O N T H E C O N V E R GE N C E OF S M O OT H S OL U TIO N S F R O M B OLT Z M A N N TO N AV IE R – STO K ES

IS A B E L L E G A L L A G H E R IS A B E L L E T R IS TA N I

O N T H E C O N V E R G E N C E O F S M O O T H S O L U T IO N S F R O M

B O LT Z M A N N T O N AV IE R – S T O K E S

S U R L A C O N V E R G E N C E D E S O L U T IO N S R É G U L IÈ R E S D E B O LT Z M A N N V E R S

N AV IE R – S T O K E S

Abstract. — In this work, we are interested in the link between strong solutions of the Boltzmann and the Navier–Stokes equations. To justify this connection, our main idea is to use information on the limit system (for instance the fact that the Navier–Stokes equations are globally wellposed in two space dimensions or when the initial data is small). In particular we prove that the life span of the solutions to the rescaled Boltzmann equation is bounded from below by that of the Navier–Stokes system. We deal with general initial data in the whole space in dimensions 2 and 3, and also with well-prepared data in the case of periodic boundary conditions.

Keywords:équation de Navier–Stokes, équation de Boltzmann.

2020Mathematics Subject Classification:35Q35, 35Q30, 76D05, 82B40, 82C40, 82D05.

DOI:https://doi.org/10.5802/ahl.40

(*) The authors thank François Golse for his valuable advice, as well as the anonymous referee for a very careful reading of the manuscript. The second author thanks the ANR EFI: ANR-17-CE40- 0030.

Résumé. — On s’intéresse dans ce travail au lien entre les solutions fortes des équations de Boltzmann et de Navier–Stokes. Pour justifier cette relation, notre idée principale est d’utiliser des informations sur le système limite (par exemple le fait que les équations de Navier–Stokes ont une solution globale unique en deux dimensions d’espace, ou quand la donnée initiale est petite). En particulier on démontre que le temps d’existence de la solution de l’équation de Boltzmann remise à l’échelle est toujours plus grand que celui du système de Navier–Stokes.

On considère des données initiales générales dans l’espace entier en dimensions 2 et 3, et nous traitons également le cas de données bien préparées dans le cas de conditions aux limites périodiques.

1. Introduction

In this paper, we are interested in the link between the Boltzmann and Navier–

Stokes equations. Before giving a (non exhaustive) presentation of past results in this context, let us recall that standard perturbative theories prove the convergence of (smooth) solutions of the Boltzmann equation to solutions to the fluid dynamics equations when the Knudsen number goes to zero, globally in time for small initial data or up to the singular time of the fluid solution in periodic settings. In this paper we propose a different approach, intertwining fluid mechanics and kinetic estimates, which enables us to prove (short-time) convergence without any smallness at initial time, and which is valid for any initial data (ill prepared or not) in the case of the whole space. The time of existence of the solution to the Boltzmann equation is bounded from below by the existence time of the fluid equation as soon as the Knudsen number is small enough (depending on norms of the initial data).

The problem of deriving hydrodynamic equations from the Boltzmann equation goes back to Hilbert [Hil02] and can be seen as an intermediate step in the problem of deriving macroscopic equations from microscopic ones, the final goal being to obtain a unified description of gas dynamics including all the different scales of description. The first justifications of this type of limit (mesoscopic to macroscopic equations) were formal and based on asymptotic expansions, given by Hilbert [Hil02]

and Chapman–Enskog [CC60]. Later on, Grad introduced a new formal method to derive hydrodynamic equations from the Boltzmann equation in [Gra63] called the moments method.

The first convergence proofs based on asymptotic expansions were given by Caflisch in [Caf80] for the compressible Euler equation. The idea there was to justify the limit up to the first singular time for the limit equation. In this setting, let us also mention the paper by Lachowicz [Lac87] in which more general initial data are treated and also the paper by De Masi, Esposito and Lebowitz [DEL89] in which roughly speaking, it is proved that in the torus, if the Navier–Stokes equation has a smooth solution on some interval [0, T_∗], then there also exists a solution to the rescaled Boltzmann equation on this interval of time. Our main theorem is actually reminiscent of this type of result, also in the spirit of [BMN97, CDGG00, Gre97, Sch94]: we try to use information on the limit system (for instance the fact that the Navier–Stokes equations are globally wellposed in two space dimensions) to obtain results on the life span of solutions to the rescaled Boltzmann equation. We would like to emphasize here that in our result, if the solution to the limit equation is global (regardless of

its size), then, we are able to construct a global solution to the Boltzmann equation, which is not the case in the aforementioned result. Moreover, we treat both the case of the torus and of the whole space.

Let us also briefly recall some convergence proofs based on spectral analysis, in the framework of strong solutions close to equilibrium introduced by Grad [Gra64]

and Ukai [Uka86] for the Boltzmann equation. They go back to Nishida [Nis78]

for the compressible Euler equation (this is a local in time result) and this type of proof was also developed for the incompressible Navier–Stokes equation by Bardos and Ukai [BU91] in the case of smooth global solutions in three space dimensions, the initial velocity field being taken small. These results use the description of the spectrum of the linearized Boltzmann equation performed by Ellis and Pinsky in [EP75]. In [BU91], Bardos and Ukai only treat the case of the whole space, with a smallness assumption on the initial data which allows them to work with global solutions in time. In our result, no smallness assumption is needed and we can thus treat the case of non global in time solutions to the Navier–Stokes equation. We would also like to emphasize that Bardos and Ukai also deal with the case of ill- prepared data but their result is not strong up tot= 0 contrary to the present work (where as in [Bri15] the strong convergence holds in an averaged sense in time).

More recently, Briant in [Bri15] and Briant, Merino–Aceituno and Mouhot in [BMAM19] obtained convergence to equilibrium results for the rescaled Boltzmann equation uniformly in the rescaling parameter using hypocoercivity and “enlargement methods”, that enabled them to weaken the assumptions on the data down to Sobolev spaces with polynomial weights.

Finally, let us mention that this problem has been extensively studied in the framework of weak solutions, the goal being to obtain solutions for the fluid models from renormalized solutions introduced by Di Perna and Lions in [DPL89] for the Boltzmann equation. We shall not make an extensive presentation of this program as it is out of the realm of this study, but let us mention that it was started by Bardos, Golse and Levermore at the beginning of the nineties in [BGL91, BGL93]

and was continued by those authors, Saint-Raymond, Masmoudi, Lions among others.

We mention here a (non exhaustive) list of papers which are part of this program:

see [GSR04, GSR09, LM10, LM01, SR09].

1.1. The models

We start by introducing the Boltzmann equation which models the evolution of a rarefied gas through the evolution of the density of particles f =f(t, x, v) which depends on time t ∈ R⁺, position x ∈ Ω and velocity v ∈ R^d when only binary collisions are taken into account. We take Ω to be the d-dimensional unit periodic boxT^d(in which case the functions we shall consider will be assumed to be mean free) or the whole space R^d in dimension 2 or 3. We focus here on hard-spheres collisions and hard potentials with cutoff interactions. The Boltzmann equation reads:

∂tf +v· ∇xf = 1

εQ(f, f)

where ε is the Knudsen number which is the inverse of the average number of collisions for each particle per unit time andQ is the Boltzmann collision operator.

It is defined as

Q(g, f) :=

Z

R^d×S^d−1

B(v−v_∗, σ) [g_∗⁰f⁰−g_∗f] dσdv_∗.

The Boltzmann collision kernel B(v−v∗, σ) only depends on the relative velocity

|v−v∗|and on the deviation angleθ through cosθ =hκ, σiwhereκ= (v−v∗)/|v−v∗| andh ·,· i is the usual scalar product in R^d. In this paper, we shall be concerned by kernelsB taking product form in its argument as:

B(v−v∗, σ) =b(cosθ)|v−v∗|^γ.

In the latter formula,b is a non-negative measurable function satisfying the following form of Grad’s cutoff assumption: there exist positive constants b₀ and b₁ such that

Z

S^d−1

b(cosθ) dσ>b₀, b(cosθ)6b₁ ∀θ ∈[0, π],

andγ ∈(0,1], we are thus dealing with hard potentials interactions and the caseγ = 1 with constantb corresponds to hard spheres collisions. Here and below, we are using the shorthand notations f = f(v), g∗ = g(v∗), f⁰ = f(v⁰) and g_∗⁰ = g(v⁰_∗). In this expression, v⁰, v_∗⁰ and v, v∗ are the velocities of a pair of particles before and after collision. More precisely we parametrize the solutions to the conservation of momentum and energy (which are the physical laws of elastic collisions):

v+v∗ =v⁰ +v⁰_∗,

|v|²+|v∗|² =|v⁰|²+|v⁰_∗|², so that the pre-collisional velocities are given by

v⁰ := v+v∗

2 + |v−v∗|

2 σ , v⁰_∗ := v+v∗

2 − |v−v∗|

2 σ , σ ∈S^d−1.

Takingε small has the effect of enhancing the role of collisions and thus whenε→0, in view of Boltzmann H-theorem, the solution looks more and more like a local thermodynamical equilibrium. As suggested in previous works [BGL91], we consider the following rescaled Boltzmann equation in which an additional dilatation of the macroscopic time scale has been done in order to be able to reach the Navier–Stokes equation in the limit:

(1.1) ∂_tf^ε+ 1

εv· ∇_xf^ε= 1

ε²Q(f^ε, f^ε) in R⁺×Ω×R^d.

It is a well-known fact that global equilibria of the Boltzmann equation are local Maxwellians in velocity. In what follows, we only consider the following global normalized Maxwellian defined by

M(v) := 1 (2π)^d2e^−|v|

2 2 .

To relate the Boltzmann equation to the incompressible Navier–Stokes equation, we look at equation (1.1) under the following linearization of orderε:

(1.2) f^ε(t, x, v) = M(v) +εM¹²(v)g^ε(t, x, v).

Let us recall that takingε small in this linearization corresponds to taking a small Mach number, which enables one to get in the limit the incompressible Navier–Stokes equation. If f^ε solves (1.1) then equivalentlyg^ε solves

(1.3) ∂_tg^ε+1

εv· ∇_xg^ε= 1

ε²Lg^ε+ 1

εΓ(g^ε, g^ε) in R⁺×Ω×R^d with

(1.4)

Lh:=M⁻¹²Q(M, M¹²h) +Q(M¹²h, M) and Γ(h₁, h₂) := 1

2M⁻¹²Q(M¹²h₁, M¹²h₂) +Q(M¹²h₂, M¹²h₁).

In the following we shall denote by Π_L the orthogonal projector onto KerL. It is well-known that

KerL= SpanM¹², v1M¹², . . . , vdM¹²,|v|²M¹².

Appendix B.2 collects a number of well-known results on the Cauchy problem for (1.3).

1.2. Notation

Before stating the convergence result, let us define the functional setting we shall be working with. For any real number ` > 0, the space H<sub>x</sub><sup>`</sup> (which we sometimes denote by H^{`</sup> or H<sup>`}(Ω)) is the space of functions defined on Ω such that

kfk²_H` x :=

Z

R^d

hξi^2`|f^b(ξ)|²dξ <∞ if Ω = R^d, or

kfk²_H`

x := ^X

ξ∈Z^d

hξi^2`|f^b(ξ)|² <∞ if Ω = T^d,

where f^bis the Fourier transform of f inx with dual variable ξ and where hξi² := (1 +|ξ|)².

We shall sometimes note F_xf for f^b. We also recall the definition of homogeneous Sobolev spaces (which are Hilbert spaces fors < d/2), defined through the norms

kfk²H˙^s(R^d) :=

Z

R^d

|ξ|^2s|f(ξ)|^{b 2}dξ and kfk²H˙^s(T^d)):= ^X

ξ∈Z^d

|ξ|^2s|f^b(ξ)|².

In the case when Ω =T^d we further make the assumption that the functions under study are mean free. Note that for mean free functions defined on T^d, homogeneous and inhomogeneous norms are equivalent. We also defineW_x^{`,∞</sup>(orW<sup>`,∞}orW^`,∞(Ω)) the space of functions defined on Ω such that

kfk_W^`,∞

x := ^X

|α|6`

sup

x∈Ω

|∂_x^αf(x)|<∞, We set, for any real numberk

L^∞,k_v :=

f =f(v)/hvi^kf ∈L^∞(R^d)

endowed with the norm

kfk_L^∞,k

v := sup

v∈R^d

hvi^k|f(v)|. The following spaces will be of constant use:

(1.5) X^`,k :=

f =f(x, v)/kf(·, v)k_Hx^` ∈L^∞,k_v , sup

|v|>R

hvi^kkf(·, v)k_Hx^` −−−→

R→∞ 0

(note that the R → ∞ property included in this definition is here to ensure the continuity property of the semi-group generated by the non homogeneous linearized Boltzmann operator [Uka86]) and we set

kfk_`,k := sup

v∈R^d

hvi^kf(·, v)

H_x^`.

Finally if X1 and X2 are two function spaces, we say that a function f belongs toX₁+X₂ if there are f₁ ∈X₁ and f₂ ∈X₂ such thatf =f₁+f₂ and we define

kfk_X1+X2 := min

f=f1+f2

fi∈X_i

kf₁k_X1 +kf₂k_X2.

1.3. Main result

Let us now present our main result, which states that the hydrodynamical limit of (1.1) as ε goes to zero is the Navier–Stokes–Fourier system associated with the Boussinesq equation which writes

(1.6)















∂_tu+u· ∇u−µ₁∆u=−∇p

∂_tθ+u· ∇θ−µ₂∆θ = 0

divu = 0

∇(ρ+θ) = 0.

In this systemθ (the temperature), ρ (the density) and p (the pressure) are scalar unknowns andu (the velocity) is ad-component unknown vector field. The pressure can actually be eliminated from the equations by applying to the momentum equation the projector P onto the space of divergence free vector fields. This projector is bounded over H_x<sup>`</sup> for all `, and in L^p_x for all 1 < p < ∞. To define the viscosity coefficients, let us introduce the two unique functions Φ (which is a matrix function) and Ψ (which is a vectorial function) orthogonal to KerL such that

M⁻¹²LM¹²Φ= |v|²

d Id−v⊗v and M⁻¹²LM¹²Ψ=v d+ 2

2 − |v|² 2

!

. The viscosity coefficients are then defined (see for instance [BGL91]) by

µ₁ := 1

(d−1)(d+ 2)

Z

Φ :LM¹²ΦM¹² dv

and µ₂ := 2 d(d+ 2)

Z

Ψ·LM¹²ΨM¹² dv . Before stating our main results, let us mention that Appendix B.3 provides some useful results on the Cauchy problem for (1.6).

Theorem 1.1. — Let` > d/2andk > d/2 +γ be given and consider(ρ<sub>in</sub>, u<sub>in</sub>, θ<sub>in</sub>) in H<sup>`</sup>(Ω) if Ω 6= R² and in H^`(Ω)∩L¹(Ω) if Ω = R². If Ω = T^d, we furthermore assume that ρin, uin, θin are mean free. Define

(1.7) ρ¯_in := 2

d+ 2ρ_in− d

d+ 2θ_in, u¯_in=Pu_in, θ¯_in:=−ρ¯_in.

Let(ρ, u, θ)be the unique solution to(1.6)associated with the initial data( ¯ρ_in,u¯_in,θ¯_in) on a time interval [0, T]. Set

(1.8) g¯_in(x, v) :=M¹²(v)

¯

ρ_in(x) + ¯u_in(x)·v+1

2(|v|²−d)¯θ_in(x)

, and define on [0, T]×Ω×R^d

(1.9) g(t, x, v) :=M¹²(v)

ρ(t, x) +u(t, x)·v+1

2(|v|²−d)θ(t, x)

.

The well prepared case. — Assume Ω =T^d or R^d, d = 2,3. There is ε₀ >0 such that for allε 6ε₀ there is a unique solutiong^ε to (1.3)in L^∞([0, T], X^`,k)with initial data g¯_in, and it satisfies

(1.10) lim

ε→0

g^ε−g

L^∞([0,T],X^`,k)= 0.

Moreover, if the solution (ρ, u, θ) to (1.6) is defined onR⁺, then ε₀ depends only on the initial data and not on T and there holds

ε→0lim

g^ε−g

L^∞(R⁺,X^`,k) = 0.

The ill prepared case. — Assume Ω =R^d, d= 2,3. For all initial data g_in in X^`,k satisfying

ρin(x) =

Z

R^d

gin(x, v)M¹²(v) dv , u_in(x) =

Z

R^d

v g_in(x, v)M¹²(v) dv , θ_in(x) = 1

d

Z

R^d

(|v|² −d)g_in(x, v)M¹²(v) dv ,

there is ε₀ > 0 such that for all ε 6 ε₀ there is a unique solution g^ε to (1.3) inL^∞([0, T], X^`,k)with initial data g_in. It satisfies for all p >2/(d−1)

(1.11) lim

ε→0

g^ε−g

L^∞([0,T],X^{`,k</sup>)+L<sup>p</sup>(R<sup>+</sup>,L<sup>∞,k</sup><sub>v</sub> (W<sub>x</sub><sup>`,∞}+H_x^`)(R^d))= 0.

Moreover, if the solution (ρ, u, θ) to (1.6) is defined onR⁺, then ε₀ depends only on the initial data and not on T and there holds

limε→0

g^ε−g

L^∞(R⁺,X^{`,k</sup>)+L<sup>p</sup>(R<sup>+</sup>,L<sup>∞,k</sup>v (Wx<sup>`,∞}+H_x^`)(R^d)) = 0.

Notice that the last assumption (that the solution (ρ, u, θ) to (1.6) is defined onR⁺) always holds when d = 2 and is also known to hold for small data in dimension 3 or without any smallness assumption in some cases (see examples in [CG06] in the periodic case, [CG10] in the whole space for instance): see Appendix B.3 for more on (1.6).

Remark 1.2. — We choose initial data for (1.3) which does not depend on ε, but it is easy to modify the proof if the initial data is a family depending onε, as long as it is compact inX^`,k.

Remark 1.3. — In the case of R², we have made the additional assumption that our initial data lie inL¹(Ω). Actually, it would be enough to suppose thatρ_in, u_in, θ_in are inL¹(Ω).

Remark 1.4. — Let us mention that if we work with smooth data, we can obtain a rate of convergence of ε¹² in (1.10) and (1.11), which is probably not the optimal rate.

Remark 1.5. — As noted in [LZ01], the original solution to the Boltzmann equa- tion, constructed asf^ε(t, x, v) =M(v) +εM¹²(v)g^ε(t, x, v), is nonnegative under our assumptions, as soon as the initial data is nonnegative (which is an assumption that can be made in the statement of Theorem 1.1).

The proof of the Theorem 1.1 mainly relies on a fixed point argument, which enables us to prove that the equation satisfied by the difference h^ε between the solutiong^ε of the Boltzmann equation and its expected limit g does have a solution (which is arbitrarily small) as long as g exists. In order to develop this fixed point argument, we have to filter the unknownh^ε by some well chosen exponential function which depends on the solution to the Navier–Stokes–Fourier equation. This enables us to obtain a contraction estimate. Let us also point out that the analysis of the operators that appear in the equation on h^ε is akin to the one made by Bardos and Ukai [BU91] and it relies heavily on the Ellis and Pinsky decomposition [EP75]. In the case of ill-prepared data, the fixed point argument needs some adjusting. Indeed the linear propagator consists in two classes of operators, one of which vanishes identically when applied to well-prepared case, and in general decays to zero in an averaged sense in time due to dispersive properties. Consequently, we choose to apply the fixed point theorem not to h^ε but to the difference betweenh^ε and those dispersive-type remainder terms. This induces some additional terms to estimate, which turn out to be harmless thanks to their dispersive nature.

2. Main steps of the proof of Theorem 1.1

2.1. Main reductions

Giveng_in∈X^{`,k</sup>, the classical Cauchy theory on the Boltzmann equation recalled in Appendix B.2 states that there is a timeT<sup>ε</sup>and a unique solutiong<sup>ε</sup>inC<sup>0</sup>([0, T<sup>ε</sup>], X<sup>`,k}) to (1.3) associated with the datag_in. The proof of Theorem 1.1 consists in proving that the life span ofg^ε is actually at least that of the limit system (1.6) by proving the convergence result (1.10). Our proof is based on a classical fixed point argument, of the following type.

Lemma 2.1. — Let X be a Banach space, let L be a continuous linear map fromX toX, and let B be a bilinear map fromX×X toX. Let us define

kLk:= sup

kxk=1

kLxk and kBk := sup

kxk=kyk=1

kB(x, y)k. IfkLk<1, then for any x₀ inX such that

kx₀k_X < (1− kLk)² 4kBk the equation

x=x₀+Lx+B(x, x)

has a unique solution in the ball of center0and radius ^1−kLk_2kBk and there is a constantC₀ such that

kxk6C₀kx₀k.

We are now going to give a formulation of the problem which falls within this framework. To this end, let us introduce the integral formulation of (1.3)

(2.1) g^ε(t) = U^ε(t)g_in+ Ψ^ε(t)g^ε, g^ε

whereU^ε(t) denotes the semi-group associated with−¹_εv·∇_x+_ε¹2L(see [BU91, Uka86]

as well as Appendix A) and where (2.2) Ψ^ε(t)(f₁, f₂) := 1

ε

Z t 0

U^ε(t−t⁰)Γf₁(t⁰), f₂(t⁰)dt⁰,

with Γ defined in (1.4). It follows from the results and notations recalled in Ap- pendix A (in particular Remark A.5) that given ¯gin ∈ X^`,k of the form (1.8) the functiong defined in (1.9) satisfies

(2.3) g(t) = U(t)¯g_in+ Ψ(t)(g, g),

where as explained in the rest of the paper, the operators U(t) and Ψ(t) (defined respectively in Remarks A.2 and A.5) are in some sense the limiting operators ofU^ε(t) and Ψ^ε(t). Formulation (2.3) is thus a way to reformulate the fluid equation in a kinetic fashion.

It will be useful in the following to assume that gin and ¯gin are as smooth and decaying as necessary inx. So we consider families (ρ^η_in, u^η_in, θ^η_in)_η∈(0,1) in the Schwartz class S_x, as well as (g_in^η)_η∈(0,1) and (¯g^η_in)_η∈(0,1) related by

(2.4) g¯^η_in(x, v) =M¹²(v)

¯

ρ^η_in(x) + ¯u^η_in(x)·v +1

2(|v|² −d)¯θ^η_in(x)

with (¯ρ^η_in,u¯^η_in,θ¯_in^η) defined by notation (1.7), with

(2.5)

ρ^η_in(x) =

Z

R^d

g_in^η(x, v)M¹²(v) dv , u^η_in(x) =

Z

R^d

v g_in^η(x, v)M¹²(v) dv , θ_in^η(x) = 1

d

Z

R^d

(|v|² −d)g^η_in(x, v)M¹²(v) dv ,

and such that

(2.6) ∀η∈(0,1), g_in^η ,g¯^η_in∈ Sx,v and kδ_in^ηk`,k +kδ¯<sub>in</sub><sup>η</sup>k`,k 6η , with δ_in^η :=g^η_in−g_in and δ¯_in^η := ¯g_in^η −g¯_in.

If Ω = R², we furthermore assume, recalling that (ρ_in, u_in, θ_in) belong to H_x^` ∩L¹_x, that

(2.7) kδ_in^ηk_L²

vL¹_x 6η .

Thanks to the stability of the Navier–Stokes–Fourier equation recalled in Appen- dix B.3 we know that

(2.8) g^η(t) :=U(t)¯g_in^η + Ψ(t)(g^η, g^η) satisfies

(2.9) lim

η→0

g^η −g

L^∞([0,T],X^`,k) = 0, uniformly inT if the solutiong is global. Moreover setting (2.10) g^ε,η :=g^ε+δ^ε,η, δ^ε,η(t) :=U^ε(t)δ^η_in there holds

(2.11) g^ε,η(t) =U^ε(t)g_in^η + Ψ^ε(t)g^ε,η−δ^ε,η, g^ε,η−δ^ε,η.

Thanks to (2.7) and the continuity of U^ε(t) recalled in Lemma 3.1 we know that (2.12) kδ^ε,ηk_L^∞_(R⁺_,X^`,k₎ .η

hence with (2.9) it is enough to prove the convergence results (1.10) and (1.11) withg^ε and g respectively replaced by g^ε,η and g^η (the parameter η will be converging to zero uniformly inε). Indeed we have the following inequality

kg^ε−gk_L^∞_([0,T],X`,k)

6kδ^ε,ηk_L^∞_(R+,X<sup>`,k</sup>)+kg−g<sup>η</sup>k<sub>L</sub><sup>∞</sup><sub>([0,T],X</sub>`,k)+kg^ε,η−g^ηk_L^∞_([0,T],X`,k), which is uniform in time if g_in (and hence also g_in^η if η is small enough, thanks to Proposition B.5) generates a global solution to the limit system. In order to achieve this goal let us now write the equation satisfied byg^ε,η−g^η. Our plan is to conclude thanks to Lemma 2.1, however there are two difficulties in this strategy. First, linear terms appear in the equation on g^ε,η−g^η, whose operator norms are of the order of norms of g^η which are not small. Those linear operators therefore do not satisfy the assumptions of Lemma 2.1. In order to circumvent this difficulty we shall introduce weighted Sobolev spaces, where the weight is exponentially small in g^η in order for the linear operator to become a contraction. The second difficulty in the ill-prepared case is that the linear propagatorU^ε−U acting on the initial data can be decomposed into several orthogonal operators (as explained in Appendix A), some of which vanish in the well-prepared case only, and are dispersive (but not small in the energy space) in the ill-prepared case. These terms need to be removed fromg^ε,η−g^η if one is to apply the fixed point lemma in the energy space. All these reductions are carried out in the following Lemma 2.2, where we prepare the problem so as to apply Lemma 2.1.

Lemma 2.2. — Letr > 4 andλ >0 be given. With the notation introduced in Lemma A.1 Remark A.2 set

δ^ε,η(t) := U_disp^ε (t)g_in^η +U^ε](t)g_in^η and δ^eε,η(t) := U^ε(t)(g_in^η −¯g_in^η)−δ^ε,η(t). Finally set

g^ε,η :=g^η+δ^ε,η and define h^ε,η_λ as the solution to the equation

(2.13) h^ε,η_λ (t) =D^ε_λ(t) +L^ε_λ(t)h^ε,η_λ (t) + Φ^ε_λ(t)(h^ε,η_λ , h^ε,η_λ )

where (dropping the dependence on η on the operators to simplify) we have written

(2.14)

D^ε_λ(t) :=e^−λR

t

0kg^ε,η(t⁰)k^r_`,kdt⁰D^ε(t)

D^ε(t) :=δ^eε,η+U^ε(t)−U(t)g¯^η_in+Ψ^ε(t)−Ψ(t)(g^η, g^η) + 2Ψ^ε(t)

g^η+ 1

2δ^ε,η−δ^ε,η, δ^ε,η

+ Ψ^ε(t)δ^ε,η−2g^η, δ^ε,η L^ε_λ(t)h:= 2Ψ^ε_λ(t)(g^ε,η−δ^ε,η, h) with

Ψ^ε_λ(t)(h₁, h₂) := 1 ε

Z t 0

e^−λR

t

t0kg^ε,η(t⁰⁰)k^r_`,kdt⁰⁰

U^ε(t−t⁰)Γ(h₁, h₂)(t⁰) dt⁰ and Φ^ε_λ(t)(h₁, h₂) := 1

εe^λ

Rt

0kg^ε,η(t⁰)k^r_`,kdt⁰Z t 0

e^−2λ

Rt

t0kg^ε,η(t⁰⁰)k^r_`,kdt⁰⁰

×U^ε(t−t⁰)Γ(h₁, h₂)(t⁰) dt⁰.

Then to prove Theorem 1.1, it is enough to prove the following convergence results:

In the well-prepared case, forλ large enough

η→0limlim

ε→0

h^ε,η_λ

L^∞([0,T],X^`,k)= 0

and in the ill-prepared case for all p > 2/(d−1)and for λ large enough limη→0lim

ε→0

h^ε,η_λ

L^∞([0,T],X^{`,k</sup>)+L<sup>p</sup>(R<sup>+</sup>,L<sup>∞,k</sup>v (W<sup>`,∞}+H_x^`)(R^d))) = 0,

where the convergence is uniform in T if g¯_in^η gives rise to a global unique solution.

Proof. — Let us set, with notation (2.8) and (2.10), he^ε,η:=g^ε,η−g^η

which satisfies the following system in integral form, due to (2.8) and (2.11) (2.15) ^eh^ε,η(t) = D^eε(t) +L^eε(t)h^eε,η+ Ψ^ε(t)(^eh^ε,η,^eh^ε,η)

where

De^ε(t) := U^ε(t)(g_in^η −¯g_in^η) +U^ε(t)−U(t)¯g_in^η + Ψ^ε(t)(δ^ε,η, δ^ε,η)

−2Ψ^ε(t)(g^η, δ^ε,η) +Ψ^ε(t)−Ψ(t)(g^η, g^η) Le^ε(t)h:= 2Ψ^ε(t)(g^η−δ^ε,η, h).

The conclusion of Theorem 1.1 will be deduced from the fact that^eh^ε,η goes to zero inL^∞([0, T], X^{`,k</sup>) (resp. in the spaceL<sup>∞</sup>([0, T], X<sup>`,k})+L^p(R⁺, L^∞,k_v (W^{`,∞</sup>+H<sub>x</sub><sup>`})(R^d)) in the well-prepared case (resp. in the ill-prepared case).

In order to apply Lemma 2.1, we would need the linear operator L^eε appearing in (2.15) to be a contraction in L^∞([0, T], X^{`,k</sup>), and the term D<sup>eε</sup>(t) to be small in L<sup>∞</sup>([0, T], X<sup>`,k}). It turns out that in the R²-case, to reach this goal, we have to introduce a weight in time (note that in the references mentioned above in this context, only the three-dimensional case is treated, in which case it is not necessary to introduce that weight). We thus introduce a function χ_Ω(t) defined by

∀t∈R⁺, χ_Ω(t) :=







1 if Ω = T^d, d= 2,3, orR³, hti¹⁴ if Ω = R².

For a givenT > 0 we define the associate weighted in time space X_T^`,k :=

f =f(t, x, v)/ f ∈L^∞1_[0,T](t)χ_Ω(t), X^`,k

endowed with the norm

kfk_X^`,k

T

:= sup

t∈[0,T]

χΩ(t)kf(t)k`,k.

In order to apply Lemma 2.1, we then need the term D^eε(t) to be small in X_T^{`,k</sup>. Concerning this fact, it turns out that the first term inD<sup>eε</sup>(t) namelyU<sup>ε</sup>(t)(g<sub>in</sub><sup>η</sup> −g¯<sub>in</sub><sup>η</sup>), which is small (in fact zero) in the well-prepared case since g<sub>in</sub><sup>η</sup> = ¯g<sub>in</sub><sup>η</sup>, contains in the case of ill-prepared data, a part which is not small in X<sub>T</sub><sup>`,k} but in a different space:

that is

δ^ε,η(t) =U_disp^ε (t)g_in^η +U^ε](t)g^η_in.

This is stated (among other estimates on ¯δ^ε,η) in the following Lemma 2.3, which is proved in Section 3.3.

Lemma 2.3. — Letp∈(1,∞] andΩ =R^d. There exist a constant C such that for all η∈(0,1) and allε ∈(0,1),

(2.16) δ^ε,η

L^p(R⁺,X^`,k) 6C .

Moreover there is a constant C such that for all η∈(0,1)and all ε∈(0,1) (2.17) U^ε](t)g_in^η

`,k 6Ce^−αεt2

where α is the rate of decay defined in (A.3), and for all η ∈ (0,1) there is a constant C_η such that for all ε∈(0,1)

(2.18) U_disp^ε (t)g_in^η

L^∞,kv Wx^`,∞ 6C_η

1∧(εt)^d−12

and U_disp^ε (t)g^η_in

`,k 6 C_η hti^d4 · In particular δ^ε,η satisfies for all η∈(0,1)

ε→0lim

δ^ε,η

X_∞^`,k 6C_η and lim

ε→0

δ^ε,η

L^p(R⁺,L^∞,kv (Wx^{`,∞</sup>+H<sub>x</sub><sup>`})(R^d)) = 0,

∀p∈(2/(d−1),∞).

Returning to the proof of Lemma 2.2, let us set

h^ε,η :=^eh^ε,η−δ^ε,η, g^ε,η :=g^η +δ^ε,η, and notice thath^ε,η satisfies the following system in integral form (2.19) h^ε,η(t) = D^ε(t) +L^ε(t)h^ε,η+ Ψ^ε(t)(h^ε,η, h^ε,η) with

D^ε(t) :=δ^eε,η+U^ε(t)−U(t)g¯_in^η +Ψ^ε(t)−Ψ(t)(g^η, g^η) + 2Ψ^ε(t)

g^η+ 1

2δ^ε,η−δ^ε,η, δ^ε,η

+ Ψ^ε(t)δ^ε,η−2g^η, δ^ε,η L^ε(t)h:= 2Ψ^ε(t)(g^ε,η−δ^ε,η, h)

with Ψ^ε(t)(h₁, h₂) := 1 ε

Z t 0

U^ε(t−t⁰)Γ(h₁, h₂)(t⁰) dt⁰.

In order to apply Lemma 2.1, we need L^ε to be a contraction, so we introduce a modified space, in the spirit of [CG10], in the following way. Sinceg^η andδ^ε,η belong toL^∞([0, T], X^`,k), then for all 26r 6∞, there holds

(2.20) g^ε,η:=δ^ε,η+g^η ∈L^r([0, T], X^`,k)

with a norm depending on T. Moreover as recalled in Proposition B.5, if the unique solution to (1.6) is global in time then in particular

(2.21) g^η ∈L^r(R⁺, X^`,k), ∀r >4.

So thanks to (2.16) we can fixr ∈(4,∞) from now on and define for all λ >0 h^ε,η_λ (t) :=h^ε,η(t) exp

−λ

Z t 0

kg^ε,η(t⁰)k^r_`,k dt⁰

.

The quantity appearing in the exponential is finite thanks to (2.16) and (2.21). The parameter λ > 0 will be fixed, and tuned later for L^ε to become a contraction.

Then h^ε,η_λ satisfies

h^ε,η_λ (t) =D^ε_λ(t) +L^ε_λ(t)h^ε,η_λ (t) + Φ^ε_λ(t)(h^ε,η_λ , h^ε,η_λ )

with the notation (2.14). This concludes the proof of the Lemma 2.3.

2.2. End of the proof of Theorem 1.1

The following results, together with Lemma 2.1, are the key to the proof of Theo- rem 1.1. They will be proved in the next sections.

Proposition 2.4. — Under the assumptions of Theorem 1.1, there is a con- stantC such that for all T >0, η >0and λ >0

limε→0kL^ε_λ(t)hk_X^`,k

T 6C

1 λ^1r +η

khk_X^`,k

T .

Proposition 2.5. — Under the assumptions of Theorem 1.1, there is a con- stantC such that for all T >0, η >0, ε >0and λ>0

Φ^ε_λ(t)(f₁, f₂)

X_T^`,k 6Cexp

λ

Z T 0

k¯g^ε,η(t)k^r_`,kdt

kf₁k_X^`,k

T kf₂k_X^`,k

T .

Proposition 2.6. — Under the assumptions of Theorem 1.1, there holds uni- formly in λ>0(and uniformly in T if ¯g_in^η gives rise to a global unique solution)

η→0limlim

ε→0

D_λ^ε(t)

X_T^`,k = 0.

Proof. — Assuming those results to be true, let us apply Lemma 2.1 to equa- tion (2.13) andX =X_T^{`,k</sup>, withx<sub>0</sub> =D<sub>λ</sub><sup>ε</sup>,L =L<sup>ε</sup><sub>λ</sub> andB= Φ<sup>ε</sup><sub>λ</sub>. Proposition 2.5, (2.16) along with (2.21) ensure that Φ<sup>ε</sup><sub>λ</sub> is a bounded bilinear operator over X<sub>T</sub><sup>`,k}, uniformly inT if ¯g_in^η gives rise to a global unique solution. Moreover choosing λlarge enough,ε small enough (depending onη, and on T except if ¯g_in^η gives rise to a global unique solution) and η small enough uniformly in the other parameters, Proposition 2.4 ensures that L^ε_λ is a contraction in X_T^{`,k</sup>. Finally thanks to Proposition 2.6 the as- sumption of Lemma 2.1 onD<sup>ε</sup><sub>λ</sub> is satisfied as soon asεandηare small enough. There is therefore a unique solution to (2.13) in X<sub>T</sub><sup>`,k}, which satisfies, uniformly in T if ¯g_in^η gives rise to a global unique solution,

(2.22) lim

η→0lim

ε→0

h^ε,η_λ

X^`,k

T

= 0.

Thanks to Lemma 2.2, this ends the proof of Theorem 1.1.

To conclude it remains to prove Propositions 2.4 to 2.6 as well as Lemma 2.3. Note that the proofs of Propositions 2.4 to 2.6 are conducted to obtain estimates uniform in T, and this information is actually only useful in the case of global solutions (which is, for example, always the case in dimension 2). Note also that, here and in what follows, we have denoted by A.B if there exists a universal constant C (in particular independent of the parametersT, ε, λ, η) such that A6CB.

Before going into the proofs of Propositions 2.4 to 2.6, we are going to state lemmas about continuity properties ofU^ε(t) and Ψ^ε(t) in the next section that are useful in the rest of the paper.

3. Estimates on U

(t) and Ψ

(t)

Let us mention that some of the following results (Lemmas 3.1, 3.2 and 3.7) have already been proved in some cases (see [BU91]) but for the sake of completeness, we write the main steps of the proofs in this paper, especially because theR²-case is not always clearly treated in previous works. The conclusions of the following lemmas hold for Ω =T^d orR^d with d= 2,3 unless otherwise specified.

3.1. Estimates on U^ε(t)

Lemma 3.1. — Let ` > 0 and k > d/2 be given. Then for all ε >0, the opera- tor U<sup>ε</sup>(t)is a strongly continuous semigroup on X<sup>`,k</sup> and there is a constantC such that for all ε∈(0,1)and all t>0

(3.1) kU^ε(t)fk_{`,k</sub> 6Ckfk<sub>`,k}, ∀f ∈X^`,k.