Mustapha Mokhtar-Kharroubi

David Seifert
email: david.seifert@sjc.ox.ac.uk

Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry

This work deals with free transport equations with partly di¤use stochastic boundary operators in slab geometry. Such equations are governed by stochastic semigroups in L 1 spaces: We prove convergence to equilibrium at the rate O t k 2(k+1)+1 (t ! +1) for L 1 initial data g in a suitable subspace of the domain of the generator T where k 2 N depends on the properties of the boundary operators near the tangential velocities to the slab. This result is derived from a quanti…ed version of Ingham's tauberian theorem by showing that Fg(s) := lim"!0 + (is + " T ) 1 g exists as a C k function on Rn f0g such that d j ds j Fg(s)

C jsj 2(j+1) near s = 0 and bounded as jsj ! 1 (0 j k) : Various preliminary results of independent interest are given and some related open problems are pointed out.

Introduction

This paper is devoted to rates of convergence to equilibrium for one-dimensional free (i.e. collisionless) transport equations with mass-preserving partly di¤use boundary operators. We provide a general L 1 theory relying on a quanti…ed tauberian theorem [START_REF] Chill | Quanti…ed versions of Ingham's theorem[END_REF]. In linear or non-linear kinetic theory, various non-local (combinations of specular and di¤use) boundary conditions are physically relevant, see e.g. [START_REF] Guo | Decay and continuity of the Boltzmann equation in bounded domains[END_REF] [START_REF] Petterson | On weak and strong convergence to equilibrium for solutions to the linear Boltzmann equation[END_REF] and the references therein. Furthermore, general free transport equations with smooth vector …elds and positive contractive boundary operators are well posed, see e.g. [3][4]. On the other hand, the existence of an invariant density and the return to this equilibrium state for solutions to free transport equations has not received much attention; see however [START_REF] Aoki | On the speed of approach to equilibrium for a collisionless gas[END_REF][5] [START_REF] Kuo | Free molecular ‡ow with boundary e¤ect[END_REF] [START_REF] Tsuji | Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall[END_REF] for the vector …eld v:r x with a Maxwell di¤use boundary operator with constant temperature; in this case, the invariant density is given by a maxwellian function. The L 1 convergence to this maxwellian equilibrium goes back to [START_REF] Arkeryd | Boltzmann asymptotics with di¤use re ‡ection boundary conditions[END_REF] while the analysis of rates of convergence was considered more recently in [START_REF] Aoki | On the speed of approach to equilibrium for a collisionless gas[END_REF] [START_REF] Kuo | Free molecular ‡ow with boundary e¤ect[END_REF] after some numerical investigations in [START_REF] Tsuji | Relaxation of a free-molecular gas to equilibrium caused by interaction with vessel wall[END_REF]; we will comment below on some results in [START_REF] Aoki | On the speed of approach to equilibrium for a collisionless gas[END_REF] [START_REF] Kuo | Free molecular ‡ow with boundary e¤ect[END_REF]. We note that collisionless transport semigroups present a lack of spectral gap which make them akin to collisional linear kinetic equations with soft potentials. More recently, the authors of [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF] provided a convergence theory to equilibrium for a general class of monoenergetic free transport equations in slab geometry with azymuthal symmetry and abstract boundary operators. In this abstract model, the existence of invariant density is characterized and shown for a general class of partly di¤use boundary operators. Our aim here is to derive a quanti…ed version (with algebraic rates) of this convergence theory from a quanti…ed version of Ingham's tauberian theorem [START_REF] Chill | Quanti…ed versions of Ingham's theorem[END_REF]. We provide a general theory based on some natural structural conditions on the boundary operators in the vicinity of the tangential velocities to the slab. To keep the ideas of this work more transparent, we restrict ourselves to monoenergetic models; (non-monoenergetic free models in slab geometry could be treated similarly, see Remark 33). Besides the main result on the rates of convergence, our construction provides us with various new mathematical results of independent interest. Several open problems are also pointed out.

We note that a special quanti…ed version of Ingham's theorem for "asymptotically analytic" C 0 -semigroups (see [START_REF] Chill | Quanti…ed versions of Ingham's theorem[END_REF] Corollary 2.12) was already used for the …rst time in kinetic theory to deal with spatially homogeneous linear Boltzmann equations with soft potentials where the generators are bounded [START_REF] Lods | Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: a semigroup approach in L 1 spaces[END_REF].

Finally, we point out that there exists a substantial literature on rates of convergence to equilibrium for collisional (linear or non-linear) kinetic equations relying mostly on entropy methods. In particular, collisional kinetic equations with soft potentials exhibit algebraic rates of convergence, see e.g. [START_REF] Canizo | On the rate of convergence to equilibrium for the linear Bolzmann equation with soft potentials[END_REF][7] [START_REF] Desvillettes | Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials[END_REF][18] [START_REF] Toscani | On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds[END_REF] and references therein.

We consider here the monoenergetic free transport equation in slab geometry with azymuthal symmetry @f @t (t; x; v) + v @f @x (t; x; v) = 0; (x; v) 2

f (0; x; v) = g(x; v) (1)
where = ( a; a) ( 1; 1)

(with a > 0). The boundary conditions are jvj f (t; a; v) = 1 jvj f (t; a; v) + 1 K 1 (j j f a (t)) (v > 0);

jvj f (t; a; v) = 2 jvj f (t; a; v) + 2 K 2 (j j f + a (t)) (v < 0) (3)
where i 0;

i 0; i + i = 1 (i = 1; 2); (5)
here f a (t) (resp. f + a (t)) denotes the restriction of f (t; a; :) (resp. f (t; a; :)) to ( 1; 0) (resp. to (0; 1)), j j f a (t) : ( 1; 0) 3 v ! jvj f (t; a; v) j j f + a (t) : (0; 1) 3 v ! f (t; a; v) and K i (i = 1; 2) are stochastic (i.e. positive and norm preserving on the positive cone) weakly compact operators K 1 : L1 (( 1; 0) ; dv) ! L 1 ((0; 1) ; dv); K 2 : L 1 ((0; 1) ; dv) ! L 1 (( 1; 0) ; dv):

The weak compactness assumption implies that K i has a kernel k i (:; :); (i = 1; 2) (see remark in [START_REF] Dunford | Linear Operators, Part I[END_REF], p. 508); it also plays a key role in several places of this work. Note that the boundary conditions are convex combinations of specular (deterministic) parts and di¤ use (random) ones modeled by K i (i = 1; 2). We point out that for the physical model in slab geometry with azymuthal symmetry, v 2 ( 1; +1) is not a "velocity" but rather the cosine of the angles of the monoenergetic velocities (of particles moving in the slab) with an oriented axis perpendicular to the slab. In particular, the tangential velocities to the slab correspond to v = 0 i.e. to the degeneracy of the vector …eld v @ @x : These tangential velocities turn out to play a natural and fundamental role in our construction. Finally, we note that the boundary conditions are local in space, i.e. we have two separated boundary conditions (one at x = a and another one at x = a) even if one can imagine much more complex models including a coupling of the ‡uxes at a and at a.

It is known that the problem (1)(2)(3)(4) is well-posed in L 1 ( ) in the sense of semigroup theory and the corresponding C 0 -semigroup is stochastic (i.e. norm preserving on the positive cone), [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF]. We deal here with the partly di¤ use model

1 + 2 > 0 (6)
only, i.e. we assume that at least one boundary condition is at least partly di¤use. It is known that under condition (6) the semigroup admits an invariant density, [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF]; (see below for the details). Furthermore, the C 0 -semigroup converges strongly to its ergodic projection as time goes to in…nity provided that Our aim here is to give a quanti…ed version of the convergence theory given in [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF]. We provide a systematic functional analytic treatment based on a quan-ti…ed version of Ingham's tauberian theorem [START_REF] Chill | Quanti…ed versions of Ingham's theorem[END_REF]; see Section 2 below. This tauberian theorem turns out to be usable in semigroup theory provided that the restrictions of the resolvent (of the generator) and some of its derivatives to a suitable subspace extend to iRn f0g and satisfy suitable estimates. We point out that for the stochastic kinetic semigroups we consider here, 0 always belongs to the spectrum of the generator; it may happen (e.g. if 1 = 1 or 2 = 1) that the whole imaginary axis is included in the spectrum of the generator. The object of this work is to show how to obtain systematically such estimates on the resolvent provided that one of the boundary conditions is completely di¤use, i.e. under suitable structural conditions to be stated below. (For the obstruction to the treatment of the general case [START_REF] Canizo | On the rate of convergence to equilibrium for the linear Bolzmann equation with soft potentials[END_REF], see Remark 31.) Note that [START_REF] Carlen | On strong convergence to equilibrium for the Boltzmann equation with soft potentials[END_REF] need not be the completely di¤use model which corresponds to 1 = 2 = 1; for instance the case of a di¤use boundary condition at x = a and a specular boundary condition at x = a is covered by our theory. Even if we do not treat completely the more general model ( 6), some of our statements are given under this general assumption and various mathematical results of independent interest are provided. The result which has motivated our whole construction is the following:

If the initial data g belongs to the domain of the generator and if Z jg(x; v)j jvj (k+1) dxdv < +1

then the solution to (1)(2)(3)(4) converges to the equilibrium state in

L 1 norm at the rate O t k 2(k+1)+1 ; (t ! +1) (8)
where k 2 N is an integer depending on natural structural properties of the boundary operators near v = 0 (i.e. near the tangential velocities to the slab).

To our knowledge, this is the …rst systematic quantitative result in collisionless kinetic theory for L1 initial datum.

Indeed, until now, the sole known quanti…ed L 1 results in collisionless kinetic theory are much better rates obtained for bounded initial datum in balls with Maxwell di¤use boundary conditions and constant boundary temperature. More precisely, in dimension 3; the rate of convergence in L 1 norm is O(t 1 ) if the initial data is radial (in space and in velocity) and is dominated by a maxwellian function (see [START_REF] Aoki | On the speed of approach to equilibrium for a collisionless gas[END_REF] Theorem 4.1); this result was improved in ( [START_REF] Kuo | Free molecular ‡ow with boundary e¤ect[END_REF] Corollary 2) where the rate is shown to be O(t d ) in dimension d 3 for bounded initial datum; (see [1][15] for additional results which we do not comment on here). We point out that Maxwell di¤use boundary conditions refer to boundary operators which are (local in space and) rank-one in velocity. Finally, we mention that quantitative time asymptotics have never been dealt with for partly di¤use boundary operators.

Let us give a more precise view on the mathematical construction behind the rate of convergence [START_REF] Cessenat | Théorèmes de trace L p pour des espaces de fonctions de la neutronique[END_REF]. Let

W 1 ( ) = f 2 L 1 ( ); v @f @x 2 L 1 ( ) (v @f @
x is understood in the sense of distributions) be endowed with the norm

kf k W1 = kf k + v @f @x where kgk = Z +a a Z +1 1 jg(x; v)j dx dv; g 2 L 1 ( ):
According to classical trace theory (see [8][9]), the elements of W 1 ( ) admit a trace on f ag ( 1; +1) and fag ( 1; +1)

belonging to the weighted L 1 -space

L 1 (( 1; +1) ; jvj dv) :
More precisely, the trace operator is surjective, continuous and admits a continuous lifting operator. For any f 2 W 1 ( ), we denote by f a (resp. f + a ) the restriction of f ( a; :) to ( 1; 0) (resp. to (0; 1)), i.e.

f a : ( 1; 0) 3 v ! f ( a; v); f + a : (0; 1) 3 v ! f ( a; v):
Similarly

f a : ( 1; 0) 3 v ! f (a; v); f + a : (0; 1) 3 v ! f (a; v):
We keep in mind that f a ; f a 2 L 1 (( 1; 0) ; jvj dv) and f + a ; f + a 2 L 1 ((0; +1) ; jvj dv) :

We de…ne also

h a (v) = jvj f a (v); h a (v) = jvj f a (v)
and keep in mind that h a ; h a 2 L 1 (( 1; 0) ; dv) and h + a ; h + a 2 L 1 ((0; +1) ; dv) :

De…ne O 1 = 1 R 1 + 1 K 1 : L 1 (( 1; 0) ; dv) ! L 1 ((0; +1) ; dv)
where

R 1 : L 1 (( 1; 0) ; dv) ! L 1 ((0; +1) ; dv)
is a specular re ‡ection operator de…ned by (R 1 ') (v) = '( v) and

O 2 = 2 R 2 + 2 K 2 : L 1 ((0; +1) ; dv) ! L 1 (( 1; 0) ; dv) where R 2 : L 1 ((0; +1) ; dv) ! L 1 (( 1; 0) ; dv)
is the re ‡ection operator de…ned by (R 2 ') (v) = '( v) while the parameters i ; i (i = 1; 2) satisfy the convexity condition [START_REF] Arkeryd | Boltzmann asymptotics with di¤use re ‡ection boundary conditions[END_REF]. The transport operator

T O : D(T O ) L 1 ( ) ! L 1 ( ); indexed by O := (O 1 ; O 2 )
; is de…ned by

T O f = v @f @x
on the domain

D(T O ) = f 2 W 1 ( ); h + a = O 1 h a ; h a = O 2 h + a :
It is known (see [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF]) that T O generates a stochastic (i.e. mass preserving on the positive cone) C 0 -semigroup (e tT O ) t 0 and, for g 2 L 1 ( );

f := ( T O ) 1 g; (Re > 0)
is given by

f (x; v) = 1 v e v (x+a) h + a + Z x a e v (x y) 1 v g(y; v) dy (v > 0) (9) f (x; v) = 1 jvj e jvj (a x) h a + Z a x e jvj (y x) 1 jvj g(y; v) dy (v < 0) (10)
with

h + a = (1 G ) 1 O 1 e 2 a jvj O 2 Z a a e v (a y) g(y; v) dy (11)
+(1 G ) 1 O 1 Z a a e jvj (y+a) g(y; v) dy h a = O 2 e 2 a jvj h + a + Z a a e v (a y) g(y; v) dy (12)
and

G f = O 1 e 2 a jvj O 2 e 2 a jvj f (13)
where the operator "e 2 a jvj " refers to the multiplication operator by the function e 2 a jvj : For the sake of simplicity, if no ambiguity may occur, the di¤erent (natural) L 1 norms as well as their corresponding operator norms are denoted by the symbol kk : Note that kG k e 4a Re (Re 0) and kG 0 k = 1: Under the general assumption [START_REF] Canizo | On the rate of convergence to equilibrium for the linear Bolzmann equation with soft potentials[END_REF], the essential spectral radii of the stochastic operators

G 0 = O 1 O 2 : L 1 ((0; +1) ; dv) ! L 1 ((0; +1) ; dv) e G 0 = O 2 O 1 : L 1 (( 1; 0) ; dv) ! L 1 (( 1; 0) ; dv)
are strictly less than 1; in particular, G 0 and e G 0 admit 1 as an isolated eigenvalue associated respectively to the eigenfunctions

h 0 2 L 1 + ((0; +1) ; dv) (14)
h 0 (v) v dv + Z 0 1 e h 0 (v) jvj dv < 1; (16)
in this case, a space homogeneous invariant density is given by

0 (v) = ( 1 v h 0 (v) (v > 0) 1 jvj e h 0 (v) (v < 0); (17)
see [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF] for all these results. We note that [START_REF] Lasota | Chaos, Fractals and Noise. Stochastic Aspects of Dynamics[END_REF] requires that the kernels of the di¤use parts K i vanish (in an appropriate sense) at v = 0. For example, ( 16) is not satis…ed in the purely di¤use case (i.e.

1 = 2 = 1) if inf (v;v 0 ) k i (v; v 0 ) > 0 (i = 1; 2): (18)
Of course, the object of this paper is meaningful only if e tT O t 0 has an invariant density. A su¢ cient condition ensuring [START_REF] Lasota | Chaos, Fractals and Noise. Stochastic Aspects of Dynamics[END_REF] is given in Theorem 5 below, (see also Remark 6). Actually, the present paper is built on much stronger structural assumptions (see below) so that the existence of the invariant density is guaranteed.

If e tT O t 0 is irreducible (a criterion is given in Theorem 7) then, under the normalization R 0 = 1, the invariant density 0 is unique and the C 0semigroup e tT O t 0 is mean ergodic with ergodic projection e tT O = P is proved in [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF] under the condition 1 2 > 0 by using a result (from [START_REF] Rudnicki | Stochastic operators and semigroups and their applications in physics and biology[END_REF]) on partially integral semigroups; (a new approach of this result is considered in [START_REF] Mokhtar-Kharroubi | A new approach of asymptotic stability of collisionless kinetic equations in slab geometry[END_REF]).

P : g ! Z g 0 ; (19)
We point out that if [START_REF] Lasota | Chaos, Fractals and Noise. Stochastic Aspects of Dynamics[END_REF] were not satis…ed then e tT O t 0 would be sweeping with respect to the sets

( a; a) [( 1; ") [ ("; +1)] (" > 0)
in the sense that the total mass of e tT O g concentrates in the vicinity of v = 0 (i.e. around the tangential velocities) as t ! +1, i.e.

Z "

1 Z +a a e tT O g (x; v) dx dv + Z 1 " Z +a a e tT O g (x; v) dx dv ! 0 (20)
as t ! +1; [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF]. Actually, the following alternative holds: e tT O t 0 is either strongly convergent if an invariant density exists or is sweeping in the sense [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF] otherwise; (i.e. a Foguel-like alternative holds, see [START_REF] Lasota | Chaos, Fractals and Noise. Stochastic Aspects of Dynamics[END_REF], Theorem 5. 10. 1, p. 130).

Thus, we are concerned here with quantitative time asymptotics of strongly convergent kinetic models; (the relevant open question for the non-convergent kinetic models is whether we can quantify their sweeping behaviour [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF], see Remark 32 (ii)). To this end, a key preliminary result is that r (G is ) < 1 (s 2 Rn f0g) (r refers to spectral radius) and

f 2 C; Re > 0g 3 ! (1 G ) 1 2 L(L 1 ( ))
extends continuously (in the strong operator topology) to iRn f0g : Various technical estimates are given in this paper. We can summarize them in two key statements. Let k 2 N; k 6 = 0; (the integer k comes from the structural assumptions).

The …rst statement is: if

Z a a Z 1 1 jg(x; v)j jvj k+1 dxdv < +1 then f 2 C; Re > 0g 3 ! ( T O ) 1 g 2 L 1 ( )
extends continuously to iRn f0g and, with

F g (s) := lim !is Re >0 ( T O ) 1 g; (21)
the map Rn f0g 3 s ! F g (s) 2 L 1 ( )

lies in C k Rn f0g ; L 1 ( ) with the uniform C k estimates d j ds j F g (s) C j+1 X p=0 (1 G is ) 1 p ! g jvj k+1 (0 j k; s 6 = 0)
where C > 0 is a constant, see Theorem 20.

The second statement is:

sup jsj (1 G is ) 1 < 1; ( > 0)
and there exists a constant C > 0 such that and there exists a constant C > 0 such that

(1 G is ) 1 C s 2 (
d j ds j F g (s) C jsj 2(j+1) (0 j k) (for small s 2 Rn f0g)
and consequently a quanti…ed version of Ingham's theorem (see Corollary 3 below) implies

e tT O g Z g 0 = O t k 2(k+1)+1 ; (t ! +1) (22)
for any initial data g 2 D(T O ) such that are checkable. Indeed jvj j R i jvj j (i = 1; 2) are always bounded while the boundedness of jvj j K i jvj j (i = 1; 2) is a condition on the kernel of K i (i = 1; 2) in the neighborhood of v = 0. On the other hand, (37)(38) are checkable only if O 1 is a kernel operator (or if O 2 is a kernel operator); this explains why the condition " 1 = 1 or 2 = 1" appears in di¤erent statements. We point out that the need for conditions on the kernels of K i (i = 1; 2) near the tangential velocities (i.e. v = 0) is not fortuitous since the existence of an invariant density already requires a condition in the same spirit, see ( 16) and [START_REF] Lods | Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: a semigroup approach in L 1 spaces[END_REF].

The fact that

k 2(k+1)+1 ! 1 2 (k ! 1) shows that if there exists C j > 0 such that d j ds j F g (s) C j jsj 2(j+1) (0 < jsj 1; j 2 N)
(this occurs if the structural assumptions are satis…ed for all k 2 N) then the quanti…ed version of Ingham's tauberian theorem provides us with the rate

O 1 t 1 2+" ; (" > 0) : (23)
It is a priori unclear whether we can reach the limit rate O 1 p t

or can go beyond this rate for the kinetic semigroups e tT O t 0 (note that much better rates of convergence occur for bounded initial datum in balls, see [1][15]). We refer to Remark 31 and Remark 32 for di¤erent open problems suggested by our construction.

Our paper is organized as follows:

In Section 2, we give a corollary of a quanti…ed version of Ingham's theorem [START_REF] Chill | Quanti…ed versions of Ingham's theorem[END_REF] which implies the rates of convergence

e tT g P g = O t k (k+1)+1 ; t ! +1
for bounded mean ergodic C 0 -semigroups e tT t 0 on a Banach space X with ergodic projection P (and generator T ) for initial data

g 2 D(T ) \ (Ker(T ) + Ran(T )) (24)
provided that F g (s) := lim "!0+ ("+is T ) 1 g (s 6 = 0) exists, lies in C k (Rn f0g ; X) for some k 2 N and satis…es the estimates

sup jsj 1 F (j) g (s) < +1 and F (j) g (s) C jsj (j+1) ; (0 j k; 0 < jsj 1):
In Section 3, we give a su¢ cient criterion for the existence of an invariant density of e tT O t 0 . A su¢ cient criterion of irreducibility of e tT O t 0 is given in Section 4. The combination of the last two results implies that the C 0semigroup e tT O t 0 is mean ergodic. Because of the importance of ( 24), a su¢ cient criterion for a given g 2 L 1 ( ) to belong to the range of T O is given in Section 5. Section 6 is devoted to (T O ) \ iR; the boundary spectrum of the generator; while 0 2 (T O ) is always true, we show that the imaginary axis is equal to the boundary spectrum at least when 1 = 1 or 2 = 1. In Section 7, we explain why

F g (s) := lim "!0+ (" + is T O ) 1 g (s 6 = 0) exists and lies in C k Rn f0g ; L 1 ( ) if R jg(x; v)j jvj (k+1) dxdv < +1 and
if the boundary ‡uxes h + a and h a given by (11)( 12) with = is (s 6 = 0) are C k functions of s 2 Rn f0g and their jth derivatives belong to suitable weighted spaces depending on j (1 j k). Such conditions depends heavily on the existence of (1 G is ) 1 (s 6 = 0) and its derivatives in s (in suitable spaces) which are thus the cornerstone of this work. The existence and estimate of (1 G is ) 1 (s 6 = 0) are postponed until Section 11. Under the general condition

1 + 2 > 0; (25)
we show that r (jG is j) < 1 (s 6 = 0) where jG is j is the linear modulus of G is (see [START_REF] Chacon | Linear modulus of a linear operator[END_REF]). The proof relies on strict comparison of spectral radii of positive operators in a context of domination [START_REF] Marek | Frobenius Theory of Positive Operators: Comparison Theorems and Applications[END_REF]. We show also the key estimates

sup j j (1 G ) 1 < +1 ( > 0; Re 0) (1 G ) 1 = O 1 jIm j 2 ! ( ! 0; Re 0; 6 = 0): (26)
The proof of ( 26) is quite involved and relies on a second order expansion about s = 0 (uniformly in " 0) of a suitable function related to R 3 s ! kG "+is k :

In Section 8, we show by induction the key estimate of the derivatives

d j ds j (1 G is ) 1 L(L 1 (dv);L 1 (jvj k 1+j dv)) C j+1 X l=0 (1 G is ) 1 l (1 j k)
by exploiting a di¤erential equation satis…ed by Rn f0g 3 s ! (1 G is ) 1 : It is at this place that we need that at least one of the boundary conditions must be completely di¤use. In Section 9, we deduce the estimate on the left ‡ux

d j h + a d j L 1 (jvj k 1+p dv) C j+1 X l=0 (1 G is ) 1 l ! g jvj k+1 (0 j k)
and a similar estimate on the right ‡ux h a : In Section 10, we sum up the previous estimates in the statement

d j ds j F g (s) C j+1 X l=0 (1 G is ) 1 l ! g jvj k+1 (0 j k; s 6 = 0) :
Finally, in Section 12, we deduce the algebraic estimates of ( 21) on iRn f0g

sup jsj d j ds j F g (s) < +1; ( > 0; 0 j k) d j ds j F g (s) C s 2(j+1) g jvj k+1 (0 j k; s ! 0)
and derive, from the quanti…ed version of Ingham's theorem, the rate of convergence

e tT O g Z g 0 = O t k 2(k+1)+1 ; (t ! +1)
for any initial data g 2 D(T O ) \ L 1 ( ; dv jvj k+1 ): As far as we know, all these functional analytic results on collisionless kinetic theory appear here for the …rst time. Some open problems suggested by our construction in slab geometry are pointed out in Remark 31 and Remark 32 below. We note that this work could be extended to non-monoenergetic free transport equations in slab geometry with more general re ‡ection operators R i (i = 1; 2), see Remark 33. However, its extension to multidimensional-space geometries is an open problem, see Remark 34. For the sake of simplicity, in all the paper, we will denote by the same symbol C various positive constants occuring in our di¤erent proofs and statements.

A quanti…ed version of Ingham' s theorem

Let X be a complex Banach space. For any f 2 L 1 (R + ; X) ; we de…ne its Laplace transform by

b f ( ) = Z +1 0 e t f (t)dt (Re > 0): Let 2 R:
We say that i is a weakly regular point for b f if there exist " > 0 and h 2 L 1 (( "; + ") ; X) such that b f ( + i:) ! h(:) in the distributional sense on ( "; + ") as ! 0 + :

The weak half-line spectrum sp w (f ) of f is de…ned as the set of all real numbers such that i is not weakly regular for b f . Then sp w (f ) is a closed subset of R and there exists

F 2 L 1 loc (Rnsp w (f ); X) such that b f ( + i:) ! F (:) in the distributional sense on Rnsp w (f ) as ! 0 + ; (27)
see e.g. [START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF] Lemma 4.9.9, p. 326. We give now a quanti…ed version of the classical Ingham's tauberian theorem (see e.g. [START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF] Theorem 4.9.5, p. 327). This version is a special case of ( [START_REF] Chill | Quanti…ed versions of Ingham's theorem[END_REF] Theorem 2.13 (a)).

Theorem 1 Let X be a complex Banach space and suppose that f belongs to L 1 (R + ; X), is Lipschitz continuous and sup t 0 R t 0 f (s)ds < +1: Suppose furthermore that sp w (f ) f0g and F (given by ( 27)) lies in C k (Rn f0g ; X) for some k 2 N: If sup jsj 1 F (j) (s) < +1 (0 j k) and if

F (j) (s) C jsj (j+1) ; ((0 j k; 0 < s 1)
for some constants C > 0; 1 then

kf (t)k = O t k (k+1)+1 ; (t ! +1) :
Now suppose that e tT t 0 is a bounded C 0 -semigroup with generator T on X, and that f (t) = e tT g (t 0) for some g 2 X: Then f is a bounded continuous function. It is Lipschitz continuous if g 2 D(T ) and has uniformly bounded primitive if g 2 Ran(T ): Recall that the Laplace of f is given by (Lf ) ( ) = R( ; T )g for Re > 0: Hence, by a calculation similar to that in ( [START_REF] Chill | Quanti…ed versions of Ingham's theorem[END_REF], Eq. (1.2)) we see that f satis…es the assumptions of Theorem 1 provided that g 2 D(T ) \ Ran(T ) and R( ; T )g extends continuously to a su¢ ciently smooth function on iRn f0g : Note that, crucially for us, this is possible for particular initial values g 2 X even if iR (T ). We obtain the following corollary of Theorem 1.

Corollary 2 Let e tT t 0 be a bounded C 0 -semigroup with generator T on a complex Banach space X and let g 2 D(T ) \ Ran(T ): Suppose that R( ; T )g (Re > 0) extends continuously to iRn f0g and that

F g (s) := lim "!0+ R(is + "; T )g lies in C k (Rn f0g ; X) for some k 2 N: If sup jsj 1 F (j) g (s) < +1 (0 j k) and if F (j) g (s) C jsj (j+1) ; (0 j k; 0 < s 1)
for some constants C > 0;

1 then

e tT g = O t k (k+1)+1 ; (t ! +1) :
In this paper, we need the following simple consequence of Corollary 2.

Corollary 3 Let e tT t 0 be a bounded mean ergodic C 0 -semigroup with generator T on a complex Banach space X with ergodic projection P: Let g 2 D(T ) \ (Ker(T ) + Ran(T )) : Suppose that R( ; T )g (Re > 0) extends continuously to iRn f0g and that

F g (s) := lim "!0+ R(is + "; T )g lies in C k (Rn f0g ; X) for some k 2 N: If sup jsj 1 F (j) g (s) < +1 (0 j k) and if F (j) g (s) C jsj (j+1) ; (0 j k; 0 < s 1)
for some constants C > 0; 1 then

e tT g P g = O t k (k+1)+1 ; (t ! +1) :
Remark 4 Theorem 1 can be complemented by the statement that if F 2 C 1 (Rn f0g ; X) and if there exists a constant C > 0 such that

F (j) (s) C j! jsj (j+1)+1 ; (j 2 N; 0 < s 1) then kf (t)k = O( ln(t) t 1
); (t ! +1), (see [START_REF] Chill | Quanti…ed versions of Ingham's theorem[END_REF] Theorem 2.13 (b)).

On existence of invariant density

We complement a result from [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF].

Theorem 5 We assume that either O 1 = K 1 and both jvj 1 K 1 and jvj 1 K 2 jvj are bounded or O 2 = K 2 and both jvj 1 K 2 and jvj 1 K 1 jvj are bounded. Then ( 16) is satis…ed and consequently e tT O t 0 has an invariant density.

Proof. Without loss of generality, we may suppose that 1 = 1 and therefore 1 = 0: We know that G 0 h 0 = h 0 and e G 0 h 0 = e h 0 . Thus

K 1 ( 2 R 2 + 2 K 2 ) h 0 = h 0 so R 1 0 h0(v) v dv < 1: By (15) e h 0 = O 2 h 0 = 2 R 2 h 0 + 2 K 2 h 0 : By assumption K 2 h 0 2 L 1 dv jvj if h 0 2 L 1 dv jvj : Since L 1 dv jvj is invariant under R 2 we have e h 0 2 L 1 dv jvj :
Remark 6 The assumptions in Theorem 5 can be weakened. For instance, we can replace the boundedness of jvj 1 K 1 by the assumption that

jvj 1 K 1 R 2 K 1 and jvj 1 K 1 K 2 K 1 are bounded. Indeed, since [K 1 ( 2 R 2 + 2 K 2 )] 2 h 0 = h 0 then h 0 2 L 1 dv jvj provided that jvj 1 K 1 ( 2 R 2 + 2 K 2 ) K 1 is bounded.
4 On irreducibility of (e tT O ) t 0

We give two complementary irreducibility criteria.

Theorem 7 We assume that either

O 1 = K 1 or O 2 = K 2 . If G 0 = O 1 O 2 : L 1 ((0; +1) ; dv) ! L 1 ((0; +1) ; dv)
is irreducible and if O 2 is strict positivity preserving in the sense that

h(v) > 0 a.e. =) (O 2 h) (v) > 0 a.e.
then (e tT O ) t 0 is irreducible.

Proof. Note that

(1 G ) 1 = 1 X j=0 G j ( > 0)
so that for any nonnegative h and h

h(1 G ) 1 h; h i hG j h; h i; (j 2 N; > 0):
Since G 0 is irreducible then for any non trivial nonnegative h and h there exists an integer j (depending a priori on h and h ) such that hG j 0 h; h i > 0: Since lim !0+ hG j h; h i = hG j 0 h; h i then hG j h; h i > 0 for small enough. Since ! hG j h; h i 2 R + is nonincreasing, an analyticity argument shows that

hG j h; h i > 0; ( > 0)
and …nally h(1 G ) 1 h; h i > 0 so (1 G ) 1 h > 0 a.e. Thus [START_REF] Chill | Quanti…ed versions of Ingham's theorem[END_REF] gives h + a > 0 a.e. for any non trivial nonnegative g and ( 12) gives h a > 0 a.e. since O 2 is strict positivity preserving. Finally (1 T O ) 1 is positivity improving or equivalently (e tT O ) t 0 is irreducible.

Remark 8 Note that G 0 is an integral operator with kernel q(v; v 0 ): The irreducibility of G 0 amounts to

Z [0;1]nS Z S q(v; v 0 )dv 0 dv > 0
for any measurable S [0; 1] such that S and [0; 1] nS have positive measure. In particular, this the case if q(v; v 0 ) > 0 a.e. Note that O 2 = 2 R 2 + 2 K 2 is automatically strict positivity preserving if 2 > 0:

Remark 9 Another irreducibility criterion is a "dual" version of Theorem 7: Assume that either

O 1 = K 1 or O 2 = K 2 . If e G 0 = O 2 O 1 : L 1 (( 1; 0) ; dv) ! L 1 (( 1; 0) ; dv)
is irreducible and if O 1 is strict positivity preserving then (e tT O ) t 0 is irreducible. Indeed, it is easy to see that

h a = I e G 1 O 2 e v 2a O 1 Z a a e jvj (y+a) g(y; v) dy + Z a a e v (a y) g(y; v) dy e G := O 2 e v 2a O 1 e jvj 2a h + a = O 1 h a = O 1 e jvj 2a h a + Z a a
e jvj (y+a) g(y; v) dy and then it su¢ ces to exchange the roles of h + a and h a and to argue as previously.

On the range of T O

According to Corollary 3, the knowledge of the range of the generator is a key point. To this end, we describe now a useful subspace of the range of T O .

Theorem 10 We assume that either O 1 = K 1 and both jvj 1 K 1 and jvj 1 K 2 jvj are bounded or O 2 = K 2 and both jvj 1 K 2 and jvj 1 K 1 jvj are bounded. We assume additionally, in the …rst case, that

G 0 = O 1 O 2 is irreducible or, in the second case, that e G 0 = O 2 O 1 is irreducible. Let g 2 L 1 ( ): If 1 jvj g 2 L 1 ( ) (28)
and if

Z g = 0 (29) then g 2 Ran(T O ): Proof. Note that (e tT O ) t 0 is a stochastic semigroup so that Z T O ' = 0; ' 2 D(T O )
and consequently (29) is a necessary condition for g 2 L 1 ( ) to belong to Ran(T O ). We consider the case By inspection of (9)(10)(11) [START_REF] Desvillettes | Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials[END_REF], for solving ( T O )f = g with = 0; it su¢ ces that (28) is satis…ed,

O 1 O 2 (b g + ) + O 1 (b g ) 2 Ran(1 G 0 ); (30)
that h + a , given by [START_REF] Chill | Quanti…ed versions of Ingham's theorem[END_REF], is such that

1 jvj h + a 2 L 1 ((0; +1) ; dv) (31)
and that

1 jvj h a := 1 jvj O 2 h + a + b g + 2 L 1 (( 1; 0) ; dv) : (32)
Note that 1 is an isolated algebraically simple eigenvalue of G 0 = O 1 O 2 associated with the eigenfunction h 0 (see ( 14)) so that

Ran(1 G 0 ) is closed in L 1 ((0; +1) ; dv) :
By the Fredholm alternative

G 0 (b g + ) + O 1 (b g ) 2 Ran(1 G 0 )
if and only if G 0 (b g + ) + O 1 (b g ) is orthogonal (for the duality pairing) to the dual eigenfunction h 0 2 L 1 ((0; +1) ; dv) i.e.

hG 0 (b g + ) + O 1 (b g ) ; h 0 i = 0 or hG 0 (b g + ) ; h 0 i + hO 1 (b g ) ; h 0 i = 0: Since hG 0 (b g + ) ; h 0 i = h b g + ; G 0 h 0 i = h b g + ; h 0 i we have hb g + + O 1 (b g ) ; h 0 i = 0:
On the other hand, G 0 : L 1 ((0; +1) ; dv) ! L 1 ((0; +1) ; dv) is integral preserving, i.e.

Z 1 0 G 0 ' = Z 1 0 ' 8' 2 L 1 ((0; +1) ; dv) 0 b g + + Z 1 0 O 1 (b g ) = 0:
Since O 1 is also integral preserving we have

Z 1 0 O 1 (b g ) = Z 0 1 b g
and …nally (30) amounts to

Z 1 0 b g + + Z 0 1 b g = 0
which is nothing but (29). Hence (30) is satis…ed. Note that

L 1 ((0; +1) ; dv) = Ker(I G 0 ) Ran(1 G 0 )
and

Ran(1 G 0 ) is invariant under G 0 . It follows that on Ran(1 G 0 ) (1 G 0 ) 1 = I + G 0 (1 G 0 ) 1 so h + a = (1 G 0 ) 1 [O 1 O 2 (b g + ) + O 1 (b g )] = O 1 O 2 (b g + ) + O 1 (b g ) + G 0 (1 G 0 ) 1 [O 1 O 2 (b g + ) + O 1 (b g )]
shows that h + a 2 L 1 (0; +1) ;

dv jvj since 1 jvj O 1 is bounded. Note that 1 jvj h a = 1 jvj O 2 h + a + b g + = 1 jvj R 2 h + a + b g + + 1 jvj K 2 h + a + b g + = 1 jvj R 2 jvj h + a jvj + b g + jvj ! + 1 jvj K 2 jvj h + a jvj + b g + jvj ! : Since h + a + b g + 2 L 1 (
0; +1) ; dv jvj we have, using our assumption,

1 jvj K 2 jvj h + a jvj + b g + jvj ! 2 L 1 ( 1; 0) ; dv jvj :
We always have

1 jvj R 2 jvj h + a jvj + b g + jvj ! 2 L 1 ( 1; 0) ; dv jvj :
This shows (32). The case 2 = 1 can be treated similarly.

Remark 11

We do not know whether (28) is a necessary condition for g to belong to Ran(T O ):

6 On the boundary spectrum of T O This section is devoted to the analysis of (T O ) \ iR: Note …rst that the type of (e tT O ) t 0 is equal to 0 since (e tT O ) t 0 is a stochastic semigroup. Thus 0 2 (T O ) since the type of (e tT O ) t 0 coincides with the spectral bound of its generator, see e.g. [START_REF] Nagel | One-Parameter Semigroups of Positive Operators[END_REF].

Theorem 12 Suppose that O 1 = K 1 and that jvj 1 K 1 is bounded (or O 2 = K 2 and jvj 1 K 2 is bounded). Then iR (T O ):
Proof. A simple inspection of ( T O ) 1 g shows that it consists of two parts, the …rst one being

H g := fv>0g Z x a e v (x y) 1 v g(y; v) dy + fv<0g Z a
x e jvj (y x) 1 jvj g(y; v) dy which is nothing but ( T 0 ) 1 g where T 0 is the classical free transport operator with the "zero incoming" boundary condition. It is well known (see [START_REF] Lehner | On the spectrum of an unsymmetric operator arising in the transport theory of neutrons[END_REF]) that

(T 0 ) = f 2 C; Re 0g ; (33)
(the proof is given there in L 2 ( ) but is the same in all L p spaces (p 1)). Let us regard this result in a slightly di¤erent way. Indeed, let + = ( a; a) (0; 1) and = ( a; a) ( 1; 0) :

We note that L 1 ( + ) and L 1 ( ) are invariant under ( T 0 ) 1 (or equivalently under (e tT0 ) t 0 ) and therefore T 0 splits as T 0 = T 0 T + 0 where T 0 are the generators of the restrictions of (e tT0 ) t 0 to the subpaces L 1 ( ): Thus

( T + 0 ) 1 g + = Z x a e v (x y) 1 v g + (y; v) dy and
( T 0 ) 1 g = Z a
x e jvj (y x) 1 jvj g (y; v) dy where g are the restrictions of g to L 1 ( ): As in [START_REF] Lehner | On the spectrum of an unsymmetric operator arising in the transport theory of neutrons[END_REF], we can show that

(T 0 ) = (T + 0 ) = f 2 C; Re 0g :
In particular lim

"!0+ (" + is T + 0 ) 1 = +1 (s 2 R) (34)
and lim

"!0+ (" + is T 0 ) 1 = +1 (s 2 R): (35)
(i) Suppose …rst that 1 = 1 and that jvj 1 K 1 is bounded. We know that ( T O ) 1 g is given for positive v by

1 v e v (x+a) h + a + ( T + 0 ) 1 g + where h + a = (1 G ) 1 O 1 e 2 a jvj O 2 Z a a
e v (a y) g(y; v) dy + Z a a e jvj (y+a) g(y; v) dy :

Note that (1 G ) 1 O 1 = O 1 + G (1 G ) 1 O 1 and G = O 1 e 2 a jvj O 2 e
2 a jvj : According to Corollary 25 (1 G ) 1 (Re > 0) extends continuously to iRn f0g (in the strong operator topology). It follows that the norm of the operator (depending on = " + is; " > 0)

L 1 ( ) 3 g ! 1 v e v (x+a) h + a 2 L 1 ( + )
remains uniformly bounded when " ! 0 + (8s 6 = 0): Finally (34) implies that lim

"!0+ sup kgk 1 (" + is T O ) 1 g L 1 ( +) = +1 (s 6 = 0)
whence is 2 (T O ) (8s 6 = 0): (ii) Suppose now that 2 = 1 and that jvj 1 K 2 is bounded. It is easy to see that (

T O ) 1 g can also be given by

f (x; v) = 1 v e v (x+a) h + a + Z x a e v (x y) 1 v g(y; v) dy (v > 0) f (x; v) = 1 jvj e jvj (a x) h a + Z a x e jvj (y x) 1 jvj g(y; v) dy (v < 0)
where

h a = I e G 1 O 2 e v 2a O 1 Z a a e jvj (y+a) g(y; v) dy + Z a a e v (a y) g(y; v) dy e G := O 2 e v 2a O 1 e jvj 2a
and

h + a = O 1 h a = O 1 e jvj 2a h a + Z a a
e jvj (y+a) g(y; v) dy :

In particular, ( T O ) 1 g is given for negative v by

1 jvj e jvj (a x) h a + ( T 0 ) 1 g : 20
By noting that

I e G 1 O 2 = O 2 + e G (1 e G ) 1 O 2 ;
and using the fact that (1 e G ) 1 (Re > 0) extends continuously to iRn f0g in the strong operator topology (see Remark 26), we see as before that (35) implies lim

"!0+ sup kgk 1 (" + is T O ) 1 g L 1 ( ) = +1 (s 6 = 0)
and is 2 (T O ) (8s 6 = 0):

Remark 13 A priori, it is not clear whether iR (T O ) for more general partly di¤ use models.

The objects to be estimated

Note that H g = ( T 0 ) 1 g does not extend to iR for all g because of (33). On the other hand, we can extend it on a suitable subspace. Indeed, let k 2 N; (k 6 = 0): It is easy to see that H g extends to the whole closed half space f 2 C; Re 0g with the C k norm estimates

@ j @ j H g (2a) j g jvj j+1 (0 j k; Re 0) (36)
provided that g jvj k+1 < +1: Actually, to estimate ( T O ) 1 g up to the imaginary axis, the key point is to estimate in C k norm the boundary terms

1 v e v (x+a) h + a ;
1 jvj e jvj (a x) h a :

Consider …rst 1 v e v (x+a) h + a :
Note that a priori h + a 2 L 1 ((0; +1]; dv): Since

@ k 1 v e v (x+a) h + a @ k = k X j=0 k j @ j @ j 1 v e v (x+a) @ k j @ k j h + a = k X j=0 k j x + a v j 1 v e v (x+a) @ k j @ k j h + a
our main concern is to estimate the norms

1 jvj @ k @ k h + a ; 1 jvj 2 @ k 1 @ k 1 h + a ; :::; 1 jvj k @ @ h + a ; 1 jvj k+1 h + a in f 2 C; Re 0; 6 = 0g :
8 Operator estimates up to the imaginary axis Note that G is de…ned on the closed half space f 2 C; Re 0g and kG k e 4a Re (Re or more precisely

Since h + a = (1 G ) 1 O 1 e
0): The derivatives of G @ p G @ p = p X j=0 p j O 1 @ j @ j e 2 a jvj O 2 @ p j @ p j e 2 a jvj = ( 2a) p p X j=0 p j O 1 1 jvj j e 2 a jvj ! O 2 1 jvj p j e
1 jvj k+1 O 1 O 2 ; 1 jvj k O 1 1 jvj O 2 and 1 jvj k O 1 O 2 1 jvj are bounded operators. ( 38
)
Remark 14 If O 1 or O 2 is weakly compact then at least one of the two is an integral operator and consequently Assumptions (37)(38) are checkable in principle.

We will show, under the condition 1 + 2 > 0; that r (G ) < 1 (Re 0, 6 = 0) and (1 G ) 1 extends continuously (in the strong operator topology) to iRn f0g ; (see Corollary 25). We are ready to give our key estimates of the derivatives of (1 G ) 1 in terms of (1 G ) 1 : Lemma 15 Suppose that (37)(38) are satis…ed. Then there exists a constant C > 0 such that for all s 2 R; s 6 = 0

d j ds j (1 G is ) 1 L(L 1 (dv);L 1 (jvj k 1+j dv)) C j+1 X l=0 (1 G is ) 1 l (1 j k): Proof. Note that (1 G ) 1 = I + G (1 G ) 1 (39) and d d ( 1
G ) 1 = (1 G ) 1 G 0 (1 G ) 1 (40) so d d ( 1
G ) 1 = I + G (1 G ) 1 G 0 I + G (1 G ) 1 = G 0 I + G (1 G ) 1 + G (1 G ) 1 G 0 I + G (1 G ) 1
and (37)(38) show that

d d (1 G ) 1 : L 1 ((0; +1]; dv) ! L 1 ((0; +1]; dv jvj k )
and that there exists a constant C > 0 such that

d d (1 G ) 1 L(L 1 (dv);L 1 (jvj k dv)) C 1 + (1 G ) 1 + (1 G ) 1 2 :
Let us show by induction that

d j d j (1 G ) 1 : L 1 ((0; +1]; dv) ! L 1 ((0; +1]; dv jvj k+1 j ) (1 j k)
and there exists a constant C > 0 such that

d j d j (1 G ) 1 L(L 1 (dv);L 1 (jvj k 1+j dv)) C 1 + j+1 X l=1 (1 G ) 1 l ! :
(41) We already know that this statement is true for j = 1: It su¢ ces to show that if 1 p < k and that if

d j d j (1 G ) 1 L(L 1 (dv);L 1 (jvj k 1+j dv)) C 1 + j+1 X l=1 (1 G ) 1 l ! (1 j p) then estimate (41) is true for j = p + 1: Let f ( ) = (1 G ) 1 :
According to (40), f ( ) satis…es the di¤erential equation

f 0 ( ) = f ( )G 0 ( )f ( ): (42)
Di¤erentiating (42) p times we get

d p+1 d p+1 f = p X q=0 p q d q d q f d p q d p q (G 0 ( )f ( )) = p X q=0 p q d q d q f p q X m=0 p q m d p q m d p q m G 0 ( ) d m d m f = p X q=0 p q d q d q f p q X m=0 p q m d p q m+1 d p q m+1 G( ) d m d m f = p X q=0 p q X m=0 p q p q m d q d q f d p q m+1 d p q m+1 G( ) d m d m f : Note that jvj k 1+j jvj k 1+j 0 (j j 0 ) shows that L 1 (jvj k 1+j dv) L 1 (jvj k 1+j 0 dv) and k'k L 1 (jvj k 1+j 0 dv) k'k L 1 (jvj k 1+j dv) 8' 2 L 1 (jvj k 1+j dv): Thus d m d m f ( ) L(L 1 (dv)) d m d m f ( ) L(L 1 (dv);L 1 (jvj k 1+m dv))
and (by assumption)

d m d m f ( ) L(L 1 (dv);L 1 (jvj k 1+m dv)) C 1 + m+1 X l=1 kf ( )k l ! (m p q) so d m d m f ( ) L(L 1 (dv)) C 1 + m+1 X l=1 kf ( )k l ! (m p q):
By (37) the derivatives d p q m+1 d p q m+1 G( ) are uniformly bounded for the natural operator norms. Similarly,

d q d q f ( ) : L 1 (dv) ! L 1 (jvj k 1+q dv) L 1 (jvj k 1+p dv) (q p)
and

d q d q f ( ) L(L 1 (dv);L 1 (jvj k 1+p dv)) d q d q f ( ) L(L 1 (dv);L 1 (jvj k 1+q dv))
so (using the assumption)

d q d q f ( ) L(L 1 (dv);L 1 (jvj k 1+p dv)) d q d q f ( ) L(L 1 (dv);L 1 (jvj k 1+q dv)) C 1 + q+1 X r=1 kf ( )k r ! (q p):
On the other hand

1 + m+1 X l=1 kf ( )k l ! 1 + q+1 X r=1 kf ( )k r ! = 1+ m+1 X l=1 kf ( )k l + q+1 X r=1 kf ( )k r + m+1 X l=1 q+1 X r=1 kf ( )k l+r :
Since m p q we have l + r m + 1 + q + 1 p + 2 and there exists C > 0 such that

d p+1 d p+1 f L(L 1 (dv);L 1 (jvj k 1+p dv)) C 1 + p+2 X l=1 kf ( )k l ! : Finally d p+1 d p+1 f L(L 1 (dv);L 1 (jvj k 1+p+1 dv)) d p+1 d p+1 f L(L 1 (dv);L 1 (jvj k 1+p dv)) C 0 @ 1 + (p+1)+1 X l=1 kf ( )k l 1 A
and hence we are done.

Estimates of boundary ‡uxes

We note that if

O 1 : L 1 ((0; +1]; dv jvj k+1 ) ! L 1 ((0; +1]; dv jvj k+1 ) is bounded i.e. if 1
jvj k+1 O 1 jvj k+1 is bounded (43)
then (39) gives

h + a = (1 G ) 1 O 1 e 2 a jvj O 2 Z a a e v (a y) g(y; v) dy + Z a a e jvj (y+a) g(y; v) dy = G (1 G ) 1 O 1 e 2 a jvj O 2 Z a a e v (a y) g(y; v) dy + Z a a e jvj (y+a) g(y; v) dy +O 1 e 2 a jvj O 2 Z a a e v (a y) g(y; v) dy + O 1 Z a a
e jvj (y+a) g(y; v) dy and

h + a L 1 (jvj (k+1) dv) 2 1 jvj k+1 O 1 O 2 (1 G ) 1 kgk + 1 jvj k+1 O 1 O 2 kgk + 1 jvj k+1 O 1 jvj k+1 g jvj k+1 :
Leibnitz's rule shows that

d p h + a d p is given by p X j=0 p j d j d j (1 G ) 1 d p j d p j O 1 e 2 a jvj O 2 Z a a e v (a y) g(y; v) dy + O 1 Z a a
e jvj (y+a) g(y; v) dy or indeed by p X j=0

( 1) p j p j

d j d j (1 G ) 1 p j X m=0 p j m (2a) m O 1 1 jvj m e 2 a jvj O 2 Z a a e v (a y) (a y) p j m g(y; v) jvj p j m dy + p X j=0 ( 1) p j p j d j d j (1 G ) 1 O 1 Z a a
e jvj (y+a) (y + a) p j g(y; v) jvj p j dy ! :

Finally, Lemma 15 implies:

Lemma 16 Suppose that (37)(38)(43) are satis…ed. There exists a constant C > 0 such that

d j h + a d j L 1 (jvj k 1+p dv) C j+1 X l=0 (1 G is ) 1 l ! g jvj k+1 (0 j k):
We deal now with h a .

Proposition 17 Suppose that (37)(38)(43) are satis…ed. If

jvj (k+1 p) O 2 jvj k+1 p is bounded (0 p k) (44)
then there exists a constant C > 0 such that

h a L 1 (jvj (k+1) dv) C " h + a L 1 (jvj (k+1) dv) + g jvj k+1 # d p h a d p L 1 (jvj k 1+p dv) C 2 4 p X j=0 d p j h + a d p j L 1 (jvj k 1+p j dv) + g jvj k+1 3
5 (1 p k):

Proof. Note that

h a = O 2 e 2 a jvj h + a + Z a a e v (a y) g(y; v) dy or h a = O 2 " jvj k+1 e 2 a jvj h + a jvj k+1 + Z a a e v (a y) g(y; v) jvj k+1 dy !#
shows that

h a L 1 (jvj (k+1) dv) jvj (k+1) O 2 jvj k+1 " h + a L 1 (jvj (k+1) dv) + g jvj k+1 # :
Leibnitz's rule gives

d p h a d p = p X j=0 p j O 2 " d j d j (e 2 a jvj d p j h + a d p j # +( 1) p O 2 Z a a e v (a y) (a y) p g(y; v) v p dy = p X j=0 p j ( 2a) j O 2 " 1 jvj j e 2 a jvj d p j h + a d p j # +( 1) p O 2 Z a a e v (a y) (a y) p g(y; v) v p dy or d p h a d p = p X j=0 p j ( 2a) j O 2 " jvj k+1 p 1 jvj k+1 p+j e 2 a jvj d p j h + a d p j # +( 1) p O 2 v k+1 p
Z a a e v (a y) (a y) p g(y; v) v k+1 dy so there exists a constant C 0 > 0 such that

d p h a d p L 1 (jvj k 1+p dv) C 0 jvj (k+1 p) O 2 jvj k+1 p 2 4 p X j=0 d p j h + a d p j L 1 (jvj k 1+p j dv) + g jvj k+1 3 5 :
This ends the proof. Finally, Proposition 17 and Lemma 16 imply:

Corollary 18 Suppose that (37)(38)(43)(44) are satis…ed. There exists a constant C > 0 such that

d j h a d j L 1 (jvj k 1+p dv) C j+1 X l=0 (1 G is ) 1 l ! g jvj k+1 (0 j k):
Remark 19 Note that (43) and (44) are checkable since they are always satis-…ed by the specular parts of the boundary operators O i (i = 1; 2) and checkable for the di¤ use parts.

10 On the resolvent on the imaginary axis

Combining Lemma 15, Lemma 16, Corollary 18, Corollary 25, (36) and using the limit F g (s) de…ned by [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF] we get:

Theorem 20 Let k 2 N and let (37)(38)(43)(44) be satis…ed. Let

Z := ( g 2 L 1 ( ); g jvj k+1 2 L 1 (
)

)
be endowed with the norm kgk Z = g jvj k+1 : For any g 2 Z;

f 2 C; Re > 0g 3 ! ( T O ) 1 g 2 L 1 ( )
extends continuously to iRn f0g as a C k function F g (:) and there exists a constant C > 0 such that

d j ds j F g (s) C j+1 X l=0 (1 G is ) 1 l ! kgk Z (0 j k; s 6 = 0) :
11 Existence and estimates of (1 G ) 1

The preceeding sections show that the existence and estimate of (1 G ) 1 for = is (s 6 = 0) are the cornerstone of this work. We start with a general result.

Theorem 21 If 1 + 2 > 0 then r (G ) < 1 (Re 0; 6 = 0):

Proof. We have G = O 1 e 2 a jvj O 2 e 2 a jvj and G 0 = O 1 O 2 : Note that O 1 O 2 is stochastic so r (G 0 ) = 1: Accordng to [21], r ess (G 0 ) < 1 if 1 + 2 > 0 so r (G 0 )
is an isolated eigenvalue of G 0 with …nite algebraic multiplicity. We know that

kG k = O 1 e 2 a jvj O 2 e 2 a jvj e 4a Re < 1 if Re > 0 since e 2 a jvj
e 2a Re : Let = i ( 2 R): Note that the (operator) modulus jG j of G (see [START_REF] Chacon | Linear modulus of a linear operator[END_REF]) is such that We complement now Theorem 21 in di¤erent directions by adding suitable assumptions.

O 1 e 2 a jvj O 2 e 2 a jvj O 1 O 2 = G 0 and O 1 e 2 a jvj O 2 e 2 a jvj 6 = G 0 ( 6 = 0) so by [19] r ( O 1 e 2 a jvj O 2 e 2 a jvj ) < r (G 0 ) = 1 whence r (O 1 e 2 a jvj O 2 e
Theorem 23 Let K i (i = 1; 2) be compact and let 1 + 2 > 0. Then:

(i) If 1 > 0, 2 > 0 and, for almost all v 00 ; k 1 (v 00 ; :) 2 L 1 ( 1; 0) then c := sup j j kG k < 1 ( > 0) : If the kernels k i (:; :) of K i (i = 1; 2) are continuous and K 1 jvj 2 K 2 is bounded then there exists b c > 0 such that kG k 1 b c jIm j 2 ( ! 0):

(ii) If 1 > 0, 2 = 0 and, for almost all v 00 ; k 1 (v 00 ; :) 2 L 1 ( 1; 0) then c := sup j j G 2 < 1 ( > 0). If the kernel k 1 (:; :) of K 1 is continuous and

K 1 jvj 2 R 2 K 1 is bounded then there exists b c > 0 such that G 2 1 b c jIm j 2 ( ! 0):
(A similar statement holds if 1 = 0 and 2 > 0): (iii) In particular, in both cases (i) and (ii) we have

sup j j (1 G ) 1 < +1 ( > 0) and (1 G ) 1 = O(jIm j 2 ) ( ! 0):
Proof. Note that kG k e 4a Re (Re 0) so we may restrict ourselves to the strip f ; 0 Re 1g : Let = " + is; " 2 [0; 1] : Without loss of generality, we may restrict ourselves to the case 1 > 0: This case subdivides into two subcases:

1 > 0 and 2 > 0 (45) or 1 > 0 and 2 = 0:

Consider …rst the case (45).

G = O 1 e 2 a jvj O 2 e 2 a jvj = ( 1 R 1 + 1 K 1 ) e 2 a jvj ( 2 R 2 + 2 K 2 ) e 2 a jvj = 1 2 K 1 e 2 a jvj K 2 e 2 a jvj + H
where

H = 1 2 R 1 e 2 a jvj R 2 e 2 a jvj + 1 2 R 1 e 2 a jvj K 2 e 2 a jvj + 1 2 K 1 e 2 a jvj R 2 e 2 a jvj :
We have

kG k 1 2 K 1 e 2 a jvj K 2 + 1 2 + 1 2 + 1 2 = 1 2 K 1 e 2 a jvj K 2 + (1 1 ) (1 2 ) + (1 1 ) 2 + 1 (1 2 ) = 1 2 K 1 e 2 a jvj K 2 + 1 1 2 29 so kG k 1 1 2 1 K 1 e 2 a jvj K 2 : (47) Note that K 1 e 2 a jvj K 2 = K 1 e 2isa jvj b K 2 where b K 2 has the kernel b k 2 (v; v 0 ) = e 2"a jvj k 2 (v; v 0 ) k 2 (v; v 0 ):
We have

Z 0 1 k 1 (v; v 0 )dv = 1; Z 1 0 b k 2 (v; v 0 )dv Z 1 0 k 2 (v; v 0 )dv = 1: Since K 1 e 2isa jvj b K 2 f = Z 0 1 dvk 1 (v 00 ; v)e 2isa jvj Z 1 0 b k 2 (v; v 0 )f (v 0 )dv 0 (48) = Z 1 0 Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 )dv f (v 0 )dv 0 then K 1 e 2isa jvj b K 2 sup v 0 2(0;1) Z 1 0 Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 )dv dv 00 :
We recall (see [START_REF] Rudin | Analyse réelle et complexe[END_REF], Thm 1.39, p. 30) that for any complex function h 2 L 1 ( );

Z hd = Z jhj d
if and only if there exists a constant such that h = jhj : It follows that

Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) dv < Z 0 1 k 1 (v 00 ; v) b k 2 (v; v 0 ) dv
otherwise there exists a constant such that

k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) = k 1 (v 00 ; v) b k 2 (v; v 0 )
so e 2isa jvj = 1 and = e 2isa jvj is not a constant. Thus, for Re 0 and 6 = 0;

Z 1 0 Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) dv dv 00 < Z 1 0 Z 0 1 k 1 (v 00 ; v) b k 2 (v; v 0 ) dv dv 00 Z 1 0 Z 0 1 k 1 (v 00 ; v)k 2 (v; v 0 ) dv dv 00 = 1:
Let us show that for any constant c > 0 sup

c jsj c 1 sup "2[0;1] sup v 0 Z 1 0 Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) dv dv 00 < 1: (49)
Let us argue by contradiction by supposing that this supremum is equal to

1: Note that b k 2 (v; v 0 ) = e
2"a jvj k 2 (v; v 0 ) and K 2 is weakly compact, i.e. fk 2 (:; v 0 ); v 0 2 (0; 1)g is a relatively weakly compact subset of the unit sphere of L 1 ( 1; 0) (at this stage we do not need the compactness of K 2 ). There exist " j ! ", v 0 j ! !, s j ! s 2 c; c 1 and g 2 L 1 ( 1; 0) such that

k 2 (:; v 0 j ) ! g weakly in L 1 ( 1; 0): (50)
and

Z 1 0 Z 0 1 k 1 (v 00 ; v)e 2is j a jvj e 2" j a jvj k 2 (v; v 0 j ) dv dv 00 ! sup c jsj c 1 sup "2[0;1] sup v 0 Z 1 0 Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) dv dv 00 = 1:
Since the sequence k 2 (:; v 0 j ) j is equiintegrable and since, for almost all v 00 ; k 1 (v 00 ; :) 2 L 1 ( 1; 0) we have

Z 0 k 1 (v 00 ; v)e
Z 0 1 k 1 (v 00 ; v)e 2" j a jvj k 2 (v; v 0 j ) dv ! Z 0 1 k 1 (v 00 ; v)e
2"a jvj g(v) dv:

Since Z 1 0 Z 0 1 k 1 (v 00 ; v)e 2is j a jvj e 2" j a jvj k 2 (v; v 0 j ) dv dv 00 Z 1 0 Z 0 1 k 1 (v 00 ; v)e 2" j a jvj k 2 (v; v 0 j ) dv dv 00 1 then Z 0 1 Z 1 0 k 1 (v 00 ; v)dv 00 e 2" j a jvj k 2 (v; v 0 j ) dv ! 1:
This last limit shows that " j ! 0 and Z 0 1

Z 1 0 k 1 (v 00 ; v)dv 00 g(v) dv = 1 or indeed Z 1 0 Z 0 1 k 1 (v 00 ; v)g(v) dv dv 00 = 1: Hence Z 1 0 Z 0 1 k 1 (v 00 ; v)e 2isa jvj g(v) dv dv 00 = Z 1 0 Z 0 1 k 1 (v 00 ; v)g(v) dv dv 00 = 1
and the inequality

Z 0 1 k 1 (v 00 ; v)e 2isa jvj g(v) dv Z 0 1 k 1 (v 00 ; v)g(v) dv implies the equality Z 0 1 k 1 (v 00 ; v)e 2isa jvj g(v) dv = Z 0 1 k 1 (v 00 ; v)g(v) dv
which is not possible since s 6 = 0. This ends the proof of (49). Hence kG "+is k < 1:

(51)

We have and sup j j

K 1 e 2isa jvj b K 2 sup v 0 2(0;1) Z 1 0 Z 0 1 k 1 (v 00 ; v)e
Z 1 Z 1 k 1 (v 00 ; v)e 2isa jvj e 2"a jvj k 2 (v; v 0 )dv dv 00 = 0 (52) uniformly in " 2 [0; 1] : If K 2 is compact then fk 2 (:; v 0 ); v 0 2 (0; 1)g is a relatively compact subset of L 1 (
(1 G ) 1 (1 c ) 1 ; ( > 0) :
Let us analyze the function

R 3 s ! sup v 0 2(0;1) Z 1 0 Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) dv dv 00 (depending on " 2 [0; 1]) in the vicinity of s = 0: Consider …rst Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) dv = s Z 0 1 k 1 (v 00 ; v) cos( 2sa jvj ) b k 2 (v; v 0 ) dv 2 + Z 0 1 k 1 (v 00 ; v) sin( 2sa jvj ) b k 2 (v; v 0 ) dv 2 and let u " (s; v 0 ; v 00 ) := Z 0 1 k 1 (v 00 ; v) cos( 2sa jvj ) b k 2 (v; v 0 ) dv 2 + Z 0 1 k 1 (v 00 ; v) sin( 2sa jvj ) b k 2 (v; v 0 ) dv 2 (" comes from b k 2 (v; v 0 ) = e
2"a jvj k 2 (v; v 0 )). We may write u " (s) or u(s) for simplicity. We note that

u " (0; v 0 ; v 00 ) = Z 0 1 k 1 (v 00 ; v) b k 2 (v; v 0 ) dv 2 Z 0 1 k 1 (v 00 ; v)k 2 (v; v 0 ) dv 2 = u 0 (0; v 0 ; v 00 ): We have @u @s = 4a Z 0 1 k 1 (v 00 ; v) cos( 2sa jvj ) b k 2 (v; v 0 ) dv Z 0 1 k 1 (v 00 ; v) jvj sin( 2sa jvj ) b k 2 (v; v 0 ) dv +4a Z 0 1 k 1 (v 00 ; v) sin( 2sa jvj ) b k 2 (v; v 0 ) dv Z 0 1 k 1 (v 00 ; v) jvj cos( 2sa jvj ) b k 2 (v; v 0 ) dv so @u @s (0; v 0 ; v 00 ) = 0: We have Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) dv = p u( ; v 0 ; v 00 ) so @ @s p u(s; v 0 ; v 00 ) = @u @s 2 p u(s; v 0 ; v 00 ) is such that @ @s p u(s; v 0 ; v 00 ) s=0 = 0 and @ 2 @s 2 p u(s; v 0 ; v 00 ) = 1 2 @ 2 u @s 2 p u ( @u @s ) 2 2 p u u = 1 2 2 @ 2 u @s 2 u @u @s 2 2 p uu so @ 2 @s 2 p u(s; v 0 ; v 00 ) =0 = 1 2 @ 2 u @s 2 (0; v 0 ; v 00 ) p u(0; v 0 ; v 00 ) :
On the other hand

@ 2 u @s 2 s=0 = 8a 2 Z 0 1 k 1 (v 00 ; v) b k 2 (v; v 0 ) dv Z 0 1 k 1 (v 00 ; v) jvj 2 b k 2 (v; v 0 ) dv ! +8a 2 Z 0 1 k 1 (v 00 ; v) jvj b k 2 (v; v 0 ) dv Z 0 1 k 1 (v 00 ; v) jvj b k 2 (v; v 0 ) dv so 1 8a 2 @ 2 u @s 2 s=0
is given by

Z 0 1 k 1 (v 00 ; v) b k 2 (v; v 0 ) dv Z 0 1 k 1 (v 00 ; v) jvj 2 b k 2 (v; v 0 ) dv ! Z 0 1 k 1 (v 00 ; v) jvj b k 2 (v; v 0 ) dv 2 :
Since we have strict inequality in the Cauchy-Schwarz inequality which is to say

Z 0 1 k 1 (v 00 ; v) jvj b k 2 (v; v 0 ) dv 2 < Z 0 1 k 1 (v 00 ; v) b k 2 (v; v 0 ) dv Z 0 1 k 1 (v 00 ; v) jvj 2 b k 2 (v; v 0 ) dv ! we see that c " (v 0 ; v 00 ) := Z 0 1 k 1 (v 00 ; v) b k 2 (v; v 0 ) dv Z 0 1 k 1 (v 00 ; v) jvj 2 b k 2 (v; v 0 ) dv ! Z 0 1 k 1 (v 00 ; v) jvj b k 2 (v; v 0 ) dv 2 > 0
and is continuous for smooth (say continuous) functions k 1 and k

2 (" comes again from b k 2 (v; v 0 ) = e 2"a jvj k 2 (v; v 0 )). Now p u(s; v 0 ; v 00 ) = p u(0; v 0 ; v 00 ) + s 2 2 @ 2 @s 2 p u( ; v 0 ; v 00 )
where 2 (0; s) or 2 (s; 0) according as s > 0 or s < 0: Write it as p u(0; v 0 ; v 00 ) p u(s; v 0 ; v 00 ) = s 2 2 @ 2 @s 2 p u( ; v 0 ; v 00 ) :

For smooth (say continuous) functions k 1 and k 2 @ 2 @s 2 p u(s; v 0 ; v 00 ) ! @ 2 @s 2 p u(s; v 0 ; v 00 ) s=0 = 1 2 @ 2 u @s 2 (0; v 0 ; v 00 ) p u(0; v 0 ; v 00 ) (as s ! 0) uniformly in (v 0 ; v 00 ) and " 2 [0; 1] : On the other hand

1 8a 2 @ 2 u @s 2 (0; v 0 ; v 00 ) = c " (v 0 ; v 00 ) so 1 2 @ 2 u @s 2 (0; v 0 ; v 00 ) p u(0; v 0 ; v 00 ) = b c " (v 0 ; v 00 ) := 4a 2 c " (v 0 ; v 00 ) p u " (0; v 0 ; v 00 ) :
is (say) continuous and bounded away from zero uniformly in " 2 [0; 1]. Hence This ends the proof in the case (45). Consider now the case (46). In this case 2 = 0 and 2 = 1 so Remark 27 In Theorem 23, the continuity assumption on the kernels k 1 and k 2 could probably be replaced by a piecewise continuity assumption; we have not tried to elaborate on this point here.

p u " (0; v 0 ; v 00 ) p u " (s; v 0 ; v 00 ) s 2 2 b c " (v 0 ; v 00 ) 2 for s small enough. Let b := inf "2[0;1] inf v 0 2(0;1) Z 1 0 b c " (v 0 ; v 00 ) 2 dv 00 > 0: Thus, Z 0 1 k 1 (v 00 ; v) b k 2 (v; v 0 ) dv Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) dv s 2 2 b c " (v 0 ; v 00 ) 2 (" 2 [0; 1]) for s small enough. Thus Z 1 0 dv 00 Z 0 1 k 1 (v 00 ; v) b k 2 (v; v 0 ) dv Z 1 0 dv 00 Z 0 1 k 1 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) dv s 2 2 Z 1 0 b c " (v 0 ; v 00 ) 2 dv 00
G = ( 1 R 1 + 1 K 1 )
12 Rates of convergence to equilibrium

We give …rst algebraic estimates of the resolvent on the imaginary axis. Remark 30 A su¢ cient criterion of irreducibility of e tT O t 0 is given in Theorem 7. The continuity of the kernels k i (:; :) (i = 1; 2) could probably be relaxed, see Remark 27.

Remark 31 A priori, the rates of convergence given in this paper depend on the condition 1 = 1 or 2 = 1: Two kinds of assumptions appear in this work: The "kernel" assumptions (37) (38) which can be checked only if 1 = 1 or

g

  jvj k+1 < +1; see Theorem 29. Apart from Theorem 21 and Theorem 23 (which hold under the general condition 1 + 2 > 0), the paper is based upon a set of structural assumptions (37)(38)(43)(44). A priori, Assumptions (43)(44), which say that jvj (k+1) O 1 jvj k+1 and jvj (k+1 p) O 2 jvj k+1 p are bounded (0 p k) ;

1 = 1 :dv jvj and b g + 2 L 1

 1121 Let g : (y; v) 2 ( a; a) ( 1; 0) ! g(y; v); g + : (y; v) 2 ( a; a) (0; +1) ! g(y; v) (0; +1) ; dv jvj :

  a y) g(y; v) dy + Z a a e jvj (y+a) g(y; v) dy then the key object to deal with is the resolvent (1 G ) 1 where G = O 1 e

1 jvj j O 2 1 jvj

 11 are uniformly bounded on f 2 C; Re 0g (for the usual operator norms) provided that O 1 p j are bounded operators (0 j p k).(37)We need also the additional conditions G : L 1 ((0; +1]; dv) ! L 1 ((0; +1]; dv jvj k+1 ) and d d G : L 1 ((0; +1]; dv) ! L 1 ((0; +1]; dv jvj k )

2 a

 2 jvj ) < 1 ( 6 = 0) and r (G ) < 1 (Re 0; 6 = 0): Remark 22 We can show similarly that r ( e G ) < 1 (Re 0; 6 = 0) where e G := O 2 e v 2a O 1 e jvj 2a :

2is j a jvj e 2 " j a jvj k 2 1 k 1 (

 2211 (v; v 0 j ) dv ! 0 ( ! 0 )uniformly in j and (50) implies Z 0

sup c jsj c 1 sup

 1

jvj k 2

 2 (v; v 0 )dv dv 00 :Let us show that lim jsj!1 K 1 e 2isa jvj b K 2 = 0 uniformly in " 2 [0; 1]. By weak compactness of K i (i = 1; 2) (and an equiintegrability argument) it su¢ ces to show that for any > 0 lim

1 Z 1 k 1

 111 1; 0) and consequently, for almost all v 00 2 (0; 1) ; n k 1 (v 00 ; :)e 2"a j:j k 2 (:; v 0 ); v 0 2 (0; 1) ; " 2 [0; 1] o is a relatively compact subset of L 1 ( 1; ): A Riemann-Lebesgue argument gives lim jsj!(v 00 ; v)e 2isa jvj e 2"a jvj k 2 (v; v 0 )dv = 0 uniformly in v 0 2 (0; 1) and " 2 [0; 1] : Finally, (52) holds by the dominated convergence theorem. Hence c := sup j j kG k < 1; ( > 0)

k 1 2 Z 1 0

 121 (v 00 ; v)e 2isa jvj b k 2 (v; v 0 ) dv s 2 b c " (v 0 ; v 00 ) 2 dv 00so that taking the in…mum in v 0 2 (0; 1) and " 2 [0; 1] on both sides

jvj k 1 (K 1 e 2 a jvj R 2 e 2 a

 12 v; v 0 )f (v 0 )dv 0 dv so that jvj K 1 has the same structure as the operator K 1 e 2 a jvj K 2 considered previously (see (48)). In particular, arguing as previously, one sees that for any 0 < c < c 0 sup c jsj c 1 sup "2[0;1]

2 l 1 f

 21 K 1 < 1 and (53)) there exists c < 1 such that G 2 l c 8l and then we can pass to the limit in(1 G l ) 1 f = (I + G l ) 1 G = (I + G l ) " l ! 0 + and s l ! b s to show that (1 G l ) 1 f ! 1 G b 1 f (l ! 1):Remark 26 We have also a similar statement with 1 e G

Theorem 28 ( 1 G;jvj 0 2 L

 2812 We assume that O 1 = K 1 or O 2 = K 2 : Let the kernels k i (:; :) of K i (i = 1; 2) be continuous. Let k 2 N and let (37)(38)(43)(44) be satis…ed. Then, for any g 2 Z;f 2 C; Re > 0g 3 ! ( T O ) 1 g 2 L 1 ( ) extends continuously to iRn f0g as a C k fucntion Rn f0g 3 s ! F g (s) 2 L 1 ( ) such that sup jsj 1 d j ds j F g (s) < +1 (0 j k)(54)and there exists a constant C > 0 such thatd j ds j F g (s) C s 2(j+1) kgk Z (0 j k; 0 < jsj 1): is ) 1 l ! kgk Z (0 j k; s 6 = 0):The fact that s ! (1 G is ) 1 is uniformly bounded outside any neighborhood of 0 shows (54). It su¢ ces to prove (55) for small s: By using Theorem 38 This ends the proof.We are now ready to prove the main result of this paper.Theorem 29 We assume thatO 1 = K 1 or O 2 = K 2 :Let the kernels k i (:; :) of K i (i = 1; 2) be continuous and let e tT O t 0 be irreducible. Let k 2 N and let (37)(38)(43)(44) be satis…ed. Ifg 2 D(T O ) and Z jg(x; v)j jvj (k+1) dxdv < +1 then e tT O g (t ! +1):Proof. The ergodic projection of e tT O t 0 is given by P g = R g 0 where 0 is given by (17) and is normalized in L 1 ( ): Then Z (g P g) 1 ( ) because of[START_REF] Lasota | Chaos, Fractals and Noise. Stochastic Aspects of Dynamics[END_REF]. The assumptions in Theorem 10 are satis…ed so g P g 2 Ran(T O ): Since g P g 2 D(T O ) then Theorem 28 and Corollary 3 end the proof.

  for small s 2 Rn f0g);

	see Theorem 23.
	It follows that
	sup jsj	d j ds j F g (s) < +1 ( > 0; 0 j k)

  e

								2 a jvj R 2 e	2 a jvj
		=	1 R 1 e	2 a jvj R 2 e	2 a jvj + 1 K 1 e	2 a jvj R 2 e	2 a jvj :
	It follows that
	G 2 =		1 R 1 e	2 a jvj R 2 e	2 a jvj + 1 K 1 e	2 a jvj R 2 e	jvj 2 a	2
	=		1 R 1 e	2 a jvj R 2 e	2 a jvj	1 R 1 e	2 a jvj R 2 e	2 a jvj
		+ 1 K 1 e	2 a jvj R 2 e	2 a jvj	1 K 1 e	2 a jvj R 2 e	2 a jvj
		+ 1 R 1 e	2 a jvj R 2 e	2 a jvj	1 K 1 e	2 a jvj R 2 e	2 a jvj
		+ 1 K 1 e	2 a jvj R 2 e	2 a jvj	1 R 1 e	2 a jvj R 2 e	2 a jvj
	=		2 1 K 1 e	2 a jvj R 2 e		2 a jvj K 1 e	2 a jvj R 2 e	2 a jvj + H
	where
	H	=	2 1 R 1 e	2 a jvj R 2 e	2 a jvj R 1 e	2 a jvj R 2 e	2 a jvj
			+ 1 1 R 1 e	2 a jvj R 2 e	2 a jvj K 1 e	2 a jvj R 2 e	2 a jvj
			+ 1 1 K 1 e	2 a jvj R 2 e	2 a jvj R 1 e	2 a jvj R 2 e	2 a jvj :
	Hence
	G 2	2 1 K 1 e	2 a jvj R 2 e	2 a jvj K 1 + (1	1 ) 2 + 2 (1	1 ) 1
	=	2 1 K 1 e	2 a jvj R 2 e	2 a jvj K 1 + [1

1 + 1 ]

2 > 0;[START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF]. The lack of spectral gaps for such collisionless kinetic models means there are no obvious rates of convergence to equilibrium.

= 1 or

= 1,(7)

= 1 (seeRemark 14) and the "non-kernel" assumptions (43)(44) which are satis…ed even by the re ‡ections conditions (seeRemark 19). (Note that Theorem 21 and Theorem 23 hold under the very general condition 1 + 2 > 0:) The extension of the theory to the general case 1 + 2 > 0 (or at least to the case 1 2 > 0) should depend on a weakening of the "kernel" assumptions which are used essentially in the proof of the key Lemma 15.

Since r (G ) < 1 for Re 0 and 6 = 0 (see Theorem 21) then for 6 = 0

and

and

Remark 24 (i) We have also a similar statement with e G := O 2 e v 2a O 1 e jvj 2a instead of G :

(ii) The compactness assumption on K i (i = 1; 2) (which is used in the study of the norm of kG k or G 2 as jsj ! 1 only) could be avoided by analyzing G 2 in the case (i) and G 3 in the case (ii) (and using Dunford-Pettis arguments). Such a proof is however too cumbersome to be presented. Note that K i (i = 1; 2) are compact if the kernels k i (:; :) of K i (i = 1; 2) are continuous.

Corollary 25 Let 1 + 2 > 0: We assume that for almost all v 00 2 (0; 1);

extends continuously (in the strong operator topology) to iRn f0g : for any " > 0 if the structural assumptions are satis…ed for all k 2 N; see [START_REF] Petterson | On weak and strong convergence to equilibrium for solutions to the linear Boltzmann equation[END_REF]. Whether one can reach the limit rate O 1 p t

(or can go beyond this rate) in the context of kinetic theory is an open problem. Note also that if there exists a constant C > 0 such that

then another quanti…ed version of Ingham's theorem (see Remark 4) gives the rate O( q ln(t) t ): However, in practice, the veri…cation of (56) seems to be out of reach.

(ii) A completely open problem is to quantify the sweeping phenomenon (20) in case of lack of invariant densities.

(iii) We know (see Theorem 12) that the imaginary axis is the boundary spectrum of the generator if 1 = 1 or 2 = 1: The extension of this result to more general partly di¤ use models is an open problem.

Remark 33 This work could be extended to non-monoenergetic free models in slab geometry @f @t (t; x; v; ) + v @f @x (t; x; v; ) = 0; (x; v; ) 2 where = ( a; a) ( 1; 1) (0; +1) with stochastic partly di¤ use boundary conditions

for v 2 (0; 1) and jvj f (t; a; v; ) = 1 jvj f (t; a; v; )+ 1

for v 2 ( 1; 0) under the convexity condition [START_REF] Arkeryd | Boltzmann asymptotics with di¤use re ‡ection boundary conditions[END_REF]. Indeed, the approach taken in [START_REF] Mokhtar-Kharroubi | On asymptotic stability and sweeping of collisionless kinetic equations[END_REF] could be extended to this model and the arguments of the present paper could be adapted accordingly. We have not tried to elaborate on this point here.

The specular re ‡ection R 1 : L 1 (( 1; 0) ; dv) ! L 1 ((0; +1) ; dv) de…ned by (R 1 ') (v) = '( v) could also be replaced by a more general deterministic boundary operator of the form (R 1 ') (v) = '( (v)) where : (0; +1) ! ( 1; 0) is a smooth measure-preserving function. A similar remark applies to R 2 :

Remark 34 We are con…dent that our formalism could extend to multidimensional (in space) geometries with partly di¤ use boundary operators. However, such an extension is not straightforward at all and faces serious additional problems we hope to be able to deal with in the near future.