the second byR 20 .
We want to get an estimate of the time integral of the first term on the r.h.s. and hence of R 2vx (ξ x , 0) 2 2,2,2 . To this end, we integrate over the time variable on the interval [0, t]. The integration of the time derivative produces a term at time t = 0 which vanishes because R 2 has 0 initial conditions and a term computed at time t. Such a term is estimated by using the Green inequality. The first boundary term is estimated by noticing that dvM v z r depends only on f − and the second boundary term is estimated by using (3.24). The result is |R 2vx | 2 (ξ x , 0)dt ≤ C dt |R 20 | 2 (ξ x , 0) + η P R 2 2 2,2 + ν where η is some constant that can be made small by assuming the parameters λ, ζ, δ, ε and n 0 sufficiently small. This is the simplest case, but the other moments R 2vx , for ξ z = 0, and R 2vy , R 24 are obtained by a similar approach, see [2], and the contribution from εζR 2 produces a term of the form εζ P J R 2 2 2,2,2 which is absorbed under the smallness assumption for the parameters. We conclude: ∞ 0 dt |R 2vz | 2 + |R 2vx | 2 + |R 2vy | 2 + |R 24 | 2 (3.29) ≤ C dt R 20 2 2,2 +η P R 2 2 2,2 + ν The momentR 20 (ξ x , ξ z ) for ξ x = 0 requires a different analysis. Below, for any function h(t, x, z, v) we denotê h(σ, ξ x , ξ z , v) = F t F x F z h(σ, ξ x , ξ z , v), andĥ z (σ, ξ x , z, v) = F t F x h(σ, ξ x , z, v).
When τ 0 → 0, β tends to the Heaviside function and its derivative to the δfunction inτ = 0. Thus the last term in the first of (3.30) vanishes because R 2 is