A MINIMIZATION FORMULATION OF A BI-KINETIC SHEATH

. The mathematical description of the interaction between a plasma and a solid surface is a major issue that still remains challenging. In this paper, we model this interaction as a stationary and bi-kinetic Vlasov-Poisson-Amp`ere boundary value problem with boundary conditions that are consistent with the physics. In particular, we show that the wall potential can be determined from the ampibolarity of the particle ﬂows as the unique solution of a non linear equation. Based on variational techniques, our analysis establishes the well-posedness of the model, provided that the incoming ion distribution satisﬁes a moment condition that generalizes the historical Bohm criterion of plasma physics. Quantitative estimates are also given, together with numerical illustrations that validate the robustness of our approach.

1. Introduction.The description of the plasma-wall interaction is a challenging issue with many practical applications, be it in the modeling of Tokamak walls or ionic engines for satellites.Thus, the mathematical study and the numerical simulation of physically consistent models is of interest.When a plasma is in contact with an isolated and partially absorbing wall, a thin net-charge layer develops spontaneously between the wall and the plasma.This layer of several Debye lengths is called a sheath [6,8,24] and it is usually understood as the way by which the plasma preserves its global neutrality.Indeed, because the electrons are a lighter species they are prone to exit the plasma at a higher rate than the heavier ions.As this phenomenon alone would result in an unstable situation, namely a positive charge built up in the core plasma, the negative charge accumulated at the isolated wall causes the electric potential to drop and repel a significant fraction of the electrons.The magnitude of the drop is then such that the flow is ambipolar, in the sense that positive and negative charges exit the core plasma at the same rate [8,24].
Plasma-sheaths have been extensively studied in the last decades [6,15,8,25,22,7], however several important questions do not have fully satisfactory answers on the mathematical level.For instance, we are not aware of a simple model that describes in a unified way the physical processes at play between the sheath and the core of the plasma.Nevertheless, a common observation that is supported by both theoretical and empirical evidence is that at the sheath entrance the average ion velocity must exceed its sound speed c s , where k is the Boltzmann constant, T e is the electronic temperature and m i is the ion mass [8].This definition of the ion sound velocity corresponds to the case where the ion temperature T i is much smaller than the electron one.Another possible definition for the ion sound velocity is c s := k(T e + T i ) m i .This inequality is often referred to as the original Bohm criterion and several variants have been developed in the scope of more general models [21,4].For instance, in the case of a plasma consisting of a Poisson equation to define the electrostatic potential φ coupled to differential equations to define the ion and electron density n i and n e , it has been shown that these densities can both be expressed as functions of φ, and that at the sheath entrance (which is commonly defined as the limit between the non neutral region and the neutral region), the value φ se of the potential must be such that The sheath-edge x se , namely the entrance of the sheath, is then often defined as the position where φ(x se ) = φ se , even though it is commonly admitted that the sheath-edge is a difficult place to define [21].Overall, the inequality expresses the idea that at the entrance of the sheath the electron density decreases more rapidly than the ion density as the electric potential drops.
TO WALL n φ φ se n i (φ) . Schematic variations of the ion and electron densities with respect to some normalized potential φ ≤ 0 decreasing toward the wall.Here the plasma is supposed to be neutral at the sheathedge x se .
As for the boundary condition on the wall, most models describe the potential as having a "floating" value that adjusts itself according to the dynamics of the system.However no clear definition of a self consistent wall potential seems available.On the mathematical side some models have been proposed but they do not fully answer the above questions, see e.g.[16,10].
In the present work we address this problem by considering a simple plasma-wall interaction model with a self-consistent potential and we show that it is well posed under the assumption that the incoming ion distribution satisfies a moment condition which generalizes the usual kinetic Bohm criterion.Moreover, our solutions share most of the properties of plasma sheaths, such as a decreasing potential and a positive charge density.In our model the ion and electron densities are solutions to one dimensional stationary Vlasov equations coupled with a self consistent Poisson equation.Boundary conditions are determined to reflect the physical properties of this simplified model: in particular, the wall potential is determined so that the Ampère equation holds for the stationary solutions.A surprising result is that the resulting potential is only well-defined for incoming ions satisfying an upper bound on their average velocity.This constraint is shown to be compatible with the Bohm criterion thanks to the large mass ratio between ions and electrons.
To allow some generality, we consider that electrons are re-emitted with probability α ≤ 1 while ions are totally absorbed.Ions and electrons are assumed to enter the plasma with given velocity distributions.Since the core of the plasma is well described by a full Maxwellian, we have chosen to consider (semi-) Maxwellian distributions for the incoming electrons.At the numerical level we then observe that the resulting velocity distribution is very close to a full Maxwellian when far from the wall, in good qualitative agreement with the results from [24, p. 75].
The plan of the paper is as follows.In section 2 we begin with a presentation of the model and write down its mathematical structure.Under a decreasing assumption on φ, we prove the formal equivalence between the Vlasov-Poisson system and a non linear Poisson equation.An important result of this section is a necessary and sufficient condition for the wall potential to be uniquely determined by the physical parameters of the problem.We also give an a priori lower bound for the wall potential.The result holds for re-emission coefficient α < 1.The case of total electron re-emission (α = 1) corresponds to a Boltzmannian electron density and turns out to be degenerate.
In section 3 we set up the mathematical framework that is used in the rest of the paper and state the main result (well posedness under a condition that generalizes the usual kinetic Bohm criterion, and quasi-neutrality estimates).The proof relies on reformulating the non linear Poisson equation as a minimization problem, and our generalized Bohm inequality appears naturally as a local convexity condition for the energy functional.
In section 4, we briefly describe the numerical method employed to solve the problem.Then we illustrate the main result with a physically based sheath problem.
Final comments about the range of applicability of this work are provided as a conclusion in section 5.

Description of the model.
2.1.Physical setting.We consider a plasma at equilibrum made of one species of ions and electrons.This plasma is assumed to be contained in a one dimensional chamber.This model only describes a portion of the chamber of length L. Physical quantities will often be denoted with upper case while normalized ones will be denoted with lower case.Our system is subject to the following physical considerations: (b) ions and electrons entering the domain with non negative velocities which are described through velocity distributions denoted respectively F in i and F in e ; (c) an arbitrary charge imbalance denoted P 0 (normalized value ρ 0 ); 5.At the wall, that is at X = L (or x = 1 in normalized variables), we consider: 2.2.Kinetic modeling of the stationary plasma wall interaction.We first write the equations in physical variables and then derive a dimensionless model.The unknowns are respectively the electric potential Φ : [0, L] → R, the ion distribution function tively N e (X) := R F e (X, V )dV ) the ionic (respectively the electronic) density at the ionic (respectively the electronic) flux at X ∈ [0, L].The equations governing the ion and electron transport in the plasma, with an electric field E = − d dX Φ are assumed to be stationary Vlasov equations and write with the boundary conditions Here q is the electric charge and m i (respectively m e ) denotes the ionic (respectively the electronic) mass.Furthermore, a formal integration of equations ( 3)-( 4) with respect to the velocity variable shows that the current density J(X) := q (Γ i (X) − Γ e (X)) must be constant in space, and so since (by ambipolarity) the current has to be zero at the wall.We further stress that equation ( 9) is necessary if one wants to construct stationary solutions compatible with the Maxwell-Ampère equation.The electric potential is determined from the densities through the Gauss law with boundary conditions Here the vacuum permittivity is ε 0 and Φ W denotes the wall potential.Its value will be determined so that equation ( 9) holds.As far as our model is concerned, we will show in Section 2.5 that Φ W can be determined from the previous physical parameters.
It is convenient to rescale the equations and to this end we introduce the dimensionless variables x, v and the dimensionless functions φ, f i and f e defined as: where k is the Boltzmann constant, T e is a reference electron temperature and c s := kT e m i the ion sound speed.We also define the dimensionless quantities The coupled boundary value problem (3)-( 11) is then equivalent to the following boundary value problem: complemented with the boundary conditions and the additional constraint (derived from the Ampère equation) We remind that the value of φ w = qΦw kTe will be determined later, see Section 2.5.
Here, we have set F in e (V )dV denotes an electron reference density and λ D := ε 0 kT e q 2 N 0 is the Debye length.
The set of equations ( 12)-( 18) is the model problem and we will refer to it as the Vlasov-Poisson-Ampère problem.It contains the main physical parameters ε, α, ρ 0 , f in e and f in i .The Vlasov-Poisson problem is made of equations ( 12)- (17) which can be considered as the main equations while the Ampère equation ( 18) can be considered as an additionnal constraint.To our knowledge, this stationary and bi-kinetic boundary-value problem has never been studied in full details.For example in [20], a model of plane diode is studied.It is consists of a one single stationary Vlasov equation for electrons coupled with the Poisson equation for the electrostatic potential, the well-posedness is studied for a large class of electron boundary conditions.In [12], the non-stationary version of the plane diode is studied.12)- (17), it is possible to reformulate as a non linear Poisson equation.When the potential φ is given both Vlasov equations for ions and electrons are linear advection equations, and their solutions are determined by transport along the characteristics of the (incoming) boundary conditions.In this section we assume φ ∈ W 2,∞ (0, 1) to be given which is a sufficient condition for the characteristics curves to be well-defined [1].Moreover, we assume φ(0) = 0, φ(1) = φ w and φ < 0. In such a configuration, we have the following formal result.
Proposition 1 (Formal equivalence).Let φ ∈ W 2,∞ (0, 1) be such that φ < 0 with φ(0) = 0 and φ(1) = φ w .Then there exists f i ,f e such that the Vlasov-Poisson system holds if and only if φ is a solution to (NLP) : We will give a formal proof of the above proposition.Especially, we do not want to discuss regularity and integrability issues.The proof is decomposed in two parts.The necessary condition is shown in Section 2.3.1 and the sufficient one is established in Section 2.3.2.
2.3.1.Necessary condition.We assume that there is φ with φ < 0, f i and f e solutions of ( 12)- (17) .Let us show that φ is solution of (NLP).To this effect, we determine an explicit representation of f i and f e by means of the characteristics curves.Electrons trajectories.The characteristics trajectories of electrons (12) are the curves which satisfy the ordinary differential system of equations (C e ) : for t ≥ s and for an arbitrary initial data (s, x, v) ∈ R × [0, 1] × R. A geometry of the characteristics is illustrated in Figure 3.Under the decreasing assumption on φ, one can identify the solutions to (C e ) with the curves (x, v) The phase-space [0, 1] × R is then splitted into two subdomains which are separated by the characteristic curve of equation For (x, v) ∈ D 1 there exists w > 0 and a characteristic curve passing through (x, v) which originates from (0, w).Conversely, for (x, v) ∈ D 2 there exists w < 0 and a characteristic curve passing through (x, v) which originates from (1, w).Since the electron distribution function f e is constant along the characteristics we can determine f e (x, v) for every (x, v) ∈ [0, 1] × R. To this end, consider the two following cases.If (x, v) ∈ D 1 one has 1  2 v 2 − mi me φ(x) = 1 2 w 2 for some w > 0. Then Figure 3. Schematic characteristic electron trajectories associated with a decreasing potential φ.The dashed line corresponds to a characteristic curve which originates at the wall with a negative velocity.Because of the boundary condition at the wall, particles following this curve were originally at x = 0 with a positive velocity.
. We thus obtain that the solution of ( 12) is given by and integrating in velocity (20) leads to f in e (w)w f in e (w)w Ion trajectories.The characteristics trajectories of ions (13) are the curves which satisfy the ordinary differential system of equations on φ the solutions to For (x, v) ∈ D 3 there exists w > 0 and a characteristic curve which originates from (0, w).Conversely for (x, v) ∈ D 4 there exists w < 0 and a characteristic curve passing through (x, v) which originates from (1, w).Since the ion distribution function f i is constant along the characteristic curves we can determine We thus obtain that the solution of ( 13) is given by For each and integrating in velocity ( 22) leads to Formulas ( 21)- (23) show that φ is indeed a solution to (NLP), which ends the proof of the necessary condition.
2.3.2.Sufficient condition.Assume there is φ decreasing and solution of (N LP ).Then φ w ≤ φ(x) ≤ 0 for all x ∈ [0, 1], f e and f i defined in ( 20), ( 22) are well defined.We can check that (φ, f i , f e ) is a weak solution of the Vlasov-Poisson system in the sense of A.1 provided f in i and f in e are regular enough.Remark 1.It is easy to see there is a little change in the geometry of the characteristics when φ is permitted to vanish on some interval.However, when φ is not non increasing, it is possible that the characteristics curves are closed and never intersect the boundaries.This can lead to the presence of trapping sets of non zeromeasure (see [3] for a definition of trapping sets) which results in ( 12)-( 17) being ill-posed.

Semi-Maxwellian electron boundary condition and charge imbalance.
As mentioned in the introduction, electrons in the core of the plasma are well described by a full Maxwellian distribution.As a matter of fact, in this work we shall consider Maxwellian boundary conditions for electrons which takes the form It gives in terms of dimensionless variables where f in e (v)dv = N 0 L is an electron reference density.The electron density ( 21) is then given for all x ∈ [0, 1] by The electron flux is constant in space and given for all x ∈ [0, 1] by Notice that the electron density is close to a Boltzmannian density but not equal.It contains a perturbation that represents the truncation of the Maxwellian distribution due to the electron loss at the wall.The Boltzmannian density corresponds to the case α = 1.In this case, the Ampère equation will be shown to be degenerate see Proposition 2 and Remark 3, below.
It will be convenient for the mathematical discussion to denote by ρ 0 the charge imbalance at x = 0.By definition it writes We observe that n 0 can be expressed as Remark 2. In the mathematical analysis Section 3, we will study the well-posedness of the above problems and consider ρ 0 as a given parameter.The value of n 0 will then be defined by (29).In order that n 0 be positive we observe that ρ 0 and f in i must be chosen such that Also in the next section, we will show that φ w only depends on ρ 0 , α and f in i and hence, so does n 0 .Also remark that where erf denotes the error function.
2.5.Equation of the wall potential.In general the potential at the wall cannot be a priori specified as a physical parameter.Therefore it is important to understand how it is determined in this model from other physical parameters.As mentioned in the introduction, the wall potential adjusts itself so that equal numbers of ions and electrons reach the wall per second.Following the idea in [24] Section 2.6 page 79, its value is determined from the ambipolarity of the flow which can be also seen as a consequence of the stationary Maxwell-Ampère equation (18).Using (22) the ion flux is constant in space and given for all x ∈ [0, 1] by and the electron flux is given in (27).Then for all ρ 0 ∈ R and α ∈ [0, 1) the ambipolarity γ i = γ e (18) yields Substituting the expression (29) of n 0 in (31) leads to the equivalent non linear relation where W : R − → R is defined by Note that here we have considered α < 1, see Remark 3 below.Then for all ρ 0 ∈ R and α ∈ [0, 1), φ w must be solution of the non linear equation (32).We remember that due to the definition of n 0 one has of course ρ 0 < R + f in i (v)dv.Therefore using standard arguments one has the following Proposition 2. Let ρ 0 ∈ R and α ∈ [0, 1).The equation (32) has a unique non positive solution φ w = φ w (ρ 0 , α) if and only if Moreover the solution is in the interval, Then using the inequality du ≤ e φw √ −φ w + 1 (see [19] page 163) we obtain A simpler and easily computable bound is then given by and we conclude using the equality Let us make a list of remarks about this result.
Remark 3. When α = 1 (which corresponds to a total re-emission of electrons at the wall) the only integrable boundary condition that satisfies (31) is f in i ≡ 0 and the model is of no interest.
Remark 4. The inequality (34) also re-writes π n e (0) and since γ i = γ e this means that ion and electrons flow must be bounded and cannot exceed the upper bound π n e (0).
Remark 5.The case of equality is equivalent to φ w = 0 is the solution to (32).
For the sake of simplicity and from now, we omit to precise the dependence of φ w (ρ 0 , α) on ρ 0 and α and simply denote φ w = φ w (ρ 0 , α).

2.6.
A variational approach to the non linear Poisson problem.We remember that (NLP) is formally equivalent to the Vlasov-Poisson system when the electrostatic potential is decreasing, see Proposition 1.In the following section we will study the well posedness of (NLP) in the case of an incoming Maxwellian electron distribution.To this end, we will consider that ρ 0 ∈ R, α ∈ [0, 1) and f in i satisfying (34) are given parameters as well as the normalized Debye length ε.The wall potential φ w ≤ 0 will then be the solution of (32) and the electron boundary condition will be of the form (25) where the reference density n 0 is defined by (29).In particular, the (NLP) problem reformulates as follows : Let x) for all x ∈ (0, 1) with Dirichlet boundary conditions φ(0) = 0 and φ(1) = φ w where From a mathematical point of view, one would notice the analogy between the non linear Poisson equation and classical motion equations of a single particle in a potential force field.Indeed, the opposite of the right hand side of (36) derives from an abstract potential function U : [φ w , 0] → R given by The non linear Poisson equation (36) rewrites in the form ).Moreover, it can eventually be re-written into a variational form.Indeed, solutions to (36) are critical point of the energy functional defined on the adequate functional space.Namely critical points of J ε are solutions of dJ ε (φ) ≡ 0 where dJ ε denotes the Fréchet derivative of J ε .Then variational techniques constitute a convenient mathematical tool to solve the non linear Poisson equation.Historically speaking, variational methods to treat stationary transport problems were used in [18] to deal with neutron diffusion problems such as the Milne problem.
3. Mathematical analysis.In this section, we study the well-posedness of the non linear Poisson problem (NLP-M) which corresponds to the Vlasov-Poisson-Ampère problem in the case where the incoming electron distribution is Maxwellian.We will use variational principles and the theory of Nemytskii's operator to study the functional J ε .The results we need are reminded in the appendix.
3.1.Mathematical setting and main result.Let us define for all α ∈ [0, 1) which is the upper bound in (34).In the case α = 1 the problem has no interest (see Remark 3).The physically interesting case corresponds to φ w < 0, so we shall only consider distributions f in i satisfying (34) with a strict inequality.Let us therefore define the functional framework that will be used in the following.We shall consider ion boundary conditions that are bounded, integrable and of finite kinetic energy so that U is well defined.We denote the set of such functions For ρ 0 ∈ R and α ∈ [0, 1) given, we define the set of admissible ion boundary conditions as well as the set of admissible potential where V 0 := φ ∈ H 1 (0, 1) | φ(0) = 0 is a Hilbert space equipped with the inner product (φ, ϕ) V0 := 1 0 φ (x)ϕ (x)dx for any (φ, ϕ) ∈ V 0 × V 0 and with the induced norm defined by φ V0 = (φ, φ) V0 = φ H 1 0 for all φ ∈ V 0 .We also denote H −1 the dual space of H 1 0 (0, 1), and we remind that the norm on H −1 is defined by Remark 6.It is important to notice that in the definition of V the decreasing assumption on φ does not appear.It is not necessary for U to be well-defined, hence we decide to relax it.It will be shown in Theorem 3.2 that the solution φ of (NLP-M) is non increasing.
Finally, since the mass ratio always satisfies m e m i < 2 π we define a critical reemission coefficient which is such that s 1 (α c ) = 1 and 0 < α c < 1.
Definition 3.1.(Sheath solutions) Let (f i , f e , φ) be a solution to the Vlasov-Poisson-Ampère system ( 12)- (18).We say that it is a sheath-type solution on (x * , 1] with 0 ≤ x * < 1 if on that interval φ is decreasing and n i > n e , and if We are now in position to state our main result. Theorem 3.2.Let α ∈ [0, α c ], ρ 0 = 0, f in i ∈ I ad (0, α) and ε > 0. Let φ w be the unique solution of (32).Assume the kinetic Bohm criterion Then the Vlasov-Poisson-Ampère system (12)-( 18) is well-posed, with a Maxwellian incoming electron distribution f in e defined by (25), (29).More precisely, there is a unique φ ε ∈ V solution of (NLP-M).In addition, 1.The densities f e and f i defined in (20) and (22) belong to The proof of this theorem is proved in Section 3.2.2.Let us make a list of general but somewhat useful remarks about this theorem.Remark 7. A sufficient condition for (45) is This inequality still re-writes in physical variables It coincides with the standard kinetic Bohm criterion, see [24,Section 2.4].
Remark 8.The re-emission coefficient α c is said to be critical because when α > α c we are not able to prove the existence of admissible boundary condition satisfying the kinetic Bohm criterion (45).On the contrary when α ≤ α c we are able to do so (see theorem 3.7 and corollary 1).
Remark 9.In practice α c is close to 1 and it allows to consider a wide range of material, even those with a high re-emission coefficient.As an example, for a Deuterium plasma m i m e = 3672 and the critical re-emission coefficient is α c ≈ 0.95.
Remark 10.In the theorem we have considered ρ 0 = 0 which corresponds to the neutrality n i (0) = n e (0).It is an usual assumption in the physics literature.
In the case ρ 0 = 0 and f in i ∈ I ad (ρ 0 , α) for some α ∈ [0, 1) and ε > 0, we are able to establish the existence of a non increasing minimizer for J ε see theorem 3.8 and proposition 8.However, since V is a strict and closed convex subset of V 0 , minimizers are not necessarily critical points.
Remark 11.The kinetic Bohm criterion (45) expresses the strict convexity of U in the vicinity of ψ = 0.Moreover, we have seen that (n i − n e )(x) = −U (φ(x)).It follows that n i − n e is a function of the electrostatic potential and we can verify In particular (45) is equivalent to d dφ (n i − n e )(0) < 0 which is an usual sheath criterion [8,21].
. This means there is essentially no ions with null velocity at x = 0.In such a configuration, minimizers of J ε are concave and non increasing solutions of the non linear Poisson equation.Thus f e and f i defined in (20) and ( 22) are weak solutions of the Vlasov equations and the Vlasov-Poisson-Ampère system is well posed.The uniqueness for the Poisson equation is proven by a reduction to a first order differential equation.Remark 13.In the limit ε → 0, the estimates (46) are mathematical expression of the quasi-neutrality.

3.2.
Well posedness of the Vlasov-Poisson-Ampère problem.Here we use the variational formulation of (NLP-M) and treat the following minimization problem.
This minimization problem is a constrained problem and we can see it is a non linear variant of the obstacle problem [23].Let us remember that where the real valued function U is defined for all ψ ∈ [φ w , 0] by This function belongs to C 1 [φ w , 0] and the Nemytskii operator associated with U is denoted T U and defined for all φ ∈ V by T U (φ)(x) := U(φ(x)) for all x ∈ [0, 1] see A.4.To prove the Theorem (3.2), we shall need some preliminary results.
3.2.1.Preliminary part.Notice that J ε is made of strictly convex part plus a perturbation that is not necessarily convex All the analysis is based on the properties of the perturbation (49).Using the theory of Nemytskii operators and more precisely Theorem A.6 we have the following.
Moreover the perturbation (49) is compact as we prove in the following.
) is the Rellich compact embedding and TU is the restriction to C 0 ([0, 1]; [φ w , 0]) of T U .Since i is compact and TU is continuous, we conclude that T U is compact.
A direct consequence of the above proposition which results from Lemma A.3 is the following.Proposition 5. Let ρ 0 ∈ R, α ∈ [0, 1) and f in i ∈ I ad (ρ 0 , α).Then T U is (sequentially) weakly lower semicontinuous.
where E ε and F were given in (48)-(49).From Proposition 3 we deduce that F is C 1 over V and since E ε is also C 1 over V thus J ε is also.For the weak lower semicontinuity, we notice that E ε is convex and continuous for the strong topology, consequently applying the Mazur lemma [9, p. 562] we deduce that E ε is sequentially weakly lower semicontinuous.Applying Proposition 5 we also deduce that F is weakly lower semicontinuous and thus J ε is.
Proof.It is convenient to make the change of variable u := −ψ and to define the We shall give a lower bound for A and B. Applying respectively inequalities (51) and (52) to the integrands of A and B with u * = 0 and η = 1 v 2 , we obtain we therefore obtain By hypothesis ρ 0 ≥ 0 hence for all u ∈ (0, −φ w ] we have that d du Ũ(u) is positive if the bracket is positive and non negative if the bracket vanishes.
For the self-consistency of the analysis we investigate the existence of admissible ion boundary conditions f in i that satisfy the kinetic Bohm criterion (45).Let us define for all α ∈ [0, 1) and f in i ∈ I ad (0, α)
Proof.Let α ∈ [0, 1).We begin with showing the necessary condition.Assume there exists f in i ∈ I satisfying Applying twice the Cauchy Schwarz inequality yields .
We are now equipped to prove the main result, Theorem 3.2.

3.2.2.
Proof of the main result.
Proof.We apply Theorem A.2.By definition V 0 is a reflexive Banach space and V is a closed convex subset.By Lemma 3.3, J ε is sequentially weakly lower semicontinuous and since U is positive (Lemma 3.6) for all φ ∈ V we have By comparison J ε (φ) → +∞ as φ V0 → +∞ and thus J ε is coercive.Therefore, there is This theorem states the existence of global minimizers but does not ensure they are criticial point of J ε .Let us give more properties of minimizers that will be useful in the sequel.
and is solution of (NLP-M).
The proof is splitted into two parts.Mainly, the first part deals with the existence of a solution the Vlasov-Poisson-Ampère system and the second part deals with the uniqueness.Existence part.We apply the proposition 6, so that the function φ ∈ [φ w , 0] → U(φ) is decreasing and U (φ) < 0 for all φ ∈ [φ w , 0).Combining theorem 3.8 and corollary 9 we obtain there is ≤ 0 for all x ∈ (0, 1) we deduce that φ ε is concave on [0, 1].Now considering f e and f i defined in (20) and (22), it is easy to see that they belongs to (L 1 ∩ L ∞ )([0, 1] × R; R + ).We can also check they are weak solution of the Vlasov equations in the sense of definition A.1.Now it easy to observe that (f i , f e , φ ε ) is a sheath type solution on (x * , 1] where x * = max{x ∈ [0, 1] / φ ε (x) = 0}.Besides, the Ampère equation ( 9) is satisfied by definition of φ w and we therefore deduce that the Vlasov-Poisson-Ampère system (12)-( 18) is well posed.In addition, it is straightforward from the equation (32) that φ w does not depend on ε and so do v → (f i (1, v), f e (1, v)), n i (1) and n e (1).We shall now prove the estimates (46).The first one is obtained from (56).The second one is obtained as follows.Since n i −n e is a continuous function and by the canonical injection C 0 [0, 1] → H −1 (0, 1), it defines a linear and continuous form on the space H 1 0 (0, 1) and we have for all ψ ∈ H 1 0 (0, 1) Using the Cauchy-Schwarz inequality and the estimate (57) we obtain Uniqueness part.The proof of the uniqueness result relies on a reduction of the non linear Poisson equation to a first order differential equation.We shall also need the following lemma whose proof is a consequence of the Cauchy-Lipschitz theorem.
then it is unique.
Proof.Since g is C 1 and g > 0, the function φ ∈ [φ w , 0] → − g(φ) is Lipschitz in φ and it suffices to apply the Cauchy-Lipschitz theorem to conclude.
We are now able to prove the uniqueness result.To this effect, let us multiply the non linear Poisson equation (38) by d dx φ ε and integrate over an arbritrary segment [x, y] ⊂ [0, 1] with 0 ≤ x < y ≤ 1.Then we obtain Further assume there is a solution concave, non increasing and different of φ ε .Since φ ε and ψ ε have the same boundary conditions and are continuous, there exist Since the previous relation ( 62) is valid for any 0 ≤ x < y ≤ 1, chosing x = x 1 and y = x 2 leads to and by a comparison argument we obtain . Eventually using the relation (62) for y = x 2 and x 1 ≤ x ≤ x 2 it is easy to notice that φ ε and ψ ε are both solutions of the Cauchy problem 2 and also that U(w(x)) ≥ U(φ ε (x 1 )) because U and w are non increasing.Finally remark that U(w( 0 and conclude by invoking lemma 3.9.

3.2.3.
Complementary study when the Bohm criterion is violated.In the theorem 3.2 we have considered ρ 0 = 0 and the kinetic Bohm criterion (45).In this section, we consider more general cases where either the kinetic Bohm criterion is violated or ρ 0 = 0. We remember that for ρ 0 ∈ R, α ∈ [0, 1) and f in i ∈ I ad (ρ 0 , α) given, U is defined for all ψ ∈ [φ w , 0] by We intend to prove that the general situation is that U admits at most two local minima and two local maxima.Ultimately, it shows that whenever the Vlasov-Poisson-Ampère is well posed with φ non increasing, the charge density x ∈ [0, 1] → (n i − n e )(x) = −U (φ(x)) can change sign at most in three distinct regions.This section is thus devoted to the study of the monotonicity of U. It is convenient to make the change of variable u := −ψ and to define the function u ∈ [0, −φ w ] → Ũ(u) := U(−u).We will assume in this section that ) and for all u ∈ (0, −φ w ), Proof.Assume Ũ attains a minimum at u * ∈ (0, −φ w ) then one has the first and second order conditions d du Ũ(u * ) = 0 and d 2 2 Ũ(u * ) ≥ 0 that is One has the decomposition for all u ∈ [0, −φ w ] Let us now give a lower bound for A. Applying the inequality (51) we have dv then we use the second order condition (67) and obtain for all u ≥ u * We also have a lower bound for B. Indeed, using the inequality (52) we have dv.
Combining A and B one finally obtains and the right hand side is exactly zero so that d du Ũ(u) ≥ 0 for all u ≥ u * .
From Proposition 10 one can establish the following result.
Proof.We do the proof by contradiction.Assume Ũ has at least two local minima at some points u 1 and u 2 belonging to (0, −φ w ).Without loss of generality we can assume u 1 < u 2 .Since Ũ is not locally constant there exists u 1 < δ < u 2 such that Ũ is decreasing over (δ, u 2 ) which is contradiction with Proposition 10.We can also prove that if Ũ has a local maxima in (0, −φ w ) then it is unique.Theorem 3.10.Let ρ 0 ∈ R, α ∈ [0, 1) and f in i ∈ I ad (ρ 0 , α) such that Ũ is not locally constant.Then Ũ admits at most two local minima over [0, −φ w ] and it also admits at most two local maxima over [0, −φ w ].
Let N ∈ N * , the discretization consists of a mesh (x i := i (N +1) ) i=0,...,N +1 with h = 1 N +1 and the approximation of the Hilbert space H 1 0 (0, 1) by a standard and BADSI, CAMPOS PINTO AND DESPR ÉS conformous P 1 finite element space V h 0 .More precisely and the admissible potential set is approximated by We finally solve the minimization problem associated with J using a fixed step gradient algorithm.Namely, given η > 0 and δ > 0, we compute iteratively φ0 h ∈ W h φn+1 h = φn h − η∇ J( φn h ) and stops when ∇ J( φn h ) H 1 0 (0,1) < δ.We have denoted ∇ J( φn h ) ∈ V h 0 the gradient of J at φn h .It is the unique solution of the variational problem (∇ J( φn h ), ψ h ) H 1 0 = d J( φn h )(ψ h ) for all ψ h ∈ V h 0 where (., .)H 1 0 is the H 1 0 (0, 1) inner product and 4.2.Numerical results.We carry out two numerical experiments.In the first one we perform numerical simulations that are in the scope of Theorem 3.2, that is in the case of a satisfied kinetic Bohm criterion.For these simulations we vary the parameters ε and α.In the second one, we perform numerical simulations with fixed values of ε and α but with an incoming ion boundary condition that violates the kinetic Bohm criterion (45).The data are : • We set the mass ratio m e m i = 1 3672 for a Deuterium plasma.It results in α c ≈ 0.95.
where η is a small parameter, σ 2 = Ti Te is the temperature ratio and Z is a macroscopic velocity adjusted with respect to the kinetic Bohm criterion (45).In the following simulations η = 10 −1 and σ = 1 2 .We also remember that f in e is given in (25).• We set the neutrality ρ 0 = 0.
• We choose a mesh size h = 2 −11 and a tolerance parameter for our gradient algorithm δ = 10 −6 .

4.2.1.
The case of a satisfied Bohm criterion.In this part we present the numerical solutions we obtained with a fixed value of Z chosen equal to 3 2 .The moments are computed numerically and we obtain : We can check numerically that both the admissibility condition (34) and the kinetic Bohm criterion (45) are satisfied.In figure 6 we have represented the ion incoming boundary condition f in i .We are now going to illustrate the behavior of the solution with respect to ε and α.We know from Theorem 3.2 that n i (x) ≥ n e (x) for all 0 ≤ x ≤ 1.The general intuition is that when ε > 0 is small, one would expect n i For the data we have chosen it also seems to be convex, however notice that from its expression (37) it is not straightforward.In addition, we observe that when ε is small a sheath of length of the order of ε develops near the wall and the sheath-edge denoted x * varies with ε.In the simulations the sheath-edge corresponds to the point where approximately |φ ε (x * )| > 10 −4 |φ w |.We have x * (1) ≈ 0, x * (0.1) ≈ 0.5, x * (0.01) ≈ 0.95.For x > x * (ε), the plasma is significantly positively charged and there is a non negligible electric field that accelerates ions and decelerates electrons.Sufficiently fast electrons reach the wall and are absorbed.For x ≤ x ε , the plasma is almost neutral and there is no appreciable electric field, particles have constant velocities.These results are in good agreement with the physics, and confirms the commonly made assumption of semi-Maxwellian electron distribution function at the wall, see for example [17].Case ε = 0.1 and varying α.We fix ε = 0.1 and vary α ∈ {0, 0.5, 0.9}.The results are qualitatively the same with the difference that some electrons are re-emitted with negative velocities, therefore we decide only to plot the electron and ion densities n α e and n α i , see figure 12.We also gather the different values of the wall potential φ w and the electron reference density n 0 with respect to α in the table 1.The ion α φ w n 0 0 -2.7 0.5 0.5 -2.1 0.5 0.9 -0.48 0.5 Table 1.Values of the wall potential and the reference density for various values of α. and electron densities seem to be respectively increasing functions of α.   4.2.2.The case of a violated Bohm criterion.We present numerical results when the kinetic Bohm criterion (45) is not satisfied.We mention that we are not in the scope of Theorem 3.2, thus we cannot ensure the existence and the uniqueness of a  15 represents respectively the macroscopic densities and the electrostatic potential.We see that when the Bohm criterion is violated, there is two boundary layer, one is at x = 0 and the other one is at x = 1.The charge density is negative near x = 0 while it is positive near x = 1.The physical interpretation of this numerical experiment is not obvious.However we mention that since the point x = 0 is assumed to represent a position somewhere in the (bulk) plasma, it seems to us that the boundary layer at x = 0 is unphysical.Consequently, this unphysical effect shows a limitation of our model.Lastly, we mention that for other values of    the macroscopic velocity Z which do not ensure the kinetic Bohm criterion (45) to be satisfied, the results are qualitatively the same.

Conclusion.
We have proposed and studied a stationary and one dimensional plasma-wall interaction model, based on a bi-kinetic description of ions and electrons.Due to the presence of the wall, the electron phase space density is represented by a truncated Maxwellian distribution.As for the ions, our model supports a large class of incoming velocity distributions f in i and we have shown that it is well posed under a moment condition on f in i which generalizes the usual Bohm criterion.Furthermore, we have identified a second condition that must be satisfied by f in i for the wall potential to be well-defined.Surprisingly enough, this second condition takes the form of an upper bound on the average velocity of the incoming ions but thanks to the large mass ratio m i m e we have verified that it is not in contradiction with the Bohm criterion.Our proof relies on a reformulation of the Vlasov-Poisson system into a non linear Poisson equation that we next study as a minimization problem.This approach also provides us with quantitative estimates for the boundary layer.A physically based sheath problem was then illustrated with numerical simulations.Results show that when the neutrality is assumed at x = 0 and when the incoming ion distribution is admissible and satisfies the kinetic Bohm criterion, then for a vanishing normalized Debye length ε a sheath of length of the order of ε develops at the wall.Out of the sheath the plasma is almost neutral while in the sheath it is not, ions are accelerated and electrons decelerated.These results provide a strong numerical evidence for Theorem 3.2 and they are in good agreement with the simulations presented in [11].We should add that this work takes full advantage of the one dimensional structure of our model.Although elementary, we hope this approach to be generic enough to be used in more general cases, including additional physics such as collision operators and magnetic fields [17,14].

Figure 2 .
Figure 2. Schematic illustration.Ions and electrons are going toward the wall, some electrons reaching the wall are re-emitted with a probability α ∈ [0, 1] while ions are totally absorbed.

2. 3 .
Reformulation as a non-linear Poisson equation.Thanks to the onedimensional structure of the Vlasov-Poisson problem (

4 Figure 4 .
Figure 4. Schematic characteristic ions trajectories associated with a decreasing potential φ.Here the dashed lines correspond to characteristic curves originating from the wall with negative velocities, and they span the darker gray region D 4 .Because of the boundary condition at the wall, no particles travel on these curves and hence f i vanishes outside D 3 (lighter gray).
the function W is continuous and increasing with lim −∞ W = 0. Consequently, W is a bijection from (−∞, 0] to (0, W(0)] and the equation (32) admits a unique solution if and only if vdv ∈ (0, W(0)] which leads to the inequality (34).Now we prove the bounds (35).The upper bound is straightforward from the definition of the domain of W. For the lower bound, after a change of variable in +∞ √ −2φw e − v 2 2 dv we obtain and are weak solutions of the Vlasov equations in the sense of Definition A.1.2. There existsx * ∈ [0, 1) such that (f e , f i , φ ε ) is a sheath-type solution on (x * , 1]in the sense of Definition 3.1.3.At the wall the values of n i , n e , φ w and the velocity distributions f i , f e do not depend on the normalized Debye length ε. 4. φ ε is C 2 [0, 1], concave and we have the quantitative estimates

Figure 5 .
Figure 5. Plots of all possible behavior of the function u ∈ [0, −φ w ] → Ũ(u).The function studied in the scope of theorem 3.2 corresponds to the dashed line of Figure (B).

Figure 6 .
Figure 6.Plot of the incoming ion boundary condition for positive velocities with Z = 3 2 .

( a )
Electrostatic potential in space for various ε.(b) Ion mean velocity in space for various ε.

Figure 11 .
Figure 11.Electrostatic potential and ion mean velocity in the domain [0, 1] for various values of ε.

( a )
Ion phase space (b) Electron phase space

( a )
Ion and electron densities in space.(b)Electrostatic potential in space.
class C 1 and (sequentially) wealky lower semicontinuous.Its Fréchet derivative is given by