Relative Entropies for Kinetic Equations in Bounded Domains (Irreversibility, Stationary Solutions, Uniqueness)

The relative-entropy method describes the irreversibility of the Vlasov-Poisson and Vlasov-Boltzmann-Poisson systems in bounded domains with incoming boundary conditions. Uniform-in-time estimates are deduced from the entropy. In some cases, these estimates are sufficient to prove the convergence of the solution to a unique stationary solution, as time goes to infinity. The method is also used to analyse other types of boundary conditions such as mass- and energy-preserving diffuse-reflection boundary conditions, and to prove the uniqueness of stationary solutions for some special collision terms.

conditions for f will be presented. We shall however detail the case where the distribution function of incoming particles is prescribed.
Notations. We recall that the spatial domain is denoted by !. From now on, we assume that ! is bounded and @! is of class C 1 . We shall denote by = ! IR d and ? = @ = @! IR d the phase space and its boundary respectively. Let d @! be the surface measure induced on @! by Lebesgue's measure. The outward unit normal vector at a point x of @! is denoted by (x). For any given x 2 @!, we set (x) = fv 2 IR d : v (x) > 0g and ? = f(x; v) 2 ? : v 2 (x)g : Finally, d (x; v) stands for the measure j (x) vj d ? (x; v) where d ? (x; v) = d @! (x) dv is the measure induced by Lebesgue's measure on ?. By a standard abuse of notations, we will not distinguish a function and its trace on the boundary.
(H2) The external electrostatic potential is assumed to be in C 2 (!). Without loss of generality we assume that 0 0.
(H3) The function has the following property Property P The function is de ned on (min x2! 0 (x); +1), bounded, smooth, strictly decreasing with values in IR + , and rapidly decreasing at in nity, so that sup x2! Z +1 0 s d=2 (s + 0 (x)) ds < +1 : We denote by ?1 its inverse function to IR extended by an arbitrary, xed, strictly decreasing function.
(H4) The collision operator Q is assumed to preserve the mass R IR d Q(g) dv = 0, and satis es the following H-theorem D g] = ? Z IR d Q(g) 1 2 jvj 2 ? ?1 (g) dv 0 ; (2) for any nonnegative function g in L 1 (IR d ).
(H5) We assume that D g] = 0 () Q(g) = 0 : The aim of this paper is to study the irreversibility of the system (1), the uniqueness of the stationary solutions and the eventual convergence to a stationary solution for large time asymptotics. The main ingredient is the derivation of a -dependent relative entropy of the time-dependent solution versus a stationary solution of the problem. In order to exhibit such a stationary solution, we introduce the map U, de ned on L 1 ( ) in the following way: for any function g 2 L 1 ( ), we denote by U g] = u the unique solution in W 1  Lemma 1.1 Let 0 and consider the set of admissible functions for Q FP; de ned by A 1 = ff 2 L 1 (IR d v ) : 0 f ?1 a.e. and v q f(1 ? f); r q f(1 ? f) 2 L 2 (IR d v )g Lemma 1.3 Assume that 2 L 1 and > 0 a.e. Then the operator Q E is bounded on L 1 \L 1 (IR d ).
Moreover, for any measurable function and for any increasing function H on IR, we have Z IR d Q E (f) (jvj 2 ) dv = 0 and H(f) = Z IR d Q E (f) H(f) dv 0 : Finally, if H is strictly increasing, the three following assertions are equivalent: 1. H(f) = 0. 2. Q E (f) = 0. 3. There exists such that f(v) = (jvj 2 ).
Consequently, any function having Property (P) satis es Assumptions (H1)-(H5). We shall see in Section 3 that the condition > 0 a.e. can be slightly weakened.
Example 5 : Electron-Electron collision operator. The Boltzmann collision operator Q B ee; , possibly including the Pauli exclusion term (case > 0) or the Fokker-Planck-Landau collision operator Q L ee; , namely The operator Q L ee; has been derived by Lemou 70] trough a grazing limit of the Boltzmann operator. A compatible in ow function takes the form (u) = + e (u? )= ?1 .
In each of the above examples, for simplicity, the velocities are taken in IR d , but we could as well consider a setting for which v, dv and 1 1.3 Outline of the paper and references We rst deal with the irreversibility due to the boundary conditions and, eventually, the collision kernel (Theorem 2.1). In the one-dimensional case and under technical regularity assumptions, the large time limit solution of the Vlasov-Poisson system, which is overdetermined on the boundary, is then characterized as the unique stationary solution (Theorem 2.5). For several models with various collision kernels corresponding to the above examples, the stationary solution is also identi ed as the unique limit for large times of the Cauchy problem (Corollary 2.4). Without self-consistent potential, a uniqueness result (Theorem 2.6) allows to identify the asymptotic solution (Theorem 2.7) in a special case corresponding to boundary conditions which are not compatible with the collision kernel.
Irreversibility driven by collisions is a well known topic 30]. On the opposite, the large time behaviour of solutions of the Vlasov-Poisson system is not very well understood. A scattering result due to Caglioti and Ma ei 25] is more or less the unique result (in the one-dimensional periodic case) which has been obtained up to now.
The large-time asymptotics of the linearized version of the Vlasov-Poisson system is known under the name of Landau damping 32,51]. The instability of the so-called BGK waves has been studied in a series of papers by Strauss, Guo and Lin 62,63,64,65,71], while the nonlinear stability has been tackled by Rein and his co-authors 9, 10, 57,22]. Also see the much more di cult case of gravitational forces 92,94,58,60,61], and 23] for recent results in the presence of a con ning potential. Some extensions to the case of electromagnetic forces (Vlasov-Maxwell system) are also available.
Without con nement in the whole space, dispersion e ects dominate for large times and asymptotics are more or less understood 68,86] although the description of the asymptotic behaviour is not very precise 48]. For bounded domains with specular re ection boundary conditions or unbounded domains with con nement 46,78,79], the stability results do not provide so much information on the solutions (which are time-reversible at least for classical solutions). Injection or di use re ection boundary conditions introduce a source of irreversibility which is the scope of our paper. We also consider the case of compatible collision terms. By compatible, we mean that the stationary solution determined by the boundary conditions belongs to the kernel of the collision operator, if there is any. This is a severe restriction for some collision kernels like the classical Boltzmann collision operator (only maxwellian functions are allowed), a case which has been studied a long time ago, at a formal level, by Darroz es and Guiraud 31]. There are other cases where compatibility is not as much restrictive, like in the case of the elastic collision operator. In case of uncompatible boundary conditions, again very little is known. Some existence results of stationary solutions have been obtained by Arkeryd and Nouri 89,4,3,2], but as far as we know, uniqueness is mainly open, and some of our results are a rst step in that direction.
Technically speaking, we are going to use weak or renormalized solutions and trace properties of these solutions which have recently been studied by Mischler 41,78,79,80], and entropy functionals which are very close to the ones which are used for nonlinear parabolic equations 27,18]. There are some deep connections between entropies for kinetic equations and for nonlinear di usions, which are out of the scope of this paper. However, to illustrate this point, we will derive a di usive limit, at a formal level (see 55,74,52,13,33,91,83] for rigorous results).
Further references corresponding to more speci c aspects will be mentioned in the rest of the paper. We will not provide all details for each proof and will systematically refer to papers in which details or similar ideas can be found. Some of the results presented here have been announced in a note 12]. This paper is organized as follows. In Section 2, we will develop at a formal level a strategy to study the long time behaviour. Namely, we will prove an entropy inequality for the Vlasov-Boltzmann-Poisson system with incoming boundary conditions and state its consequences on the long time behaviour and the stationary solutions. Section 3 is devoted to the application of the strategy to the various examples cited above. For uncompatible boundary conditions, the uniqueness of the stationary solutions of the equation corresponding to a special BGK approximation of the Boltzmann collision operator, when there is no self-consistent potential, and a corresponding large time convergence result are proved in Section 4. In Section 5, we extend the relative entropy approach to other types of boundary conditions. Technical results (proof of Theorem 2.5, statements on the Bolza problem) and general considerations (nonlinear stability, di usive limits and relations between relative entropies for kinetic equations and for nonlinear parabolic equations) have been postponed to Appendices A-D.

Strategy and results
In this section we shall expose our strategy for the study of irreversibility and the large time asymptotics. One of the main di culties is the lack of uniform in time estimates. For instance, the total mass is not conserved since particles are continuously injected into the domain. By introducing a relative entropy, we shall obtain a priori estimates and then use them in order to pass to the limit. All computations are done at a formal level. Rigorous proofs corresponding to the various examples of Section 1 are postponed to Section 3. Theorem 2.1 Assume that f 0 2 L 1 \ L 1 is a nonnegative function such that f 0 jM] < +1. Let f be a smooth su ciently decaying solution of (1) and assume that and Q satisfy Assumptions Z ! D f](x; t) dx (6) where D f] is de ned in (2) and + is the boundary relative entropy ux given by

Relative entropy and irreversibility
Here, smooth means for instance C 1 and su ciently decaying means that all integrations by parts involved in the formal computation below can be done rigorously. Depending on Q, weaker conditions will be required for f: see Section 3. For weak or renormalized solutions, the equality in (6) will be replaced by an inequality. Proof. We rst deduce from the Vlasov-Boltzmann equation (1) (8) We recall that the above identity requires the use of mass conservation @ @t + div x j = 0 ; where (x; t) = R IR d f(x; v; t) dv and j(x; t) = R IR d vf(x; v; t) dv, which in turn gives d dt 1  Taking the sum of (7) (in which = ) and (8), and noticing that u t Since gjh] is always nonnegative, the above theorem provides a uniform in time control on f(t). Like in whole space problems 7, 46,39], the relative entropy f(t)jM] provides a Lyapunov functional for the study of the large time behaviour, which can also be used to study the nonlinear stability 92, 22, 23] (see Appendix C). An important di erence with whole space problems and with previous studies of boundary value problems 26,19] is that the total mass is not conserved (see Section 5 for boundary conditions preserving the mass).

The large time limit
Integrating the entropy dissipation inequality with respect to time provides the following inequality  (9) (this is an equality for classical solutions). Since the left hand side is the sum of three nonnegative terms (under Assumption (H4)), each of them is bounded by the right hand side. In order to investigate the large time behaviour of the solution (f; ), we consider an arbitrary increasing and diverging sequence (t n ) of positive real numbers and de ne (f n (x; v; t); n (x; t)) = (f(x; v; t + t n ); (x; t + t n )) : (10) It is clear from the above estimates that The last inequality provides a uniform in time estimate for f n as well as a uniform H 1 bound for n .
The remainder of the method consists in proving that 1. According to the Dunford-Pettis criterion, up to the extraction of a subsequence, (f n ; n ) weakly converges in L 1 loc (dt; L 1 ( )) L 1 loc (dt; H 1 0 (!)) towards a solution (f 1 ; 1 ) of (1), 2. The limit function f 1 satis es sup t2IR f 1 (t)jM] C and Z IR + f 1 (s)jM] ds = ? Z IR Z ! D f 1 (x; ; s)] dx ds = 0 : Depending on the a priori estimates, (1) will be satis ed by (f 1 ; 1 ) either as a weak solution or even in the sense of renormalized solutions. Item 2 above allows to show that f 1 = M on ? + and that Q(f 1 ) = 0 under Assumption (H5). Therefore (f 1 ; 1 ) is a solution of and (x; t) = 0 ; (x; t) 2 @! IR ; sup t2IR f(t)jM] C : (11) Notice that the time variable t lies in the whole real line and that the boundary conditions on f 1 are overdetermined, since f 1 is given on the whole boundary ? and not only on ? ? . A second source of overdetermination for the system (11) is the condition Q(f 1 ) = 0 (when Q is not identically vanishing).
As we shall see in Section 3, this program can be completed for each of the examples of Section 1.
When Q 0, Q = Q E or 6 = 0 in Examples 2, 3 and 5, if f 0 is bounded in L 1 , f(t) is also uniformly bounded in L 1 , and we may easily pass to the limit. The other examples (including the case = 0) require additional work (using for instance renormalized solutions). Up to this question which is a little bit delicate, the irreversibility result of Theorem 2.1 provides a characterization of the large time limit that we can summarize in the following formal result (it is formal in the sense that we assume the convergence of the collision term, which is a property that has to be proved case by case).
Corollary 2.2 Assume that f 0 2 L 1 \L 1 is a nonnegative function such that f 0 jM] < +1. Under Assumptions (H1)-(H5), consider an unbounded increasing sequence (t n ) n2IN . If (f n ; n ) de ned by (10) weakly converges to some (f 1 ; 1 ) in L 1 loc (dt; L 1 ( )) L 1 loc (dt; H 1 0 (!)) and if Q(f n ) D 0 ! Q(f 1 ), then (f 1 ; 1 ) is a solution of (11) (which belongs to the kernel of Q for any (t; x) 2 IR ! and is such that f j? +(x; v; t) = (jvj 2 =2 + 0 (x)) for any t 2 IR + , (x; v) 2 ? + ). Proof. We have to prove the convergence of r x n r v f n to r x n r v f n as n ! +1. If f n is uniformly bounded in L 1 , by interpolation (see 73,67]) with the kinetic energy, n = R IR d f n dv is bounded in L 1 (dt; L q (IR d )) with q = 1 + 2=d. Using the compactness properties of r ?1 , it is easy to pass to the limit in the self-consistent term. Without uniform bounds, one uses renormalized solutions 41, 78, 79, 80] and (11) only holds in the renormalized sense (compactness for n is a consequence of averaging lemmas). u t 2.3 Are the solutions of the limit problem stationary ?
In this paragraph, we provide some rigorous results ensuring the stationarity of the solutions of the limit problem (11). If f 1 2 Ker Q depends only on jvj 2 (examples 2, 3 and 4), we apply the following Lemma 2.3 Let f 2 L 1 loc be a solution of the Vlasov equation in the renormalized sense. If f is even (or odd) with respect to the v variable, then it does not depend on t.
The proof is straightforward. The operator @ t conserves the v parity while v r x ? (r x + r x 0 ) r v transforms the v parity into its opposite. u t Corollary 2.4 Let f be a solution of (11) with Q = Q E , Q FP; , Q , Q ee; + Q E , Q ee; + Q , Q ee; + Q FP; or a linear combination of these operators (with nonnegative coe cients). Then f does not depend on t, and is nothing else than the function M de ned in (4) under the additional assumption that there are no closed characteristics if Q = Q E .
The proof is an immediate application of Lemma 2.3. Indeed, using the H-Theorem, we deduce that the kernel of a (nonnegative) linear combination of the above collision operators is equal to the intersection of the kernels. Therefore, any function f satisfying Q(f) = 0 is even with respect to v.
Step 1 : the electric eld is repulsive at x = 0. Along the characteristics, the total energy satis es @ @t 1 2 jV j 2 + (X; t) + 0 (X) = @ @t (X; t) : As a consequence, there exists v M > 0 depending on k + 0 k L 1 and k@ t k L 1 such that the following for all x 2 (0; 1) and s 2 IR, ?1 < T in (s; x; v) < s < T e (s; x; v) < +1, We claim that this ensures the existence of a constant C 1 > 0 such that (x; t) = R IR d f(x; v; t) dv C 1 , which implies, thanks to the Poisson equation, the existence of a positive constant C 2 > 0 such that for any t 2 IR, @ @x (0; t) C 2 : To prove our claim, we rst deduce from the Vlasov equation and the boundary condition that f(x; v; t) = f(0; V in (x; v; t); T in (x; v; t)) = 1 2 jV in (x; v; t)j 2 : In view of the above estimates on V in and due to the decay of , we get the estimate (x; t) Z +1 v M 1 2 jvj 2 + C M dv =: C 1 > 0 : The conclusion then holds with C 2 = 1 2 C 1 using Step 2 : Analysis of the characteristics in a neighborhood of (0; 0; t). Since the electric eld @ @x is (uniformly in t) positive in a neighborhood of x = 0 + , there exists x M 2 (0; 1) such that for every x 0 2 (0; x M ) and every t 0 2 IR ?1 < T in (t 0 ; x 0 ; 0) < t 0 < T e (t 0 ; x 0 ; 0) < +1 and X in (t 0 ; x 0 ; 0) = X e (t 0 ; x 0 ; 0) = 0 : Saying that f is constant along the characteristics means f(X in ; V in ; T in ) = f(X e ; V e ; T e ). Besides, we deduce from the boundary conditions (11) that f(X in ; V in ; T in ) = 1 Theorem 2.6 Assume that 0 and consider two nonnegative solutions f 1 and f 2 of (13) such that for any (x; v) 2 , f i (x; v) F D (x; v) = + e ( 1 2 jvj 2 + 0 (x)? )= ?1 (for i = 1; 2). Then f 1 = f 2 .
Note here that we do not make any assumption on 0 saying for instance that there are no closed characteristics. The proof of Theorem 2.6 is deferred to Section 4. Let us denote by f s the unique stationary solution of (13). A computation similar to the one of Theorem 2.6 provides the following result on large time asymptotics. Theorem 2.7 Assume that 0. If (14) is satis ed, then any solution of with an intial data f 0 such that 0 f 0 F D -weakly converges in L 1 ( ), as time tends to +1, towards the unique stationary solution f s of (13). bounded, then f(t) is bounded as well according to sup jf(t)j max(sup jf 0 j; sup ). The L 1 bound will be useful for passing to the limit. Throughout this section, we shall assume that (H6) f 0 2 L 1 x;v \ L 1 x;v ; jvj 2 f 0 2 L 1 x;v and f 0 jM] < +1 : Moreover, we require that (H7) Z +1 0 s (d+1)=2 (s) ds < +1 ; so that R IR d jvj 3 (jvj 2 ) dv makes sense, and we assume that (H8) 0 is bounded on ?A; +1) for any A > 0 : We shall rst prove two preliminary results and then state a theorem which covers the results of Theorem 2.1 and Corollary 2.2 at once. Proof. Since ?( ?1 ) 0 (u) = ?( 0 ?1 (u)) ?1 and ? 0 (u) C according to (15), we have, for any u 2 0; A], ?( ?1 ) 0 (u) 1=C. Consequently, These estimates allow us to prove rigourously a result on the large time behaviour for Q E . The existence of solutions can be found in 11,1,78,79,80]. In these references, stability results are also proved for renormalized solutions. Theorem 3.3 Assume that (H6)-(H8) hold and that is symmetric, measurable, nonnegative. The Vlasov-Poisson-Boltzmann system (1) with Q = Q E or Q = 0 admits a weak solution f 2 L 1 (IR + ) such that kf( ; ; t)k L 1 max kf 0 k L 1; inf @! 0 : The sequence (f n ; n ) de ned by (10) converges up to the extraction of a subsequence, -weakly in L 1 (IR + ) L 1 loc (IR + ; H 1 0 ( )), towards a solution (f 1 ; 1 ) of (11).
Proof. The L 1 estimate is straightforward in case Q = 0. For Q = Q E , we may use the fact that The existence proof of f goes as follows. For a given , we remark that f has to solve @ t f + v r x f ? (r x + r x 0 ) r v f + f = Q + (f) : The mapping f 7 ! g de ned by ( @ t g + v r x g ? (r x + r x 0 ) r v g + g = Q + (f) ; g jt=0 = f 0 ; g j ? = ( 1 2 jvj 2 + 0 ) : is contractive in L 1 ((0; T) ). It has a unique xed point which can be computed with an iteration scheme. Starting the iteration procedure from a nonnegative initial point, the Maximum Principle is satis ed at each step, which implies that the solution f is nonnegative if f 0 0 and 0. On the other hand, for any K 2 IR, K ? f also satis es the same equation with f 0 and ( 1 2 jvj 2 + 0 ) replaced by K ? f 0 and K ? ( 1 2 jvj 2 + 0 ) respectively, which proves the Maximum Principle for a solution of (1).
Let (f n (x; v; t); n (x; t)) = (f(x; v; t + t n ); (x; t + t n )) with lim n!+1 t n = +1 and consider the limit as n ! +1. According to (9)  where fjg] = R h f log f g ? f + g i dx dv + 1 2 jrU f ? g]j 2 dx. In 80], it is proved that any sequence of renormalized solutions of the Vlasov-Poisson-Fokker-Planck system which satis es the above entropy inequality has a subsequence which weakly converges in L 1 towards a renormalized solution of the same system. We apply this stability result to a sequence f n (x; v; t) = f(x; v; t + t n ). Then, we have a uniform bound in L 1 for f n . Besides, a Cauchy Schwartz inequality leads to R +1 0 ( R jvfn+ rvfnjdvdx) 2 R fndvdx ds R +1 0 R 1 fn jvf n + r v f n j 2 dvdx ds = R +1 tn R 1 f jvf + r v fj 2 dvdx ds ! 0 as n ! +1 which proves that lim n!+1 kvf n ( ; s) + r v ( ; s)f n k 2 L 1 ( ) = 0. This shows that the weak limit f 1 of a converging subsequence is a Maxwellian: f 1 = (x; t)M (v). Exactly as in 21], f 1 is a renormalized solution of the Vlasov-Poisson-Fokker-Planck system whose unique Maxwellian solution is given by (4) with (u) = C e ?u= . Thus we obtain the

Example 3: semiconductor BGK model
In this paragraph, we are going to give detailed estimates which allow us to prove directly that the large time limit is in the kernel of the collision operator, without proving the convergence of Q (f n ) to Q (f 1 ) in D 0 . We deal either with the standard BGK model ( = 0) or with the BGK model for fermions ( > 0): where is such that there exists two positive constants 0 and 1 for which  which implies that Q (f 1 ) = 0 : We have then proved that (f 1 ; 1 ) is a solution of the stationary Vlasov-Poisson system and f 1 is a Fermi-Dirac function. By Lemma 2.3, it is stationary. We deduce from Theorem 2.6 that f 1 is equal to M and there is therefore no need to extract a subsequence. u t

Example 5: Boltzmann or Fokker-Planck-Landau collision operators
The Boltzmannn equation has been extensively studied during the last 15 years, so we shall only brie y sketch how the case with a Poisson coupling and injection boundary conditions can be dealt with. The main di erence with standard approaches is that the total mass is not xed. Theorem 3.9 Let 0 2 L 1 ( ) be such that r 0 2 W 1;1 ( ) and consider a solution f of the Vlasov-Poisson-Boltzmannn system with a Boltzmann or a Fokker-Planck-Landau collision term such that, with the notations of Section 2, f( ; ; t)jM] f 0 jM] : Then t 7 ! M(t) := kf( ; ; t)k L 1 ( ) is uniformly bounded in L 1 (IR + ) and (f n ) n2IN de ned by (10) weakly converges in L 1 (IR + loc ) to (M; U M]).
Note here that the condition on 0 is certainly not optimal: apart from regularity conditions which have to do with the de nition of the characteristics (see 42,72]), the right condition should be given in terms of the existence of a lower bound for the functional which de nes U M] or equivalently in terms of the existence of a lower bound for R (f) dxdv (see 46] for a discussion of the notion of con nement and 46,47] for the equivalence of these conditions).
Proof. Consider rst the case > 0 (statistics of fermions) and assume that 0 0. Since 0 f 1 a.e., for almost all t > 0, the function f( ; ; t) is bounded in L 1 ( ) as soon as R f(x; v; t) jvj 2 dxdv is bounded uniformly with respect to t. Let us prove that f( ; ; t) is also relatively compact. This also proves that R jvj 2 f(x; v; t) dxdv is uniformly bounded and gives an upper bound for M(t): fjM] M(t) log (M(t)) ? C M(t) ; for some C 2 IR. The weak compactness in L 1 then follows by Dunford-Pettis' criterion (see 50]). u t Up to these preliminary estimates, the method is more or less standard and we will only refer to the existing literature. In case = 0, for the Boltzmann collision operator, one has to use the notion of In this Section, we consider the case without self-consistent potential (no Poisson coupling), when the collision operator is the BGK approximation of the Boltzmann collision operator for fermions given by The function is a bounded positive function. We claim that this implies that h = 0. Indeed, let (x 0 ; v 0 ) 2 with jv 0 j large enough in such a way that any characteristics with initial conditions in (x; v) 2 B r (x 0 ) B r (v 0 ) (with r > 0) is open. Let be a nonnegative smooth function which is strictly positive on B r (x 0 ) B r (v 0 ). The solution of v r x ? r x 0 r v ? = ; = 0 on ? ; (17) is nonnegative and does not vanish on B r (x 0 ) B r (v 0 ). Let us give a short proof of this fact. Assume that the characteristics which is given by @X @t = V ; @V @t = ?r x 0 (X) ; exists on a maximal interval (T in (x; v); T e (x; v)) 3 0. If such a characteristics is open, this means that either T in (x; v) > ?1 and X(x; v; T in (x; v)) 2 @!, or T e (x; v) < +1 and X(x; v; T e (x; v)) 2 @!. Note that since is a steady state, the problem is autonomous, so we dont need to introduce a speci c initial time (with the notations of (12), X(t; x; v; s) is replaced by X(t ? s; x; v; 0) = X(x; v; t ? s)).
Let (x; v; t) = R t 0 (X(s); V (s)) ds. If  The above method is also usefull for the study of large time asymptotics. It gives the convergence to the unique stationary solution and shows the connection with relative entropy formulations which have been extensively used throughout the rest of this paper. We denote by h the function f ? f s . Z Q 0 (h)H 0 h m dxdv = 0 : According to the same strategy as in Section 3, we de ne h n (x; v; t) = h(t + t n ; x; v) where t n is an arbitrary diverging sequence and deduce that up to the extraction of a subsequence, the sequence h n -weakly converges in L 1 ((0; T) ) towards a function h 1 such that Q 0 (h 1 ) = 0, @ t h 1 +v r x h 1 ? r x 0 r v h 1 = 0 and h 1 = 0 on @ . The rst identity implies that h 1 is a Maxwellian, which is even with respect to v and is therefore stationary in view of Lemma 2.3. Theorem 2.6 then implies that h 1 = 0.
The nonlinear case. As in the proof of Theorem 2.6, a simple computation gives d dt Z jhj dxdv + Z ? + jhj d ?
Z Q (f) ? Q (f s )] sgn(h) dxdv = 0 : The -weak limit h + 1 in L 1 ((0; T) ) of a subsequence of h + n = jh( ; ; + t n )j de ned as above satis es @ t h + 1 + v r x h + 1 ? r x 0 r v h + 1 + 1 h + 1 = Z IR d 1 j(h + 1 ) 0 j dv 0 where f 1 is, up to the extraction of a further subsequence, the limit of f( ; ; + t n ). The convergence in the collision term holds for the same reason as in the proof of Theorem 2.6. On the other hand h + 1 (x; v; t) = 0 for all t > 0, (x; v) 2 @ . As in the proof of Theorem 2.6 again, this implies that h + 1 = 0. But since jf 1 ? f s j h + 1 , we deduce that f 1 = f s . u t

Other boundary conditions
This section is devoted to further considerations on relative entropies corresponding to various types of boundary conditions. The case of di use re ection boundary conditions is studied with some details: after a de nition of such boundary conditions, which are such that the total mass is preserved, stationary solutions are found using a variational approach. These solutions are then used to de ne a relative entropy, which describes the irreversibility, and gives the uniqueness of the stationary solution when there is no closed characteristics, exactly like in the case of injection boundary conditions. Conditions preserving the energy and the mass are then introduced and further remarks are done concerning other types of possible boundary conditions.

Di use re ection boundary conditions (DRBC)
Here we consider as in Section 1 the full Vlasov-Poisson-Boltzmann system 8 > > > > > < > > > > > : and di use re ection boundary conditions for f. These conditions are de ned as follows. For any (x; t) 2 @! IR + , let f(x; v; t) v (x) dv : (19) Assuming that is de ned on IR, satis es (P) and is such that lim s!?1 (s) = +1, there exists a unique function : @! IR + ! IR for which + (x; t) = Z P ? (x) 1 2 jvj 2 + 0 (x) ? (x; t) jv (x)j dv 8 (x; t) 2 @! IR + : (20) With the notation m f (x; v; t) := 1 2 jvj 2 + 0 (x) ? (x; t) 8 (x; t) 2 @! IR d IR + ; we shall say that f is subject to di use re ection boundary conditions (DRBC) if and only if f(x; v; t) = m f (x; v; t) ; 8 t 2 IR + ; 8 (x; v) 2 ? ? : (21) Note that under this condition, the total mass is preserved: dv is a signed measure such that d~ = d on ? , with the notations of Section 1. From now on, we denote by M the L 1 -norm of f: M = kf( ; ; t)k L 1 ( ) 8 t 2 IR + : Under DRBC conditions, we shall now prove the existence of a stationary solution corresponding to any given mass by the mean of a variational approach. This solution then allows us to de ne a relative entropy, which we shall use to prove the uniqueness of the stationary solution. This relative entropy also describes the irreversibility and the large time asymptotics as in the case of injection boundary conditions. See the concluding remark of this section for further comments on the denomination: relative.  (20) of (x; t). According to (19)  (here we omit the dependence of in t since f does not depend on t either). In the following, we shall do as if and 0 were of class C 2 . For lower regularity, one has to take advantage of the uniqueness of the characteristics according to 42]. Consider two points x 1 and x 2 in @! such that the segment (x 1 ; x 2 ) is a subset of !: there exists a characteristics connecting x 1 to x 2 (Bolza problem in IR d : see 95] and Appendix B), i.e. a solution of dX dt = V ; dV dt = ?r( (X) + 0 (X)) ; X(0) = x 1 ; V (0) = v 1 such that for some t > 0, x(t) = x 2 , for some well chosen v 1 , with jv 1 j large enough. Since f is constant along the characteristics, f(x 2 ; v 2 ) = f(x 1 ; v 1 ). Because f only depends on the energy on the boundary (up to ) and since is strictly decreasing, we have: 1 2 jv 2 j 2 + 0 (x 2 ) ? (x 2 ) = 1 2 jv 1 j 2 + 0 (x 1 ) ? (x 1 ) :

A variational formulation in
But on the other hand, does not depend on t and the energy also is conserved along the characteristics: 1 2 jv 2 j 2 + 0 (x 2 ) = 1 2 jv 1 j 2 + 0 (x 1 ) : This is possible if and only if (x 2 ) = (x 1 ). See Lemma B.1 in Appendix B for more details on how to nd v. It remains to check that any two points of a C 1 connected domain in IR d can be connected by a nite number of segments in !, whose extremities are in @!. This is the purpose of Lemma B.2 in Appendix B. Thus (x) de ned by (19) does not depend on x and we can conclude by applying By an argument similar to the one of Corollary 5.3, it is then easy to prove that any stationary solution is necessarily of the form (25).

Remarks on the boundary conditions
To the boundary conditions for f correspond various well known situations of thermodynamics (see 8]).
In the case of injection (resp. di use re ection) boundary conditions, the temperature and the chemical potential (resp. the temperature and the mass) are xed, so that the energy and the mass (resp. the energy and the chemical potential) of the system uctuate: this is the grand canonical (resp. canonical) framework and the relative entropy can be identi ed with a grand potential (resp. free energy) function. The stationary state is uniquely de ned in both cases.
When the energy and the mass are xed (microcanonical framework), the relative entropy can be identi ed with an entropy function (in the usual sense of thermodynamics, up to a sign convention), but a di culty arises from the lack of uniqueness results of stationary states (see 38,17]). Other cases formally enter in our relative entropy formulation: for instance, if the volume is not xed, one could prescribe the pressure by requiring the equality of the incoming and outgoing uxes corresponding to the rst moment in the velocity.
Remark 5.5 Why we use the denomination relative for the entropy arises from the following reason.
In the three examples of boundary conditions studied in this paper (injection boundary conditions, diffuse re ection boundary conditions with xed temperature and di use re ection boundary conditions preserving mass and energy), the function is entirely de ned by , but we further impose that the minimum of is reached by the unique stationary solution corresponding to the boundary conditions. This in turn determines the Lagrange multipliers associated to the constraints. In that sense, the entropy is therefore relative to this stationary solution. The relative entropy functional can be interpreted { from a probabilistic point of view { as a conditional expectation, or simply as a measure of the distance to the stationary state (at least when it is unique). This notion of distance is also the one which appears when measuring the stability by the Casimir-energy method or in case of di usion equations with compatible nonlinearities (see 27,18]), as we shall see in Appendices C and D.
A End of the proof of Theorem 2.5 Let : 0; 1] IR ! IR, (x; t) 7 ! (x; t) be an analytic function in x with C 1 in time coe cients.
x (0; t 0 ), (iii) @ t @ n+1 x (0; t 0 ) = 0. Proof. We rst remark that (iii) is a direct consequence of (ii) and of the fact that e + 2n+1 (1) = e ? 2n+1 (1). In order to prove (i) and (ii), we insert the expansion of e " in (27) and identify the terms of the same power in ". From the zeroth order term, we obtain d dx e 0 = 2 @ x (0; t 0 ). The formulae for e 1 (i.e. (ii) with n = 0) follow from the order 1 term. For the higher order terms, we proceed by induction.
Namely, let n 2 IN be given and assume that (i), (ii) and (iii) hold up to the order n. Let us prove that they hold for n + 1. Terms of order " 2n+2 in the right hand side of (27) are obtained by taking n + 1 derivatives of @ x with respect to x, n derivatives with respect to x and two with respect to t, n ? 1 derivatives with respect to x and 4 with respect to t, , 2n + 2 with respect to t. Noticing that (iii) holds up to n and for all times t 0 2 IR, we deduce that the only non vanishing term in this expansion is the rst one. This leads to (i) for the index n + 1. In order to prove (ii), we proceed analogously. The only non vanishing term of order 2n+3 is the one corresponding to n+1 derivatives with respect to x and one derivative with respect to t. All the other terms involve t derivatives of @ k x (0; t) with k n, and are therefore vanishing in view of (iii). This leads to (ii) (with n replaced by n + 1).
u t A straightforward consequence is the following Corollary that we use in the proof of Theorem 2.5.

B Two technical lemmata for the Bolza problem
The Bolza problem is a standard question of mechanics. For a given potential and for any given pair (x 1 ; x 2 ) 2 ! 2 of points, does there exist a trajectory which connects x 1 to x 2 , for an appropriate initial velocity v 1 ? In this Appendix, we are going to prove two lemmata which are of interest for the proof of Corollary 5.3. We consider rst the Bolza problem for two points x 1 , x 2 2 @! such that the segment (x 1 ; x 2 ) is contained in !, and then prove that two arbitrary points of the boundary of ! can be connected by a nite number of such segments, under the assumption that ! is a connected and bounded domain. For simplicity, we assume that is of class C 2 , so that we deal with classical characteristics, but an extension based on the uniqueness of weaker notions of characteristics (see 42,72]) is easy to establish. Let x 1 ; x 2 2 @!. We shall say that (x 1 ; x 2 ) satisfy Property (S) if and only if (x i ) (x 2 ? x 1 ) 6 = 0, i = 1; 2, and (x 1 ; x 2 ) = ftx 1 + (1 ? t)x 2 : t 2 (0; 1)g !.
Let u 0 = x 2 ?x 1 jx 2 ?x 1 j . We may notice that if (S) is satis ed, there exists an > 0 such that 8 u 2 S d?1 ; ju ? u 0 j < =) fx 1 + tu : t > 0g \ ! is a neighborhood of (x 1 ; x 2 ) in ! : Lemma B.1 Let be a bounded C 2 potential de ned on ! and consider x 1 ; x 2 2 @! such that (x 1 ; x 2 ) has Property (S). Then there exists an A > 0 such that 8 a > A 9 v 1 2 a jS d?1 j IR d ; for which the characteristics de ned in ! by d 2 X dt 2 = ?r (X) ; X(0) = x 1 ; dX dt (0) = v 1 ends at x 2 .
Proof. For " > 0 and u 2 S d?1 , we denote by X ";u (t) the characteristics de ned by d 2 X ";u dt 2 = ?r (X ";u ) ; X ";u (0) = x 1 ; dX ";u dt (0) = 1 " u in IR d (we extend by a bounded C 2 function to IR d ). Consider the time rescaling: "s = t, and the rescaled characteristics Y ("; u; s) = X ";u ( s " ). @ 2 Y @s 2 = ?" 2 r (Y ) ; Y ("; u; 0) = x 1 ; @Y @s ("; u; 0) = u : It is straightforward to prove that @Y @s ("; u; s) ? u " q 2 k k L 1 (IR d ) : De ne S("; u) = inffs > 0 : Y ("; u; s) 2 @!g and Z("; u) = Y ("; u; S("; u)). By the above estimate, it is clear that, Z(" = 0; u 0 ) = x 2 . The function Z is of class C 2 on a neighborhood of (0; u 0 ) 2 IR + S d?1 , and it is easy to check that r u Z(0; u 0 ) is invertible. The conclusion holds by the implicit functions theorem. u t Lemma B.2 Let x; y 2 @!. Assume that ! is a C 1 bounded and connected domain in IR d and is of class C 2 on !. Then there exists a nite sequence of points x 1 = x, x 2 ,... x i , x i+1 ,... x n?1 , x n = y in @! such that (x i ; x i+1 ) has Property (S) for i = 1; 2; :::n ? 1. Proof. We shall rst prove an in nitesimal version of Lemma B.2. Let x; y 2 @! and denote by (x) and (y) the unit outgoing normals at x and y respectively. Because of the regularity of @!, for " > 0 small enough, if jx ? yj < ", there exists an > 0 such that fz 2 B(x; ") n fxg : z ? x jz ? xj (x) < ? g ! and fz 2 B(y; ") n fyg : z ? y jz ? yj (y) < ? g ! : Next, consider U = fu 2 S d?1 : u (x) + < 0 and u (y) + < 0g for " > 0 small enough so that U is not empty (for jx ? yj < " small enough, j (x) ? (y)j is as small as we want). Moreover, in the limit " ! 0, we can take arbitrarily small. For any u 2 U, we may therefore consider Z = fz(u) : u 2 Ug ; where z(u) = x+t(x; u)u and t(x; u) = infft > 0 : x+tu 2 ! c or (y; x+tu)\! c 6 = ;g. By Sard's theorem, there exists at most a countable number of points u in U for which either (z(u) ? x) (z(u)) = 0 or (z(u) ? x) (z(u)) = 0, which ends the proof: there exists a u 2 U such that both (x; z(u)) ! and (z(u); y) ! have Property S. By compactness of @!, if x and y are in the same connected component of @!, it is then easy to nd a nite sequence of points x 1 = x, x 2 ,...x i , x i+1 ,... x n?1 , x n = y in @! with jx i+1 ? x i j < " for " > 0 small enough such that Lemma B.2 holds. If x and y are in two di erent connected components of @!, the extension is straightforward and left to the reader.u t