Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff

Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff

L. Desvillettes and B. Wennberg March 3, 2003

Abstract

For regularized hard potentials cross sections, the solution of the spatially ho- mogeneous Boltzmann equation without angular cutoff lies in Schwartz’s space S(R^N). The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role.

1 Introduction

We consider in this work the spatially homogeneous Boltzmann equation (Cf. [3])

∂f

∂t(t, v) =Q(f, f)(t, v), (1) wheref :R^N →R+is the nonnegative density of particles which at timetmove with velocityv, and the bilinear operator in the right-hand side is defined by

Q(g, f)(v) = Z

R^N

Z

S^N−1

f(v⁰)g(v⁰_∗)−f(v)g(v_∗)

b(cosθ, v−v_∗)dσdv_∗. (2) In this formula,v⁰, v⁰_∗andv, v_∗are the velocities of a pair of particles before and after a collision,

v⁰ = v+v_∗

2 +|v−v_∗| 2 σ, v_∗⁰ = v+v_∗

2 −|v−v_∗| 2 σ,

whereσ∈S^N−1. Throughout the paper,θdenotes the angle betweenσandv−v_∗.

Our goal is to prove that for a large class of collision cross sections b, and for all initial dataf_in ≥0with finite mass, energy and entropy, i.e. satisfying the

Assumption 1 : Z

R^Nfin(v) 1 +|v|²+|log(fin(v))|

dv <+∞,

there exists a solution of the homogeneous Boltzmann equation (1) withf(0,·) = fin

such that whent > 0, f(t,·) ∈ S(R^N). Apart from certain technical conditions that are discussed below, the main assumption onbis that near θ = 0, it looks like |θ|^−γ, with1 < γ < 3. Such a behavior, which naturally appears when the interaction be- tween the particles has a long range, is called “non cutoff”. It means that all the grazing collisions (those for whichθis close to0) are taken into account. Under such a condi- tion, the collision operator is expected to behave essentially as a fractional power of the Laplacian:

Q(f, f) =−Cf(−∆)^(γ−1)/2f+more regular terms, whereC_f >0depends only on quantities which are somehow controlled.

Relevant existence results were obtained in [2], [13], and [8]. Previous works have demonstrated partial regularity under rather general conditions (Cf. [1], [5], [6], [9], [10], [13]) or C^∞ regularity as here, but under severe restrictions on the equation : namely, the cross section did not depend onv−v_∗(Maxwellian molecules assumptions) and the solution was radially symmetric (Cf. [4]).

In order to keep the paper rather short, we refer to the quoted papers for a more complete history of the problem, and for discussions on the physical relevance.

The precise conditions that we impose on the cross sectionb=b(cosθ, w)are the following:

Assumption 2 : We suppose that

w∈R^Ninf, θ∈[−δ,δ]

|θ|^γb(cosθ, w)

(1 +|w|²)^α ≥K (3)

for someK >0,α∈]0,1[,γ ∈]1,3[,δ ∈]0, π[, and that for allp∈N, sup

w∈R^N, θ∈[−π,π]

|θ|^3−ε|D_pb(cosθ, w)|

(1 +|w|²)^rp ≤C_p¹, (4) whereε >0is a given number,r_p, C_p¹ >0are given constants, andD_pis any derivative of orderp with respect to the variable w.

Note that the usual regularized hard potentials without angular cutoff satisfy As- sumption 2; those are cross sections of the form

b(cosθ, w) =b₁(w)b₂(θ),

whereb₁ is a smooth and strictly positive function such that b₁(w) ∼w→+∞ |w|^α for α∈]0,1[andb₂is a function such thatb₂(θ)∼θ→0|θ|^−γforγ ∈]1,3[.

We can now state our main theorem :

Theorem 1 : Letb be a cross section satisfying Assumption 2 and f_in be an initial datum satisfying Assumption 1. Then, there exists a solution to eq. (1), (2) with initial datumfinlying inL^∞([t₀,+∞[;S(R^N))for allt₀>0.

Let us now rapidly discuss the assumptions and the conclusion of this theorem. Note first that the initial data are only assumed to belong to the space of functions satisfying the natural bounds coming from physics (finite mass, energy and entropy). It is likely that the assumption of finite entropy can be somewhat relaxed (Cf. recent works by Villani (Cf. [11])). The assumptions on the dependence with respect toθof the cross section are also quite satisfying, and probably close to being optimal (Cf. [1] to get an idea of what really optimal assumptions might be). The situation is however not so good as far as the kinetic part of the cross section (that is, its dependence with respect tov−v_∗) is concerned. First, we basically assume that this dependence is smooth, and this is not true for inverse power laws (Cf. [3] for example). While for such a cross section (having a singularity nearv=v_∗) some smoothing effect certainly occurs (Cf. [1] for example), it is not clear whether a complete smoothing of the solution can appear (the study of the Landau equation with that kind of cross section would suggest that the complete smoothing should indeed appear, Cf. [7]). Secondly, we also assume that (the kinetic part of) the cross section is strictly positive. This is also not true for inverse power laws.

The study of this problem in [1] suggests that this assumption of strict positivity could maybe be relaxed. Most probably however, to look for a result for “true” hard potentials (that is, coming from inverse power laws) would lead to tremendous technicalities (the proof of our theorem is already quite technical), which we leave to future works.

Finally, we would like to put the stress on the following facts: the conclusion of Theorem 1 most probably does not hold when soft potentials or Maxwellian molecules are concerned (no gain of moments is expected in such a situation) or when the singular- ity in the angular variable is removed (no gain of smoothness is expected in this case).

Under our assumptions, we think that maybe the solution of the Boltzmann equation is even smoother thanS, it might belong to some Gevrey space for example.

Since our computations are rather long, we first present the proof of Theorem 1 in the case when the dimension isN = 2and when in Assumption 2, eq. (4), the term

|θ|^3−εis replaced by|θ|^2−ε(that is, the singularity in the angular variable is moderately strong). These simplifications enable to present complete proofs: this is done in Sec- tion 2. Then, in Section 3, we explain how to modify the proof to get the result in the general case. Finally, for the sake of completeness, we give in an appendix the proof of a (more or less standard) interpolation lemma which is crucial in our proof.

2 The simplified case

In this section, we prove Theorem 1 in the case when the dimension isN = 2and when eq. (4) in Assumption 2 is replaced by

sup

w∈R², θ∈[−π,π]

|θ|^2−ε|Dpb(cosθ, w)|

(1 +|w|²)^rp ≤C_p¹, (5) where (as in eq. (4))ε >0is a given number,r_p, C_p¹ > 0are given constants, andD_p is any derivative of orderp with respect tow.

When velocities are restricted to two dimensions, it is possible to parametrize the pre- and postcollisional velocities by a rotation of the relative velocityv−v_∗:

v⁰ = v+v_∗ 2 +R_θ

v−v_∗ 2

, (6)

v_∗⁰ = v+v_∗ 2 −R_θ

v−v_∗ 2

, (7)

whereR_θdenotes a rotation by the angleθ. Then the integral overS^N−1 in (2) can be replaced byR_π

−πdθ, which simplifies many of the subsequent calculations.

Thanks to the change of variableθ7→θ±π, which exchangesv⁰andv_∗⁰, the collision operator can be written

Q(f, f)(v) = Z

R²

Z π/2

−π/2

f(v⁰)f(v⁰_∗)−f(v)f(v_∗)

×

b(cosθ, v−v_∗) +b(cos(θ−π), v−v_∗) 1_[0,π/2](θ) +b(cos(θ+π), v−v_∗) 1_[−π/2,0](θ)

dθdv_∗.

As a consequence, it is enough (as far as the Boltzmann equation ∂_tf = Q(f, f) is concerned) to assume thatb(cos(·), v−v_∗)has its support included in[−π/2, π/2].

We shall use in the sequel the following (easy) consequences of eq. (5) (C_p²andC_p³ are constants which depend only onC_p¹) :

sup

w∈R²

DpRπ/2

−π/2

b cosθ, ^w

cos^θ₂

₁

(cos^θ₂)² −b(cosθ, w)

dθ

(1 +|w|²)^rp ≤C_p², (8)

sup

w∈R², θ∈[−π/2,π/2]

|θ|^2−ε

D_pb cosθ, ^w

cos^θ₂

(1 +|w|²)^rp ≤C_p³. (9) The proof of Theorem 1 (under the assumptions of this section) runs as follows: in Section 2.1, we split quantities like

Z

R²(D_kQ(g, f))(v)D_kf(v)dv

(whereD_kis any derivative of orderk) in a certain number of terms. Each term is then estimated in Section 2.2. Finally, we gather the estimates in Section 2.3 to conclude the proof of our theorem.

2.1 Decomposition

First, we observe that (6) and (7) imply the following useful formulas : v= 1

cos₂^θ R₋θ

2v⁰+ tanθ 2R^π

2v_∗, (10)

v⁰−v= sinθ 2Rθ

2+^π₂(v−v_∗), (11)

v⁰−v_∗= cosθ 2Rθ

2(v−v_∗). (12)

It is in these formulas that the simplifications related to the two-dimensional case are obvious: For a fixed angle of deflectionθ, all relations between velocities before and after collision are given by some fixed linear operator, and hence these transformations are smooth.

Denoting byτ_hthe translation operator (that is,τ_hf(x) =f(x+h)), we write down the invariance ofQwith respect to translations in the form

(τ_hQ(g, f))(v) =Q(τ_hg, τ_hf)(v).

In the following we denote any generic differential operator of orderkbyD_k, and when it is necessary to indicate which variables it is acting on, we writeD_k;v, for example.

A consequence of the translation invariance is that a Leibnitz formula holds for the collision operator: for any derivativeD_kof orderk, there exist derivativesD_lof orderl such that

(D_kQ(g, f))(v) =Q(g, D_kf)(v) +

k−1

X

l=0

C_k^lQ(D_k−lg, D_lf)(v), and

Z

R²(D_kQ(g, f))(v)D_kf(v)dv = Z

R²D_kf(v)Q(g, D_kf)(v)dv+

k−1

X

l=0

C_k^lB_kl, (13) where

B_kl = Z

R²D_kf(v)Q(D_k−lg, D_lf)(v)dv. (14) We then use the pre/post-collisional change of variables v, v_∗, θ 7→ v⁰, v_∗⁰,−θ, and get

B_kl = Z

R²

Z

R²

Z π/2

−π/2D_kf(v)

D_lf(v⁰)D_k−lg(v_∗⁰)

−Dlf(v)Dk−lg(v_∗)

b(cosθ, v−v_∗)dθdv_∗dv (15)

= Z

R²

Z

R²

Z _π/2

−π/2

D_kf(v⁰)−D_kf(v)

D_lf(v)D_k−lg(v_∗)b(cosθ, v−v_∗)dθdv_∗dv.

The next step in our calculation is to carry out the change of variablesw=v⁰(with the variablesθandv_∗ fixed; this has been used eg. in [1]). From (10) and (12), we get the following formulas,

w= v+v_∗ 2 +R_θ

v−v_∗ 2

, (16)

v= 1 cos^θ₂ R₋θ

2w+ tanθ 2R^π

2v_∗, (17)

|w−v_∗|= cosθ

2|v−v_∗|, (18)

dw= (cosθ

2)²dv, (19)

which then give B_kl=

Z

R²

Z

R²

Z _π/2

−π/2D_kf(w)D_lf 1 cos^θ₂ R₋θ

2w+ tanθ 2Rπ

2v_∗

×Dk−lg(v_∗)b cosθ,w−v_∗ cos^θ₂

1

(cos^θ₂)² dθdv_∗dw

− Z

R²

Z

R²

Z _π/2

−π/2Dkf(v)Dlf(v)Dk−lg(v_∗)b(cosθ, v−v_∗)dθdv_∗dv.

Writing thenvinstead ofwand using the notation

˜ v= 1

cos^θ₂ R₋θ

2v+ tanθ 2R^π

2v_∗, we get

B_kl= Z

R²

Z

R²

Z _π/2

−π/2D_kf(v)D_k−lg(v_∗)

×

Dlf(˜v)b cosθ,v−v_∗ cos^θ₂

1

(cos^θ₂)² −Dlf(v)b(cosθ, v−v_∗)

dθdv_∗dv.

Next, the termsB_klare split asB_kl=B_kl¹ +B²_kl, where B_kl¹ =

Z

R²

Z

R²

Z _π/2

−π/2D_kf(v) [D_lf(˜v)−D_lf(v)]D_k−lg(v_∗)

×b cosθ,v−v_∗ cos^θ₂

1

(cos^θ₂)²dθdv_∗dv B_kl² =

Z

R²

Z

R²

Z π/2

−π/2D_kf(v)D_lf(v)D_k−lg(v_∗)

×

b cosθ,v−v_∗ cos^θ₂

1

(cos^θ₂)² −b(cosθ, v−v_∗)

dθdv_∗dv.

2.2 Estimates

We introduce the weightedL^p and Sobolev spaces and their following norms:

Definition 1 : For allp∈[1,+∞[andr >0, we define the spaceL^pr(R^N)by its norm

||f||^p_Lp r(R^N)=

Z

R^N|f(x)|^p(1 +|x|²)^rp/2dx. (20)

For allk ∈ Nand r > 0, we also define the weighted Sobolev spaceH_r^k(R^N)by its norm

||f||²_Hk

r(R^N)= X

|α|≤k

Z

R^N|∂_αf(x)|²(1 +|x|²)^rdx. (21) We begin with an estimate of the termsB_kl² (k∈N,l= 0,1, .., k−1).

Lemma 1 : Fork∈N,l= 0,1, .., k−1, one has the estimate

|B_kl²| ≤C_p²||g||_L¹_2rp||f||²_Hk

rp, (22)

for some constantC_p² >0.

Proof :We recall that B_kl² =

Z

R²

Z

R²

Z _π/2

−π/2Dkf(v)Dlf(v)Dk−lg(v_∗) (23)

×

b cosθ,v−v_∗ cos^θ₂

1

(cos^θ₂)² −b(cosθ, v−v_∗)

dθdv_∗dv,

so that (afterk−l integrations by part, and denoting by D_k−l;2 a derivative of order k−lwith respect to the second variable (v−v_∗))

|B_kl²| ≤ Z

R²

Z

R²Dkf(v)Dlf(v)g(v_∗)

×Dk−l;2

Z _π/2

−π/2

b cosθ,v−v_∗ cos^θ₂

1

(cos^θ₂)² −b(cosθ, v−v_∗)

dθ dv_∗dv

≤C_p² Z

R²

Z

R²|Dkf(v)| |Dlf(v)|g(v_∗) (1 +|v−v_∗|²)^rpdv dv_∗

≤C_p²||g||L¹_2rp ||f||²_Hrpk .

We now study the term B_kl¹. Integrating by parts k−l times with respect to the variablev_∗, we get

B_kl¹ =(−1)^k−l Z

R²

Z

R²

Z _π/2

−π/2D_kf(v)g(v_∗)

× D_k−l;v∗

[D_lf(˜v)−D_lf(v)]b cosθ,v−v_∗ cos^θ₂

1

(cos^θ₂)² dθdv_∗dv (24)

=

k−l

X

m=0

(−1)^mC_k^m_−lB_klm¹ , with

B_klm¹ = Z

R²

Z

R²

Z _π/2

−π/2Dkf(v)g(v_∗)Dm;v_∗[Dlf(˜v)−Dlf(v)]

×D_{k−l−m;v∗}b cosθ,v−v_∗ cos^θ₂

1

(cos^θ₂)² dθdv_∗dv.

In those formulas,D_r;v∗denotes a derivative of orderrwith respect to the variablev_∗.

Lemma 2 : We suppose thatk∈ N,l = 0, .., k−1, andm= 1, .., k−l. Then there exists a constantC_klm⁴ >0such that

|B_klm¹ | ≤C_klm⁴ ||g||_L¹₂

rk−l−m||f||²_Hk rk−l−m. Proof: Note first that for a given functionh, one has (form≥1)

|Dm;v_∗[h(˜v)−h(v)]| ≤ |sinθ

2|^m|Dmh(˜v)|.

Then,

|B_klm¹ | ≤ Z

R²

Z

R²

Z π/2

−π/2|D_kf(v)|g(v_∗)|sinθ

2|^m|D_m+lf(˜v)|

× 1

|cos^θ₂|^k−l−m+2

Dk−l−mb cosθ,v−v_∗

cos^θ₂ dθdv_∗dv

≤ Z _π/2

−π/2|sinθ

2|^m |θ|^ε−2

|cos^θ₂|^k−l−m+2 Z

R²

Z

R²

1 2g(v_∗)

|D_kf(v)|²+|D_m+lf(˜v)|²

×C_k³_−l−m(1 +|v−v_∗|²)^rk−l−mdvdv_∗dθ

≤ C_k³_−l−m 2

Z _π/2

−π/2|sinθ

2|^m |θ|^ε−2

|cos₂^θ|^k−l−m+2 dθ

× Z

R²

Z

R²g(v_∗)|D_kf(v)|²(1 +|v|²)^rk−l−m(1 +|v_∗|²)^rk−l−mdvdv_∗ +C_k³_−l−m

2

Z _π/2

−π/2|sinθ

2|^m |θ|^ε−2

|cos^θ₂|^k−l−m+2 (25)

× Z

R²

Z

R²g(v_∗)|D_m+lf(˜v)|²(1 + (cosθ

2)²|˜v−v_∗|²)^rk−l−m(cosθ

2)²d˜vdv_∗dθ, where the factor(cos^θ₂)² in the last term is simply the Jacobian in the transformation v→˜v(see formula (12)). Finally, we see (recalling thatm+l≤k) that

|B_klm¹ | ≤C_klm⁴ ||g||_L1

2rk−l−m||f||²_Hk rk−l−m, with

C_klm⁴ =C_k³_−l−m Z _π/2

−π/2|sinθ

2|^m |θ|^ε−2

|cos₂^θ|^k−l−m+2 dθ,

and Lemma 2 is proven.

We now turn to the case whenm= 0. We have to estimate the term B_kl¹₀=

Z

R²

Z

R²

Z _π/2

−π/2D_kf(v)g(v_∗) [D_lf(˜v)−D_lf(v)]

× D_k−l;v∗b cosθ,v−v_∗ cos^θ₂

1

(cos^θ₂)² dθdv_∗dv.

Lemma 3 : There exists a constantC_kl⁵ >0such that,

|B_kl¹₀| ≤C_kl⁵ ||g||L¹_2rk−l+1||f||²_Hk rk−l+ 12

.

Proof: We first note that

D_lf(˜v)−D_lf(v) = (˜v−v)· Z ₁

0 ∇D_lf((1−s)v+sv)˜ ds.

Then,

|B_kl¹₀|= Z

R²

Z

R²

Z π/2

−π/2D_kf(v)g(v_∗)

(˜v−v)· Z ₁

0 ∇D_lf((1−s)v+s˜v)ds

×D_k−lb cosθ,v−v_∗ cos^θ₂

1

(cos^θ₂)^k−l+2dθdv_∗dv

≤ Z

R²

Z

R²

Z _π/2

−π/2g(v_∗)1 2

|D_kf(v)|²+ Z ₁

0 |∇D_lf((1−s)v+s˜v)|²ds

× |˜v−v|C_k³_−l(1 +|v−v_∗|²)^rk−l|θ|^ε−2 1

(cos^θ₂)^k−l+2 dθdv_∗dv.

Using formula (11), we see that

|˜v−v|=|sinθ

2| |v˜−v_∗|, so that

|B_kl¹₀| ≤1 2

Z

R²

Z

R²

Z _π/2

−π/2g(v_∗)|D_kf(v)|²|v˜−v_∗|(1 +|v−v_∗|²)^rk−l

× C_k³_−l|sinθ

2| |θ|^ε−2 1

(cos^θ₂)^k−l+2dθdv_∗dv +1

2 Z

R²

Z

R²

Z _π/2

−π/2g(v_∗)|˜v−v_∗| Z ₁

0 |∇D_lf((1−s)v+s˜v)|²ds (26)

× C_k³_−l(1 +|v−v_∗|²)^rk−l|sinθ

2| |θ|^ε−2 1

(cos^θ₂)^k−l+2 dθdv_∗dv.

We now introduce the variableu = (1−s)v+sv. Its Jacobian is given by the˜ formula

du= (1 +s²(tanθ 2)²)dv .

Then,

|B_kl¹₀| ≤ 1 2

Z

R²

Z

R²

Z π/2

−π/2g(v_∗)|D_kf(v)|²

1 +|tanθ 2|

|v_∗|+ 1 cos^θ₂ |v|

×C_k³_−l(1 +|v|²)^rk−l(1 +|v_∗|²)^rk−l|sin^θ₂| |θ|^ε−2

(cos^θ₂)^k−l+2 dθdv_∗dv +1

2 Z

R²

Z

R²

Z _π/2

−π/2

Z ₁

0 g(v_∗)|∇D_lf(u)|²|v˜−v_∗|(1 +|v−v_∗|²)^rk−l

×C_k³_−l|sin^θ₂| |θ|^ε−2 (cos^θ₂)^k−l+2

ds

1 +s²(tan^θ₂)² dθdv_∗du. (27) Noticing that

v=

(1−s)Id+ s cos^θ₂ R₋θ

2

₋₁

u−stanθ 2R^π

2v_∗

, we see that

|v|² ≤4u−stanθ 2R^π

2v_∗²

≤8 (|u|²+|v_∗|²).

In the same way,

˜ v=

s Id+ (1−s) cosθ 2Rθ

2

₋₁

u+ (1−s) sinθ 2Rθ

2+^π₂v_∗

, so that

|v˜|² ≤4u+ (1−s) sinθ 2Rθ

2+^π₂v_∗² ≤8 (|u|²+|v_∗|²). (28) Then, we obtain the estimate

|B_kl¹₀| ≤C_kl⁵ ||g||L¹_2rk−l+1||f||²_Hk rk−l+ 12

, with (for example)

C_kl⁵ = C_k³_−l 2

Z _π/2

−π/2

1 +|tan θ 2|+ 1

cos^θ₂ + 2√

8 (18)^rk−l

|sin^θ₂| |θ|^ε−2

(cos^θ₂)^k−l+2 dθ. (29) This concludes the proof of Lemma 3.

Finally, we estimate the main term (that is, the term R

Q(g, D_kf)(v)D_kf(v)dv, which is crucial for the gain of smoothness). The computations are done here for any dimensionN, since they are identical for all dimensions.

Lemma 4 : LetN ≥2. There exists a constantC₆ >0depending only onf_inand a constantC₇ >0such that

− Z

R^NQ(g, D_kf)(v)D_kf(v)dv ≤ −C₆||D_kf||²_H(γ−1)/2 +C₇||f||L¹₂||f||²_Hk 2 . Proof: We compute (for allg≥0)

− Z

Q(g,D_kf)(v)D_kf(v)dv

=1 2

Z

R^N

Z

R^N

Z

S^N−1b g(v_∗)Dkf(v) (Dkf(v⁰)−Dkf(v))dσdv_∗dv

=−1 2

Z

R^N

Z

R^N

Z

S^N−1b g(v_∗)

Dkf(v⁰)−Dkf(v) ₂

dσdv_∗dv +1

2 Z

R^N

Z

R^N

Z

S^N−1b g(v_∗)

(Dkf(v⁰))²−(Dkf(v))²

dσdv_∗dv.

Then we note that thanks to Assumption 2 (more precisely, to eq. (3)), we have b(cosθ, v−v_∗)≥1_{θ∈]−δ,δ[}|θ|^−γ,

so that according to Corollary 2.1 and Proposition 3 of [1], Z

R^N

Z

R^N

Z

S^N−1b g(v_∗)

D_kf(v⁰)−D_kf(v) ₂

dσdv_∗dv ≥Cg||D_kf||_H(γ−1)/2 , withC_g depending only on (an upper bound of) the entropy and on theL¹₁norm ofg.

But those quantities are controlled by the same quantities forfin when g = f and f satisfies the Boltzmann equation under our assumptions.

On the other hand, according to Corollary 1.2 of [1] (cancellation lemma), we know that

1 2

Z

R^N

Z

R^N

Z

S^N−1b g(v_∗)

(D_kf(v⁰))²−(D_kf(v))²

dσdv_∗dv

≤C₇||g||_L1

2||(Dkf)²||_L1

2 ≤C₇||g||_L1 2||f||²_Hk

2 . This concludes the proof of lemma 4.

2.3 A differential inequality

In this section, we denote byCany strictly positive constant which can be replaced by a smaller strictly positive constant, and byDany constant which can be replaced by a larger one.

Gathering the estimates of the previous section (that is, formula (13) and Lemmas 1 to 4) and summing with respect to all derivatives of orderk, we see that forf solu- tion of the Boltzmann equation under our assumptions (and supposing without loss of generality that the sequence r_k is nondecreasing and is such thatr₀ ≥ 2), we get the differential inequality :

d

dt||f||²_Hk ≤ −C||f||²_Hk+(γ−1)/2 +D||f||²_Hk

rk+1/2||f||L¹_2rk+1.

Using Proposition 1 of the appendix, and supposing thatk ≥N, we see that for some δ∈]0,1[and somes >0,

d

dt||f||²_Hk ≤ −C||f||²_Hk+(γ−1)/2+D||f||²⁻_{Hk+(γ−1)/4}^δ ||f||¹⁺_L1s^δ.

According to [14], for example, we can suppose that for alls >1, the quantity||f||L¹_s

is bounded on all compact sets of]0,+∞[(that is, all moments inL¹ are immediately gained). Then, we obtain

d

dt||f||²_Hk ≤ −C||f||²_Hk+(γ−1)/2 +D||f||²⁻_{Hk+(γ−1)/4}^δ . For allε >0, one can findD_ε>0such that the inequality

(1 +|ξ|²)^k+(γ−1)/4 ≤ε(1 +|ξ|²)^k+(γ−1)/2+Dε(1 +|ξ|²)^−N−1 holds. Then,

||f||²⁻_{Hk+(γ−1)/4}^δ = Z

R^N|fˆ(ξ)|²(1 +|ξ|²)^k+(γ−1)/4dξ ₁₋^δ

2

≤

ε Z

R^N|fˆ(ξ)|²(1 +|ξ|²)^k+γ−12 dξ+Dε

Z

R^N|fˆ(ξ)|²(1 +|ξ|²)^−N−1dξ ₁₋^δ

2

≤ε^1−δ2 Z

R^N|fˆ(ξ)|²(1 +|ξ|²)^k+γ−12 dξ ₁₋^δ

2 +Dε^1−δ2 m^2−δ, wheremis the massR

f dvoff. Finally, we obtain the differential inequality d

dt||f||²_Hk ≤ −C||f||²_Hk+(γ−1)/2 +D ε^1−δ2||f||²⁻_{Hk+(γ−1)/2}^δ +D D¹⁻

δ2

ε m^2−δ,

so that there existsD⁰ >0such that d

dt||f||²_Hk ≤ −C

2 ||f||²_Hk+(γ−1)/2+D⁰. Using (for example) Jensen’s inequality, we see that

||f||²⁺_Hk^k+N+1γ−1 = Z

R^N|f(ξ)ˆ |²(1 +|ξ|²)^kdξ

₁₊ ^γ−1

2(k+N+1)

≤ Z

R^N|fˆ(ξ)|²(1 +|ξ|²)^k+γ−12 dξ × Z

R^N|fˆ(ξ)|²(1 +|ξ|²)^−N−1dξ

_2(k+N+1)^γ−1

≤m^k+N+1γ−1 ||f||²_Hk+(γ−1)/2, so that the differential inequality can be rewritten

d dt

||f||²_Hk

≤ −C

2 m^k+N+1γ−1

||f||²_Hk

₁₊ ^γ−1

2(k+N+1)

+D⁰.

Using a standard argument (first used by Nash for parabolic equations), we see that for allkbig enough (and consequently for allk), f lies inH^k as soon ast > 0. By interpolation (thanks to Proposition 1 for example), we see that (still whent >0),f lies inH_s^kfor allk, s >0, and therefore lies inS.

Note that in the previous computation, one should use approximate solutions of the Boltzmann equation in order to give a completely rigorous proof. For example, solutions of the equation

∂_tf_ε=Q(f_ε, f_ε) +ε∆_vf_ε, f_ε(0,·) =f_in∗φ_ε,

where φ_ε is a sequence of mollifiers, can be used. This point does not lead to any difficulties.

Thus, we conclude the proof of Theorem 1 (in the particular case whenN = 2and when (4) is replaced by (5) in Assumption 2).

3 The general case

In this section, we explain how to modify the proofs described in the previous section to get Theorem 1 in any dimension and for all cross sections satisfying Assumption 2.