Homogenization and diffusion approximation of the Vlasov–Poisson–Fokker–Planck system: A relative entropy approach

Abstract

We are concerned with the analysis of the approximation by diffusion and homogenization of a Vlasov–Poisson–Fokker–Planck system. Here we generalize the convergence result of (Comm. Math. Sci. 8 (2010), 463–479) where the same problem is treated without the oscillating electrostatic potential and we extend the one dimensional result of (Ann. Henri Poincaré 17 (2016), 2529–2553) to the case of several space dimensions. An averaging lemma and two scale convergence techniques are used to prove rigorously the convergence of the scaled Vlasov–Poisson–Fokker–Planck system to a homogenized Drift-Diffusion-Poisson system.

Keywords

Vlasov–Poisson–Fokker–Planck system Drift-Diffusion equation diffusion approximation homogenization renormalized solution two scale convergence averaging lemma

1. Introduction

In this paper we study the approximation by diffusion and homogenization of a Vlasov–Poisson–Fokker–Planck system (VPFP for short) in the presence of a spatially oscillating electrostatic potential. This system is a kinetic description of a physical plasma whose particles change only slightly their moment during collision events. The scaled Vlasov–Fokker–Planck equation which is defined on the phase space $Ω = ω \times R^{d}$ where $d ⩾ 2$ , reads as follows $\begin{matrix} (1) & \partial_{t} f^{ε} + \frac{1}{ε} (v \cdot \nabla_{x} f^{ε} - \nabla_{x} ϕ_{T}^{ε} \cdot \nabla_{v} f^{ε}) - \frac{L_{FP} (f^{ε})}{ε^{2}} = 0, (x, v) \in Ω, \end{matrix}$ where ε is a small parameter related to the mean free path and $f^{ε} (t, x, v)$ denotes the scalar distribution of particles. The time variable t is nonnegative, the position x belongs to a bounded and regular domain ω and the velocity v belongs to $R^{d}$ . This equation has to be complemented with initial and incoming boundary conditions. We denote by $n (x)$ the outward unit normal vector at the position $x \in \partial ω$ and $\begin{matrix} Γ^{\pm} = {(x, v) \in \partial Ω; \pm v \cdot n (x) > 0} . \end{matrix}$ The Fokker–Planck operator is given by $\begin{matrix} (2) & L_{FP} (f) = \nabla_{v} \cdot (v f + \nabla_{v} f) . \end{matrix}$ The initial data is assumed to be known and depend on the mean free path ε: $\begin{matrix} (3) & f^{ε} (t = 0, x, v) = f_{I}^{ε} (x, v), (x, v) \in Ω, \end{matrix}$ where, the assumption satisfied by $f_{I}^{ε}$ is detailed later on.

The electrostatic potential $ϕ_{T}^{ε} (t, x)$ is the sum of a prescribed potential with multiscale variations $ϕ_{H}$ and a Hartree potential $ϕ_{P}^{ε}$ : $\begin{matrix} (4) & ϕ_{T}^{ε} (t, x) = ϕ_{H} (x, \frac{x}{ε}) + ϕ_{P}^{ε} (t, x) . \end{matrix}$ The two-scale potential $ϕ_{H} (x, y)$ has both macroscopic and microscopic variations and it is given regular and cell-periodic function with respect to the fast variable $y = \frac{x}{ε}$ . For simplicity we assume that the cell-period is the d-dimensional unit cube $Y = {[0, 1]}^{d}$ and $ϕ_{H}$ is time independent. The potential $ϕ_{P}^{ε}$ is self-consistent, it is obtained by solving the following homogeneous Poisson equation $\begin{matrix} (5) & \{\begin{matrix} - Δ_{x} ϕ_{P}^{ε} (t, x) = ρ_{ε} (t, x) = \int_{R^{d}} f_{ε} (t, x, v) d v, \\ ϕ_{P}^{ε} {(t, x)}_{| \partial ω} = 0 . \end{matrix} \end{matrix}$ The incoming boundary conditions are assumed to be well prepared: $\begin{matrix} (6) & f^{ε} (t, x, v) = f_{b} : = ρ_{b} (t, x) M (v) exp (ϕ_{e} (x) - ϕ_{H} (x, \frac{x}{ε})), (x, v) \in Γ^{-}, \end{matrix}$ where $ρ_{b} (t, x)$ is a boundary data, $ϕ_{e} (x)$ is an effective potential defined in (12) and M is the normalized Maxwellian with zero mean velocity: $\begin{matrix} (7) & M (v) = \frac{e^{- \frac{| v |^{2}}{2}}}{{(2 π)}^{\frac{d}{2}}} . \end{matrix}$ The study of kinetic transport equations related to gas dynamics, plasmas, semiconductors, neutron transport, and other systems has developed rapidly in recent years because of its role as a mathematical tool in many applications in areas such as engineering, chemistry, biology, medicine and materials science. In the past few years there were many works dealing with the diffusion approximation of the kinetic VPFP system (see, for instance, [9,12,15,22,24,26]). The diffusion limit for the VPFP system has been studied on the whole space by Poupaud and Soler [22] in 2000 for a small enough time interval in dimension 2. In 2005, Goudon [12] established the global-in-time convergence in dimension 2 with bounds on the entropy and energy of the initial data and in 2010, El Ghani and Masmoudi generalized these results in [9] and proved the global-in-time convergence in higher dimensions with similar initial bounds, using the concept of DiPerna–Lions renormalized solutions [6] which are also called free energy solutions [7] and techniques related to averaging lemma. This procedure was introduced by Masmoudi and Tayeb in [17] to study the diffusion limit of a semiconductor Boltzmann–Poisson system (see, for instance, [2,4,16,21] for some related results). Recently, Wu, Lin and Liu [26] and Herda [15] treats with this method the case of a multi-species model. Let us mention the work of El Ghani [8], where the notion of renormalized solutions was also used in the context of a Vlasov–Maxwell–Fokker–Planck system.

In this paper, we study the diffusion and homogenization approximation of the VPFP system in the presence of a spatially oscillating electrostatic potential. This generalizes the study done in [9] where the same problem is treated without the oscillating electrostatic potential. The study of diffusion and homogenization was also done in [3,13,14,17,18,23,25] for the semiconductor Boltzmann–Poisson system and in [19] for the Spherical Harmonics Expansion model. In the present paper, the major difficulty is to get enough compactness in order to be able to pass to the limit in the nonlinear terms. The notion of two-scale convergence [1,10,20,23,25] and averaging lemma are used to prove the convergence of the VPFP system towards a homogenized Drift-Diffusion-Poisson system in the case $d ⩾ 2$ for general assumptions on $f_{I}^{ε}$ giving only a renormalized sense for the solution $f^{ε}$ . We point out here that the presence of an oscillating potential (with macroscopic and microscopic variations) can represent the technology of superlattices [5]. Indeed, superlattices are obtained by growing periodically successive slices of two materials resulting in a periodic potential $ϕ_{H}$ . Here, we assume that the thickness of the potential barriers are of the order of magnitude of diffusive regime. More precisely, when the oscillation period is of the same order of magnitude as a small parameter ε, the asymptotic leads to a phenomenon of homogenization. Note that, the case $d = 1$ is studied by M.L. Tayeb in [24] for a given velocity field with spatial macroscopic and microscopic variations by using the $L^{1}$ -contraction property of the Fokker–Planck operator and a two-scale Hybrid-Hilbert expansion.

1.1. Preliminaries

Before given the spectral properties of the Fokker–Planck operator, we remark that it can be rewritten in the following form: $\begin{matrix} (8) & L_{FP} (f) = \nabla_{v} \cdot (M \nabla_{v} (\frac{f}{M})) . \end{matrix}$ Let us define the Hilbert space $L_{M}^{2} (R^{d})$ as $\begin{matrix} (9) & L_{M}^{2} (R^{d}) = {f \in L^{2} (R^{d}) \int_{R^{d}} f^{2} \frac{d v}{M} < + \infty}, \end{matrix}$ with its inner product $\begin{matrix} {⟨ f, g ⟩}_{L_{M}^{2} (R^{d})} = \int_{R^{d}} f g \frac{d v}{M} . \end{matrix}$ The operator $L_{FP}$ acting on $L_{M}^{2} (R^{d})$ is unbounded, with domain $\begin{matrix} (10) & D (L_{FP}) = {f \in L_{M}^{2} (R^{d}) / \nabla_{v} \cdot (M \nabla_{v} (\frac{f}{M})) \in L_{M}^{2} (R^{d})} \end{matrix}$ and it satisfies the following:

Lemma 1.1 ([9]).

The operator $L_{FP}$ is continuous on $L^{1} (d v)$ and satisfies

$- L_{FP}$ is self adjoint on $L_{M}^{2} (R^{d})$ .

$K er (L_{FP}) = R M$ .

$R (L_{FP}) = {g \in L_{M}^{2} (R^{d}) / \int_{R^{d}} g (v) d v = 0}$ .

For all $g \in R (L_{FP})$ , there exists $f \in D (L_{FP})$ such that $L_{FP} (f) = g$ . This solution is unique under the solvability condition $\int_{R^{d}} f (v) d v = 0$ . We denote that $f = L_{FP}^{- 1} (g)$ .

$L_{FP}$ satisfies the following H-theorem for all $f \in L^{1} (d v) \cap D (L_{FP})$ , $\begin{matrix} \int_{R^{d}} L_{FP} (f) log (\frac{f}{M}) d v = - \int_{R^{d}} f {| \nabla_{v} log (\frac{f}{M}) |}^{2} d v . \end{matrix}$

1.2. Notations and main result

We will use in the sequel the subscript ${(\cdot)}_{#}$ to mean that we consider functions defined on the whole space in y and Y-periodic with respect to y. Indeed, $C_{#} (Y)$ and $C_{#}^{\infty} (Y)$ denote respectively continuous and infinitely differentiable functions defined on Y and extended on $R^{d}$ by Y-periodicity. For $p > 1$ , $L_{#}^{p} (Y)$ will denote the space of functions of $L_{loc}^{p} (R^{d})$ and Y-periodic. For an open subset $Ω \subset R^{d}$ , $L^{p} (Ω; C_{#} (Y))$ is the space of functions of $L^{p} (Ω)$ with value in $C_{#} (Y)$ . The following spaces $D^{'} (Ω; C_{#}^{\infty} (Y)), D (Ω; C_{#}^{\infty} (Y)), \dots$ are defined in the same manner. We also use the notation $D_{#} : = D ((0, T) \times Ω; C_{#}^{\infty} (Y))$ to denote the space of test functions which are periodic in Y. For Y-periodic functions on y, the notations ${(f)}_{ε}, {(\nabla_{y} f)}_{ε}, \dots$ will be used instead of $f (t, x, \frac{x}{ε}, v), \nabla_{y} f (t, x, \frac{x}{ε}, v), \dots$ .

We will also use the notation $\begin{matrix} ϕ_{H}^{ε} (x) = {(ϕ_{H})}_{ε} (x) : = ϕ_{H} (x, \frac{x}{ε}) . \end{matrix}$ We define the Maxwellians $M_{ϕ_{H}}$ and $M_{ϕ_{T}^{ε}}$ by $\begin{matrix} (11) & \{\begin{matrix} M_{ϕ_{H}} (x, y, v) = M (v) exp (ϕ_{e} (x) - ϕ_{H} (x, y)), \\ M_{ϕ_{T}^{ε}} (t, x, v) = M (v) exp (ϕ_{e} (x) - ϕ_{T}^{ε} (t, x)), \end{matrix} \end{matrix}$ where $\begin{matrix} (12) & ϕ_{e} (x) : = - log (\int_{Y} e^{- ϕ_{H} (x, y)} d y) \end{matrix}$ is the homogenized effective potential introduced to normalize $M_{ϕ_{H}}$ with respect to both variable v and y (i.e. $\int_{y, v} M_{ϕ_{H}} = 1$ ).

We define the total mass and the kinetic energy by $\begin{matrix} M^{ε} (t) = \int_{Ω} f^{ε} (t, x, v) d v d x, K^{ε} (t) = \int_{Ω} \frac{| v |^{2}}{2} f^{ε} (t, x, v) d v d x . \end{matrix}$ The charge and current densities associated to the distribution $f^{ε}$ are defined by $\begin{matrix} ρ^{ε} (t, x) = \int_{R^{d}} f^{ε} (t, x, v) d v, J^{ε} (t, x) = \frac{1}{ε} \int_{R^{d}} v f^{ε} (t, x, v) d v . \end{matrix}$ We will also use the following notations $\begin{matrix} (13) & \{\begin{matrix} \tilde{f^{ε}} (t, x, v) = f^{ε} (t, x, v) e^{ϕ_{H}^{ε}}, \\ \tilde{ρ^{ε}} (t, x) = ρ^{ε} (t, x) e^{ϕ_{H}^{ε}}, \\ \tilde{J^{ε}} (t, x) = J^{ε} (t, x) e^{ϕ_{H}^{ε}} . \end{matrix} \end{matrix}$

Assumptions.
Throughout this study we shall use the following assumptions:
There exists a constant $C > 0$ independent of ε such that $\begin{matrix} f_{I}^{ε} ⩾ 0 and \int_{Ω} f_{I}^{ε} (1 + \frac{| v |^{2}}{2} + | log (f_{I}^{ε}) |) d v d x < C . \end{matrix}$

The potential $ϕ_{H} ⩾ 0$ and belongs to $W^{2, \infty} (\overline{ω} \times Y)$ .

The boundary data $ρ_{b}$ is bounded from below and above: There exist ${\tilde{c}}_{b}$ and $c_{b}$ such that $0 < {\tilde{c}}_{b} ⩽ ρ_{b} (\cdot, x) ⩽ c_{b}$ for all $x \in \partial ω$ .

Let us now state our main result:
Theorem 1.2.
Assume that assumptions $A1$ , $A2$ and $A3$ are satisfied. Let $(f^{ε}, ϕ_{P}^{ε})$ be a renormalized solution in the sense of Definition 3.1 of the VPFP system ( 1 )–( 5 ) and which satisfies in addition the properties of Proposition 3.3 . Then, $\begin{array}{rcl} \tilde{f^{ε}} : = f^{ε} e^{ϕ_{H}^{ε}} & \to & ρ M (v) e^{ϕ_{e} (x)} in L^{1} (0, T; L^{1} (Ω)), \\ ϕ_{P}^{ε} & \to & ϕ_{P} in L^{2} (0, T; W^{1, p} (ω)), \forall 1 ⩽ p < 2 . \end{array}$ In particular, $ρ^{ε}$ converges weakly in $L^{1} (0, T; L^{1} (ω))$ towards ρ and $(ρ, ϕ_{P})$ is the solution of the homogenized Drift-Diffusion-Poisson system $\begin{matrix} (14) & \{\begin{matrix} \partial_{t} ρ + \nabla_{x} . J (ρ, ϕ_{P} + ϕ_{e}) = 0, \\ J (ρ, ϕ_{P} + ϕ_{e}) = - D (x) (\nabla_{x} ρ + ρ \nabla_{x} (ϕ_{P} + ϕ_{e})), \\ - △_{x} ϕ_{P} = ρ, \\ {(ρ, ϕ_{P})}_{| \partial ω} = (ρ_{b}, 0), \\ ρ (t = 0, x) = ρ_{I} (x) = \int_{R^{d}} f_{I} (x, v) d v, \end{matrix} \end{matrix}$ where the function $ϕ_{e}$ and the matrix $D$ are given, respectively, by ( 12 ) and ( 29 ) and $f_{I}$ is the weak limit of $f_{I}^{ε}$ .

The proof of Theorem 1.2 is as follows. Section 2, is devoted to the formal analysis of the asymptotic, using two-scale Hilbert expansions. In Section 3, we shall recall the existence result of renormalized solutions to the VPFP system. Then, in Section 4, we establish some a priori uniform estimates. In Section 5, we prove the compactness of $\tilde{f^{ε}}$ and $\tilde{ρ^{ε}}$ by using an averaging lemma. This result will be essential to pass to the limit in the continuity equation which will be done in Section 6.
2. Formal expansion for the Vlasov–Fokker–Planck equation

In this section, we shall present the formal analysis allowing to describe the limit model. To this aim, we assume that $f^{ε}$ has the following two-scale Hilbert development: $\begin{matrix} (15) & f^{ε} (t, x, v) = f_{0} (t, x, \frac{x}{ε}, v) + ε f_{1} (t, x, \frac{x}{ε}, v) + ε^{2} f_{2} (t, x, \frac{x}{ε}, v) + \dots, \end{matrix}$ where the coefficients $f_{i} \in L^{2} ((0, T) \times Ω, L_{#}^{2} (Y))$ are Y-periodic with respect to y. Let us consider the linear Vlasov–Fokker–Planck equation associated with a given potential $ϕ_{H}^{ε}$ : $\begin{matrix} (16) & \partial_{t} f^{ε} + \frac{1}{ε} (v \cdot \nabla_{x} f^{ε} - \nabla_{x} ϕ_{H}^{ε} \cdot \nabla_{v} f^{ε}) = \frac{L_{FP} (f^{ε})}{ε^{2}} . \end{matrix}$ All the coefficient $f_{i}$ , $i = 0, 1, 2, \dots$ , are periodic with respect to the third variable $y = \frac{x}{ε}$ . If we insert the two-scale development (15) in the previous equation, we can replace $\nabla_{x}$ by $\nabla_{x} + \frac{1}{ε} \nabla_{y}$ where $y = \frac{x}{ε} \in {[0, 1]}^{d}$ . Hence, (16) becomes $\begin{matrix} (17) & \partial_{t} (\sum_{i} ε^{i} f_{i}) + (v \cdot \nabla_{x} - \nabla_{x} ϕ_{H} \cdot \nabla_{v}) (\sum_{i} ε^{i - 1} f_{i}) = - \sum_{i} ε^{i - 2} L_{ϕ_{H}} (f_{i}), \end{matrix}$ where the cell operator $\begin{matrix} (18) & L_{ϕ_{H}} = v \cdot \nabla_{y} - \nabla_{y} ϕ_{H} \cdot \nabla_{v} - L_{FP} . \end{matrix}$ Letting identifying the coefficients of the same powers on ε, we infer: $\begin{array}{l} (19) & ε^{- 2} : L_{ϕ_{H}} f_{0} = 0, \\ (20) & ε^{- 1} : - L_{ϕ_{H}} f_{1} = v \cdot \nabla_{x} f_{0} - \nabla_{x} ϕ_{H} (x, y) \cdot \nabla_{v} f_{0}, \\ (21) & ε^{0} : - L_{ϕ_{H}} f_{2} = \partial_{t} f_{0} + v \cdot \nabla_{x} f_{1} - \nabla_{x} ϕ_{H} (x, y) \cdot \nabla_{v} f_{1} . \end{array}$ To obtain the evolution equation satisfied by $f_{0}$ , we need to study the spectral properties of $L_{ϕ_{H}}$ . This operator acting on the Hilbert space $\begin{matrix} L_{M_{ϕ_{H}}}^{2} = {f \in L_{loc}^{2} (d y d v / M_{ϕ_{H}}) / f : Y -periodic with respect to y} \end{matrix}$ equipped with the inner product $\begin{matrix} {⟨ f, g ⟩}_{L_{M_{ϕ_{H}}}^{2}} = \int_{R^{d}} \int_{Y} f g \frac{d y d v}{M_{ϕ_{H}}} . \end{matrix}$ The operator $L_{ϕ_{H}}$ acting on $L_{M_{ϕ_{H}}}^{2}$ is unbounded with domain $\begin{matrix} D (L_{ϕ_{H}}) = {f \in L_{M_{ϕ_{H}}}^{2} / v \cdot \nabla_{y} f - \nabla_{y} ϕ_{H} \cdot \nabla_{v} f \in L_{M_{ϕ_{H}}}^{2}} \end{matrix}$ and satisfies the following:

Proposition 2.1.
The operator $L_{ϕ_{H}}$ is maximal monotone on $L_{M_{ϕ_{H}}}^{2}$ and satisfies
$ker (L_{ϕ_{H}}) = R M_{ϕ_{H}}$ .

$R (L_{ϕ_{H}}) = {g \in L_{M_{ϕ_{H}}}^{2} / \int \int_{Y \times R^{d}} g (y, v) d y d v = 0}$ .

For all $g \in R (L_{ϕ_{H}})$ , there exists $f \in D (L_{ϕ_{H}})$ such that $L_{ϕ_{H}} f = g$ . This solution is unique under the solvability condition $\begin{matrix} \int \int_{Y \times R^{d}} f (y, v) d y d v = 0 . \end{matrix}$ We denote it $f = L_{ϕ_{H}}^{- 1} g$ .

Proof.
The monotonicity of $L_{ϕ_{H}}$ is a consequence of the entropy inequality $\begin{matrix} \int \int_{Y \times R^{d}} L_{ϕ_{H}} (f) \frac{f}{M_{ϕ_{H}}} d y d v = - \int_{Y} e^{ϕ_{H} - ϕ_{e}} (\int_{R^{d}} L_{FP} (f) \frac{f}{M} d v) d y . \end{matrix}$ In particular, $\begin{matrix} - \int_{R^{d}} L_{FP} (f) \frac{f}{M} d v = \int_{R^{d}} M {| \nabla_{v} (\frac{f}{M}) |}^{2} d v ⩾ 0 . \end{matrix}$ Hence, $\begin{array}{rcl} {⟨ L_{ϕ_{H}} (f), f ⟩}_{L_{M_{ϕ_{H}}}^{2}} & = & {⟨ L_{ϕ_{H}}^{} (f), f ⟩}_{L_{M_{ϕ_{H}}}^{2}} \\ = & - \int_{Y} e^{ϕ_{H} - ϕ_{e}} (\int_{R^{d}} L_{FP} (f) \frac{f}{M} d v) d y ⩾ 0 . \end{array}$ This implies that $L_{ϕ_{H}}$ and its adjoint $L_{ϕ_{H}}^{}$ are monotone. Moreover, since $L_{ϕ_{H}}$ is closed and has a dense domain in $L_{M_{ϕ_{H}}}^{2}$ , then $L_{ϕ_{H}}$ is maximal monotone.

Let us now prove the properties of $L_{ϕ_{H}}$ : To prove point 1, let $f \in ker (L_{ϕ_{H}})$ , we have $\begin{array}{rcl} \int \int_{Y \times R^{d}} L_{ϕ_{H}} (f) \frac{f}{M_{ϕ_{H}}} d y d v = - \int_{Y} e^{ϕ_{H} - ϕ_{e}} (\int_{R^{d}} L_{FP} (f) \frac{f}{M} d v) d y = 0 . \end{array}$ This implies that $\begin{matrix} \int_{R^{d}} L_{FP} (f) \frac{f}{M} d v = 0 . \end{matrix}$ Now by using Lemma 1.1, we deduce that $f = ρ (t, x, y) M (v)$ . Inserting this expression in $L_{ϕ_{H}} (f) = 0$ , we obtain $\nabla_{y} ρ + ρ \nabla_{y} ϕ = 0$ . Therefore, $ker (L_{ϕ_{H}})$ is spanned by $M_{ϕ_{H}}$ .

To prove point 2 and 3, we remark that an integration of the equation $L_{ϕ_{H}} (f) = g$ with respect to y and v show that the condition $\begin{array}{rcl} (22) & \int \int_{Y \times R^{d}} g (y, v) d y d v = 0 \end{array}$ is a necessary condition. We shall show now that this condition is sufficient, namely for ε tending to zero and g satisfying (22), the solution of $\begin{matrix} \{\begin{matrix} ε f^{ε} + L_{ϕ_{H}} (f^{ε}) = g, \\ f^{ε} : Y -periodic with respect to y \end{matrix} \end{matrix}$ is uniformly bounded. To this aim, we proceed by contradiction and assume that $f^{ε}$ is unbounded in $L_{M_{ϕ_{H}}}^{2}$ . Denoting $\begin{array}{rcl} R^{ε} = {‖ f^{ε} ‖}_{L_{M_{ϕ_{H}}}^{2}}, F^{ε} = f^{ε} / R^{ε} and G^{ε} = g / R^{ε} . \end{array}$ Then, the sequence ${(F^{ε})}_{ε}$ is bounded in $L_{M_{ϕ_{H}}}^{2}$ and $\begin{array}{rcl} (23) & ε F^{ε} + (v \cdot \nabla_{y} - \nabla_{y} ϕ_{H} \cdot \nabla_{v}) F^{ε} = L_{FP} (F^{ε}) + G^{ε} . \end{array}$ Besides, $F^{ε}$ converges weakly to a function $F \in ker (L_{ϕ_{H}})$ . After integration of the above equation, the condition (22) leads to $F \equiv 0$ . Now, multiplying (23) by $F^{ε} / M_{ϕ_{H}}$ , integrating with respect to y and v, using Jensen’s inequality, and taking advantage of the identity $\begin{matrix} (v \cdot \nabla_{y} - \nabla_{y} ϕ_{H} \cdot \nabla_{v}) M_{ϕ_{H}} (t, x, y) = 0, \end{matrix}$ we obtain $\begin{matrix} ε - \int \int_{Y \times R^{d}} L_{ϕ_{H}} (F^{ε}) \frac{F^{ε}}{M_{ϕ_{H}}} d y d v ⩽ {‖ G^{ε} ‖}_{L_{M_{ϕ_{H}}}^{2}} . \end{matrix}$ So that $\begin{matrix} ε + \int \int_{Y \times R^{d}} M_{ϕ_{H}} {| \nabla_{v} (\frac{F^{ε}}{M_{ϕ_{H}}}) |}^{2} d y d v ⩽ {‖ G^{ε} ‖}_{L_{M_{ϕ_{H}}}^{2}} . \end{matrix}$ Then, we get $\begin{matrix} ε + \int \int_{Y \times R^{d}} M_{ϕ_{H}} {| \nabla_{v} (\frac{F^{ε} - ρ_{F^{ε}} M_{ϕ_{H}}}{M_{ϕ_{H}}}) |}^{2} d y d v ⩽ {‖ G^{ε} ‖}_{L_{M_{ϕ_{H}}}^{2}}, \end{matrix}$ where $\begin{matrix} (24) & ρ_{F^{ε}} (y) : = \int_{R^{d}} F^{ε} (y, v) d v . \end{matrix}$ One can deduce that $\begin{matrix} (25) & F^{ε} - M (v) ρ_{F^{ε}} \to 0 in L_{M_{ϕ_{H}}}^{2} . \end{matrix}$ As a consequence, $F^{ε}$ converges to $ρ (t, x, y) M (v)$ . Therefore, the condition (22) leads to $\begin{matrix} \int \int_{Y \times R^{d}} F^{ε} d y d v = 0 \end{matrix}$ and the passage to the limit gives $\int_{Y} ρ d y = 0$ . Moreover, since for all $v \in R^{d}$ $\begin{matrix} (\nabla_{y} ρ + ρ \nabla_{y} ϕ_{H}) v M (v) = 0, \end{matrix}$ then $\begin{matrix} ρ = k (t, x) exp (- ϕ_{H} (x, y)), \end{matrix}$ which implies that $ρ \equiv 0$ .

Now, we want to prove that $ρ_{F^{ε}}$ is relatively compact in $L^{2} (Y)$ . Multiplying (23) by v, we obtain $\begin{matrix} (26) & ε v F^{ε} + (v \otimes v) \cdot \nabla_{v} F^{ε} - (v \otimes \nabla_{y} ϕ_{H}) \cdot \nabla_{v} F^{ε} = v L_{FP} (F^{ε}) + v G^{ε} . \end{matrix}$ Taking $F^{ε} = ρ_{F^{ε}} M (v) + θ^{ε}$ , the equation (26) becomes $\begin{array}{rcl} (27) & ε v ρ_{F^{ε}} M (v) + (v \otimes v) M (v) (\nabla_{y} ρ_{F^{ε}} + \nabla_{y} ϕ_{H} ρ_{F^{ε}}) = S^{ε}, \end{array}$ where $\begin{array}{rcl} S^{ε} : = v G^{ε} + v L_{FP} (θ^{ε}) - ε v θ^{ε} - (v \otimes v) \cdot \nabla_{y} θ^{ε} + (\nabla_{y} ϕ_{H} \cdot \nabla_{v} θ^{ε}) v . \end{array}$ Integrating (27) with respect to v and since $\begin{matrix} \int_{R^{d}} (v \otimes v) M (v) d v = d I_{R^{d}}, \end{matrix}$ we get $\begin{matrix} d (\nabla_{y} ρ_{F^{ε}} + ρ_{F^{ε}} \nabla_{y} ϕ_{H}) = \int_{R^{d}} S^{ε} d v . \end{matrix}$ Since, the sequence $θ^{ε}$ converge to zero in $L_{M_{ϕ_{H}}}^{2}$ and $ϕ_{H} \in W^{1, \infty} (Y)$ , then $\int_{R^{d}} S^{ε} d v$ belongs to a compact subset of ${[H^{- 1} (Y)]}^{d}$ . Thanks to the identity $\begin{matrix} \nabla_{y} ρ_{F^{ε}} = \frac{1}{d} \int_{R^{d}} S^{ε} d v - ρ_{F^{ε}} \nabla_{y} ϕ_{H}, \end{matrix}$ one can deduce that $(ρ_{F^{ε}})$ is relatively compact in $L^{2} (Y)$ . Consequently, $F^{ε}$ converges (up to an extraction of a subsequence) strongly in $L_{M_{ϕ_{H}}}^{2}$ . This is not possible because $‖ F^{ε} ‖_{L_{M_{ϕ_{H}}}^{2}} = 1$ and its weak limit is equal to zero. □

Let us continue the derivation of the fluid system. According to (19) and the above proposition, the weak limit of $f^{ε}$ has the form: $\begin{matrix} f_{0} (t, x, y, v) = ρ (t, x) M_{ϕ_{H}} (x, y, v) . \end{matrix}$ As a consequence, a simple computation of the right hand side of (20) gives $\begin{array}{rcl} (28) & f_{1} (t, x, y, v) = - (\nabla_{x} ρ + ρ \nabla_{x} ϕ_{e}) (t, x) \cdot L_{ϕ_{H}}^{- 1} (v M_{ϕ_{H}}) (x, y, v) . \end{array}$ Let us define the diffusion matrix $D$ by $\begin{array}{rcl} (29) & D (x) : = \int \int_{Y \times R^{d}} v \otimes χ (x, y, v) d y d v, \end{array}$ where χ is the unique solution in ${[R (L_{ϕ_{H}}) \cap D (L_{ϕ_{H}})]}^{d}$ of $\begin{matrix} (30) & L_{ϕ_{H}} χ = v M_{ϕ_{H}} . \end{matrix}$ Integrating (21) with respect to y and v, the solvability condition gives the drift-diffusion equation associated to the potential $ϕ_{e}$ : $\begin{matrix} \partial_{t} ρ + \nabla_{x} \cdot J (ρ, ϕ_{e}) = 0, \end{matrix}$ where $\begin{matrix} J (ρ, ϕ_{e}) (t, x) = - D (x) (\nabla_{x} ρ + ρ \nabla_{x} ϕ_{e}) (t, x) . \end{matrix}$ We remark that, this limit equation is associated with an effective potential $ϕ_{e}$ collecting some microscopic information induced by the rapidly oscillating potential $ϕ_{H}$ .
3. Existence of renormalized solution

Let us now give the definition of renormalized solution.

Definition 3.1.
We say that $(f^{ε}, ϕ_{P}^{ε})$ is a renormalized solution to the VPFP system (1)–(5) if the function ${\tilde{f}}^{ε} : = f^{ε} e^{ϕ_{H}^{ε}}$ satisfies
For all $β \in C^{1} (R^{+})$ , $| β (t) | ⩽ C (\sqrt{t} + 1)$ , $| \sqrt{t} β^{'} (t) | ⩽ C$ and $| t β^{″} (t) | ⩽ C$ , $β ({\tilde{f}}^{ε})$ is a weak solution of $\begin{matrix} (31) & \{\begin{matrix} ε \partial_{t} β ({\tilde{f}}^{ε}) + v \cdot \nabla_{x} β ({\tilde{f}}^{ε}) - \nabla_{v} \cdot (\nabla_{x} ϕ_{T}^{ε} β ({\tilde{f}}^{ε})) \\ = \frac{L_{FP} ({\tilde{f}}^{ε})}{ε} β^{'} ({\tilde{f}}^{ε}) + v \cdot {(\nabla_{x} ϕ_{H})}_{ε} {\tilde{f}}^{ε} β^{'} ({\tilde{f}}^{ε}) + \frac{1}{ε} v \cdot {(\nabla_{y} ϕ_{H})}_{ε} {\tilde{f}}^{ε} β^{'} ({\tilde{f}}^{ε}), \\ β {({\tilde{f}}^{ε})}_{| Γ^{-}} = β ({\tilde{f}}_{b}), \\ β ({\tilde{f}}^{ε}) (t = 0) = β ({\tilde{f}}_{I}^{ε}) . \end{matrix} \end{matrix}$

For all $λ > 0$ , $θ_{ε, λ} : = \sqrt{f^{ε} + λ M e^{- ϕ_{H}^{ε}}}$ satisfies $\begin{array}{rcl} (32) & ε \partial_{t} θ_{ε, λ} + v \cdot \nabla_{x} θ_{ε, λ} - \nabla_{v} \cdot (\nabla_{x} ϕ_{T}^{ε} θ_{ε, λ}) = \frac{1}{2 ε θ_{ε, λ}} L_{FP} ({\tilde{f}}^{ε}) + \frac{λ M}{2 θ_{ε, λ}} v \cdot \nabla_{x} ϕ_{P}^{ε} e^{- ϕ_{H}^{ε}}, \end{array}$
where the notation ${(\cdot)}_{ε}$ means that we take the expression between brackets at point $(t, x, \frac{x}{ε}, v)$ .
Remark 3.2.
The combination of the Fokker–Planck term and the Poisson term makes the existence of weak solutions to the VPFP system (1)–(5) with uniform bounds a difficult problem and we were not able to construct such kind of solutions. We notice then, that the entropy bound given in (34) is not enough to give a sense to the product $f^{ε} \nabla_{x} ϕ_{T}^{ε}$ . We stress that we need to use two types of renormalization for the Vlasov–Fokker–Planck equation (1), namely $β_{δ} ({\tilde{f}}^{ε})$ (where $β_{δ}$ was defined in Section 5) to get the compactness of $\tilde{ρ^{ε}}$ and $θ_{ε, λ}$ to pass to the limit in $J^{ε}$ .

The next proposition states the existence of renormalized solution. Proposition 3.3.
The VPFP system ( 1 )–( 5 ) has a renormalized solution in the sense of Definition 3.1 which satisfies in addition
The continuity equation: $\begin{array}{rcl} (33) & \partial_{t} ρ^{ε} + \nabla_{x} \cdot J^{ε} & = & 0 . \end{array}$

The entropy inequality: $\begin{array}{l} {[\int_{Ω} f^{ε} (log f^{ε} + \frac{| v |^{2}}{2} + ϕ_{H}^{ε}) d v d x + \frac{1}{2} {‖ \nabla_{x} ϕ_{P}^{ε} ‖}_{L^{2}}^{2}]}_{0}^{t} + \frac{4}{ε^{2}} \int_{0}^{t} \int_{Ω} {| \nabla_{v} \sqrt{f^{ε} e^{\frac{| v |^{2}}{2}}} |}^{2} e^{- \frac{| v |^{2}}{2}} \\ (34) & ⩽ - \frac{1}{ε} \int_{0}^{t} \int_{Γ^{+} \cup Γ^{-}} f^{ε} (log f^{ε} + \frac{| v |^{2}}{2} + ϕ_{T}^{ε}) (v \cdot n (x)) . \end{array}$

Proof.
The first estimate can be deduced by integrating (1) with respect to v. The second estimate can be deduced by multiplying (1) by $\frac{| v |^{2}}{2}$ , integrating the result with respect to x and v. Thus we obtain $\begin{array}{l} \frac{d}{d t} \int_{Ω} \frac{| v |^{2}}{2} f^{ε} d v d x + \int_{Ω} (v \cdot \nabla_{x} ϕ_{T}^{ε}) f^{ε} d v d x + \frac{1}{ε} \int_{Γ^{+} \cup Γ^{-}} \frac{| v |^{2}}{2} f^{ε} (v \cdot n (x)) \\ (35) & = - \frac{1}{ε^{2}} \int_{Ω} (v f^{ε} + \nabla_{v} f^{ε}) \cdot v d v d x \end{array}$ the third term on the left-hand side of (35) can be written as follows: $\begin{array}{rcl} \int_{Ω} (v \cdot \nabla_{x} ϕ_{T}^{ε}) f^{ε} d v d x & = & \frac{d}{d t} \int_{Ω} f^{ε} ϕ_{H}^{ε} d v d x + \frac{1}{2} \frac{d}{d t} \int_{w} {| \nabla_{x} ϕ_{P}^{ε} |}^{2} d x + \frac{1}{ε} \int_{Γ^{+} \cup Γ^{-}} f^{ε} ϕ_{T}^{ε} (v \cdot n (x)) . \end{array}$ Finally, the entropy inequality is formally obtained by multiplying the equation (1) by $log f^{ε}$ and integrating the result with respect to x and v. We get $\begin{array}{l} \frac{d}{d t} \int_{Ω} f^{ε} log f^{ε} d v d x + \frac{1}{ε} \int_{Γ^{+} \cup Γ^{-}} f^{ε} log f^{ε} (v \cdot n (x)) = - \frac{1}{ε^{2}} \int_{Ω} (v f^{ε} + \nabla_{v} f^{ε}) \cdot \frac{\nabla_{v} f^{ε}}{f^{ε}} d v d x . \end{array}$ Therefore, summing up these relations yields $\begin{array}{l} \frac{d}{d t} (\int_{Ω} f^{ε} (log f^{ε} + \frac{| v |^{2}}{2} + ϕ_{H}^{ε}) d v d x + \frac{1}{2} {‖ \nabla_{x} ϕ_{P}^{ε} ‖}_{L^{2}}^{2}) \\ + \frac{1}{ε^{2}} \int_{Ω} (v f^{ε} + \nabla_{v} f^{ε}) \cdot (v + \frac{\nabla_{v} f^{ε}}{f^{ε}}) d v d x \\ = - \frac{1}{ε} \int_{Γ^{+} \cup Γ^{-}} f^{ε} (log f^{ε} + \frac{| v |^{2}}{2} + ϕ_{T}^{ε}) (v \cdot n (x)) . \end{array}$ Then, since $\begin{array}{rcl} \frac{1}{ε^{2}} \int_{Ω} (v f^{ε} + \nabla_{v} f^{ε}) \cdot (v + \frac{\nabla_{v} f^{ε}}{f^{ε}}) d v d x & = & \frac{1}{ε^{2}} \int_{Ω} {(v \sqrt{f^{ε}} + 2 \nabla_{v} \sqrt{f^{ε}})}^{2} d v d x \\ = & \frac{4}{ε^{2}} \int_{Ω} {| \nabla_{v} \sqrt{f^{ε} e^{\frac{| v |^{2}}{2}}} |}^{2} e^{- \frac{| v |^{2}}{2}} d v d x . \end{array}$ Which leads to the desired results. □
Remark 3.4.
Notice that the assumption of well-prepared incoming boundary data is essential to control the entropy production terms coming from the boundary in order to get some uniform estimates on the moments of $f^{ε}$ .

4. Uniform estimates

The aim of this section is the derivation of a priori estimates, uniform with respect to ε. Precisely, we will to prove the following proposition.

Proposition 4.1.
Assume that assumptions $A1$ , $A2$ and $A3$ are satisfied. Let $(f^{ε}, ϕ_{p}^{ε})$ be a renormalized solution of the VPFP system ( 1 )–( 5 ) which satisfies the conclusions of Proposition 3.3 . Then, $\begin{array}{l} M^{ε} (t) + \frac{1}{2} K^{ε} (t) + \frac{2}{ε^{2}} \int_{0}^{t} \int_{Ω} {| \nabla_{v} \sqrt{f^{ε} e^{\frac{| v |^{2}}{2}}} |}^{2} e^{- \frac{| v |^{2}}{2}} d v d x d s \\ (36) & + \frac{1}{2} {‖ \nabla_{x} ϕ_{P}^{ε} ‖}_{L^{2}}^{2} + \int_{0}^{t} {‖ J^{ε} (s) ‖}_{L^{1} (ω)}^{2} d s ⩽ C_{T} . \end{array}$
Proof.
We denote the total relative entropy $E^{ε}$ by $\begin{array}{rcl} (37) & E^{ε} (t) & = & \int_{Ω} f^{ε} (log f^{ε} + \frac{| v |^{2}}{2} + ϕ_{H}^{ε}) (t) + \frac{1}{2} {‖ \nabla_{x} ϕ_{P}^{ε} ‖}_{L^{2} (ω)}^{2} (t) . \end{array}$ Using the inequality $a log (\frac{a}{b}) ⩾ a - b$ for $a, b > 0$ , we obtain $\begin{array}{rcl} \int_{Ω} f^{ε} (log f^{ε} + \frac{| v |^{2}}{2}) d v d x & = & \int_{Ω} f^{ε} log (\frac{f^{ε}}{e^{- \frac{| v |^{2}}{4}}}) d v d x + \frac{1}{2} K^{ε} (t) \\ ⩾ & M^{ε} (t) - | ω | \int_{R^{d}} e^{- \frac{| v |^{2}}{4}} d v + \frac{1}{2} K^{ε} (t) \end{array}$ which implies that $\begin{matrix} M^{ε} (t) + \frac{1}{2} K^{ε} (t) ⩽ \int_{Ω} f^{ε} (log f^{ε} + \frac{| v |^{2}}{2}) d v d x + | ω | \int_{R^{d}} e^{- \frac{| v |^{2}}{4}} d v . \end{matrix}$ From Proposition 3.3, we deduce that $\begin{array}{l} M^{ε} (t) + \frac{1}{2} K^{ε} (t) + \frac{4}{ε^{2}} \int_{0}^{t} \int_{Ω} {| \nabla_{v} \sqrt{f^{ε} e^{\frac{| v |^{2}}{2}}} |}^{2} e^{- \frac{| v |^{2}}{2}} d v d x d s + \frac{1}{2} {‖ \nabla_{x} ϕ_{P}^{ε} ‖}_{L^{2}}^{2} \\ (38) & ⩽ E^{ε} (0) + | ω | \int_{R^{d}} e^{- \frac{| v |^{2}}{4}} d v - \frac{1}{ε} \int_{0}^{t} \int_{Γ^{-} \cup Γ^{+}} f^{ε} (log f^{ε} + \frac{| v |^{2}}{2} + ϕ_{T}^{ε}) | v \cdot n (x) | . \end{array}$ One can rewrite the boundary fluxes in the following manner $\begin{array}{l} \int_{Γ^{-} \cup Γ^{+}} f^{ε} (log f^{ε} + \frac{| v |^{2}}{2} + ϕ_{T}^{ε}) (v \cdot n (x)) \\ = \int_{Γ^{+}} | v \cdot n (x) | f^{ε} log (\frac{f^{ε} (v)}{f^{ε} (- v)}) + \int_{Γ^{+}} | v \cdot n (x) | (f^{ε} (v) - f^{ε} (- v)) [log (\frac{ρ_{b} (t, x)}{{(2 π)}^{d / 2}}) + ϕ_{e} (x)] \\ ⩾ \int_{Γ^{+}} | v \cdot n (x) | (f^{ε} (v) - f^{ε} (- v)) (1 + E_{F} (t, x)), \end{array}$ where $E_{F}$ is a macroscopic quasi-Fermi level given by $\begin{matrix} E_{F} (t, x) = log (\frac{ρ_{b} (t, x)}{{(2 π)}^{d / 2}}) + ϕ_{e} (x) . \end{matrix}$ Multiplying (33) by $(1 + E_{F} (t, x))$ and integrating by parts, we obtain $\begin{array}{l} \frac{1}{ε} \int_{0}^{t} \int_{Γ^{+}} (1 + E_{F} (t, x)) (f^{ε} (v) - f^{ε} (- v)) | v \cdot n (x) | \\ (39) & = \int_{0}^{t} \int_{ω} \partial_{t} E_{F} ρ^{ε} + \int_{0}^{t} \int_{ω} \nabla_{x} E_{F} \cdot J^{ε} - {[\int_{ω} (1 + E_{F}) ρ^{ε}]}_{0}^{t} . \end{array}$ Combining (39) and according to assumption $A3$ , (38) becomes $\begin{array}{l} M^{ε} (t) + \frac{1}{2} K^{ε} (t) + \frac{4}{ε^{2}} \int_{0}^{t} \int_{Ω} {| \nabla_{v} \sqrt{f^{ε} e^{\frac{| v |^{2}}{2}}} |}^{2} e^{- \frac{| v |^{2}}{2}} + \frac{1}{2} {‖ \nabla_{x} ϕ_{P}^{ε} ‖}_{L^{2}}^{2} \\ ⩽ E^{ε} (0) + | ω | \int_{R^{d}} e^{- \frac{| v |^{2}}{2}} d v - \int_{0}^{t} \int_{ω} \frac{\partial_{t} ρ_{b}}{ρ_{b}} ρ^{ε} - \int_{0}^{t} \int_{ω} \nabla_{x} E_{F} \cdot J^{ε} + {[\int_{ω} (1 + E_{F}) ρ^{ε}]}_{0}^{t} \\ ⩽ C_{T} (1 + \int_{0}^{t} M^{ε} (s) d s + \int_{0}^{t} {‖ J^{ε} (s) ‖}_{L^{1} (ω)} d s), \end{array}$ where $C_{T}$ depends only on T. Let us estimate the current density in the following way $\begin{matrix} \int_{0}^{t} {‖ J^{ε} (s) ‖}_{L^{1} (ω)} d s = \frac{1}{ε} \int_{0}^{t} \int_{ω} | \int_{R^{d}} (v \sqrt{f^{ε}} + 2 \nabla_{v} \sqrt{f^{ε}}) \sqrt{f^{ε}} | . \end{matrix}$ The Young inequality $(α a^{2} + \frac{1}{4 α} b^{2} ⩾ a b, \forall α > 0)$ , gives $\begin{array}{rcl} \int_{0}^{t} {‖ J^{ε} (s) ‖}_{L^{1} (ω)} d s & ⩽ & \frac{α}{ε^{2}} \int_{0}^{t} \int_{Ω} {| v \sqrt{f^{ε}} + 2 \nabla_{v} \sqrt{f^{ε}} |}^{2} d v d x d s + \frac{1}{4 α} \int_{0}^{t} \int_{Ω} f^{ε} d s \\ (40) & ⩽ & \frac{4 α}{ε^{2}} \int_{0}^{t} \int_{Ω} {| \nabla_{v} \sqrt{f^{ε} e^{\frac{| v |^{2}}{2}}} |}^{2} e^{- \frac{| v |^{2}}{2}} + \frac{1}{4 α} \int_{0}^{t} M^{ε} (s) d s, \end{array}$ where α does not depend on ε (for examples $α = \frac{1}{2}$ ). Then, one can deduce $\begin{array}{l} M^{ε} (t) + \frac{1}{2} K^{ε} (t) + \frac{2}{ε^{2}} \int_{0}^{t} \int_{Ω} {| \nabla_{v} \sqrt{f^{ε} e^{\frac{| v |^{2}}{2}}} |}^{2} e^{- \frac{| v |^{2}}{2}} d v d x d s + \frac{1}{2} {‖ \nabla_{x} ϕ_{P}^{ε} ‖}_{L^{2}}^{2} \\ + \int_{0}^{t} {‖ J^{ε} (s) ‖}_{L^{1} (ω)}^{2} d s ⩽ C_{T} (1 + \frac{3}{2} \int_{0}^{t} M^{ε} (s) d s) . \end{array}$ The Gronwall inequality leads to a uniform bound of $M^{ε}$ , $K^{ε}$ , $‖ \nabla_{x} ϕ_{P}^{ε} ‖_{L^{2}}^{2}$ and $\frac{2}{ε^{2}} \int_{0}^{t} \int_{Ω} | \nabla_{v} \sqrt{f^{ε} e^{\frac{| v |^{2}}{2}}} |^{2} e^{- \frac{| v |^{2}}{2}} d v d x d s$ . Then, we get from (40) the $L^{1}$ -bound on $J^{ε}$ . □
Corollary 4.2.
Assume that assumption A1 is satisfied. Then, $| \nabla_{v} \sqrt{f^{ε}} |^{2}$ is bounded in $L^{1} ((0, T) \times Ω)$ .
Proof.
We note that $\begin{array}{rcl} {| \nabla_{v} \sqrt{f^{ε}} |}^{2} = \frac{1}{4} {| v \sqrt{f^{ε}} + 2 \nabla_{v} \sqrt{f^{ε}} |}^{2} - \frac{1}{4} | v |^{2} f^{ε} - v \sqrt{f^{ε}} \cdot \nabla_{v} \sqrt{f^{ε}} \end{array}$ and integrating with respect to t, x and v and using Proposition 4.1 and Cauchy–Schwartz inequality. Thus we obtain the estimate. □

The proof of the next corollary and proposition can be found in [16].
Corollary 4.3.
The renormalized solution satisfies $\begin{matrix} \int_{Ω} f^{ε} (1 + | v |^{2} + | log f^{ε} |) + \int_{0}^{t} \int_{Γ^{+}} f^{ε} (1 + | v |^{2} + | log f^{ε} |) | v \cdot n (x) | ⩽ C_{T} . \end{matrix}$ Moreover, $f^{ε}$ and its trace $f_{| Γ^{+}}^{ε}$ are weakly relatively compact in $L^{1} ((0, T) \times Ω)$ and $L^{1} ((0, T) \times Γ^{+}, | v \cdot n (x) | d t d σ_{x} d v)$ , respectively.
Proposition 4.4.
The renormalized solution $(f^{ε}, ϕ_{P}^{ε})$ satisfies
$ρ^{ε}$ is weakly relatively compact in $L^{1} ((0, T) \times w)$ .

$\nabla ϕ_{P}^{ε}$ is relatively compact in $L^{2} (0, T; L^{p} (ω))$ for all $1 ⩽ p < 2$ .

Let us define $\begin{matrix} r_{ε} = \frac{1}{ε \sqrt{M}} (\sqrt{f^{ε}} - \sqrt{ρ^{ε} M}) . \end{matrix}$
Proposition 4.5.
$r_{ε}$ is such that $| r_{ε} |^{2} M$ is bounded in $L^{1} ((0, T) \times Ω)$ , $ε | r_{ε} |^{2} | v |^{2} M$ is bounded in $L^{1} ((0, T) \times Ω)$ and $\sqrt{ε} | r_{ε} |^{2} | v | M$ is bounded in $L^{1} ((0, T) \times Ω)$ .
Proof.
See ([9], Proposition 5.5). The only difference is that here we are in a bounded and regular domain in x. □

The difficulty in this paper arises from the coupling with the homogeneous Poisson equation and how to get enough compactness to pass to the limit in the nonlinear terms.
5. Compactness of modified density

Proposition 5.1.
The modified density ${\tilde{ρ}}^{ε} (t, x) : = ρ^{ε} (t, x) e^{ϕ_{H}^{ε}}$ is relatively compact in $L^{1} ((0, T) \times ω)$ and there exists $\tilde{ρ} \in L^{1} ((0, T) \times ω)$ such that, up to extraction of a subsequence if necessary, $\begin{array}{l} {\tilde{ρ}}^{ε} \to \tilde{ρ} in L^{1} ((0, T) \times ω) and a.e., \\ {\tilde{f}}^{ε} \to \tilde{ρ} M in L^{1} ((0, T) \times Ω) and a.e. \end{array}$

In the sequel, we will use the following $\begin{matrix} β (s) = \frac{s}{1 + s}, β_{δ} (s) = \frac{1}{δ} β (δ s), \forall s > 0 . \end{matrix}$ We recall that for all fixed parameter $δ > 0$ , we have
$0 ⩽ β_{δ} (s) ⩽ min (s, \frac{1}{δ})$ ,

$| β_{δ} (s) | ⩽ C_{δ} (\sqrt{s} + 1)$ ,

$| β_{δ} (s) - β_{δ} (t) | ⩽ min (| s - t |, \frac{| \sqrt{s} - \sqrt{t} |}{\sqrt{δ}})$ ,

$| \sqrt{s} β_{δ}^{'} (s) | ⩽ C_{δ}$ ,

$| s β_{δ}^{'} (s) - t β_{δ}^{'} (t) | ⩽ min ((1 + δ) | s - t |, C_{δ} | \sqrt{s} - \sqrt{t} |)$ ,

$| s β_{δ}^{″} (s) | ⩽ C_{δ}$ .
We remark that if we want to prove Proposition 5.1, we only need to show for all $δ > 0$ , the compactness of the density associated to ${(β_{δ} ({\tilde{f}}^{ε}))}_{ε}$ . Lemma 5.2 ([16]).

Let $h^{ε}$ be a bounded sequences of $L^{2} ((0, T) \times Ω)$ , $h_{0}^{ε}$ and $h_{1}^{ε}$ are bounded in $L^{1} ((0, T) \times Ω)$ . Assume that $h^{ε}$ satisfies $\begin{array}{rcl} (41) & ε \partial_{t} h^{ε} + v \cdot \nabla_{x} h^{ε} = h_{0}^{ε} + \nabla_{v} \cdot h_{1}^{ε} . \end{array}$ Then, for all $ψ \in C_{0}^{\infty} (R^{d})$ , $\begin{array}{rcl} (42) & {‖ \int_{R^{d}} (h^{ε} (t, x + z, v) - h^{ε} (t, x, v)) ψ (v) d v ‖}_{L_{t, x}^{1}} \to 0 when z \to 0 uniformly in ε, \end{array}$ where $h^{ε}$ has been prolonged by 0 for $x \notin ω$ .

Sketch of the proof of Proposition 5.1.
The proof is done in two steps. The first step consists in proving the compactness of ${\tilde{ρ}}^{ε}$ with respect to the space variable x. This is a consequence of the previous averaging lemma (see also [11]) by using the sequence $h^{ε} : = β_{δ} ({\tilde{f}}^{ε})$ . This distribution satisfies the following transport equation: $\begin{array}{rcl} (43) & ε \partial_{t} β_{δ} ({\tilde{f}}^{ε}) + v . \nabla_{x} β_{δ} ({\tilde{f}}^{ε}) & = & h_{0}^{ε} + \nabla_{v} \cdot h_{1}^{ε}, \end{array}$ where $\begin{array}{l} h_{0}^{ε} : = - \frac{(v {\tilde{f}}^{ε} + \nabla_{v} {\tilde{f}}^{ε})}{ε} \nabla_{v} {\tilde{f}}^{ε} β_{δ}^{″} ({\tilde{f}}^{ε}) + v \cdot {(\nabla_{x} ϕ_{H})}_{ε} {\tilde{f}}^{ε} β_{δ}^{'} ({\tilde{f}}^{ε}) \\ + \frac{v \cdot {(\nabla_{y} ϕ_{H})}_{ε}}{ε} ({\tilde{f}}^{ε} β_{δ}^{'} ({\tilde{f}}^{ε}) - {\tilde{ρ}}^{ε} M β_{δ}^{'} ({\tilde{ρ}}^{ε} M)), \\ h_{1}^{ε} : = (\nabla_{x} ϕ_{P}^{ε} + {(\nabla_{x} ϕ_{H})}_{ε}) β_{δ} ({\tilde{f}}^{ε}) + \frac{(v f^{ε} + \nabla_{v} f^{ε})}{ε} β_{δ}^{'} (f^{ε}) + {(\nabla_{y} ϕ_{H})}_{ε} (\frac{β_{δ} ({\tilde{f}}^{ε}) - β_{δ} ({\tilde{ρ}}^{ε} M)}{ε}) . \end{array}$ In a second step, we show the compactness in time for the modified density ${\tilde{ρ}}^{ε}$ . If we use the continuity equation and multiplying by $e^{ϕ_{H}^{ε}}$ , we see that the modified densities $(\tilde{ρ^{ε}}, \tilde{J^{ε}})$ is a weak solution of $\begin{matrix} \partial_{t} \tilde{ρ^{ε}} + \nabla_{x} \cdot \tilde{J^{ε}} = {(\nabla_{x} ϕ_{H} + \frac{1}{ε} \nabla_{y} ϕ_{H})}_{ε} \cdot \tilde{J^{ε}} \end{matrix}$ due to the presence of the singular term $\frac{1}{ε} \nabla_{y} ϕ_{H}^{ε}$ it is not clear how to obtain directly compactness in time for $\tilde{ρ^{ε}}$ . To overcome this difficulty, we use the two-scale convergence. We refer to ([17], Proposition 5.1) for the details. □

6. Passing to the limit

Now, we would like to pass to the limit ( $ε \to 0$ ). This is done by using a two-scale convergence. Since the Maxwellian is an $L^{1}$ function and its dependence on x and v are separated. So we can pass to the limit by dividing the oscillating test function $ψ^{ε}$ by the oscillating relative Maxwellian $M_{ϕ_{H}^{ε}}$ and using the fact that the cell operator is linear. According to the previous section, there exists $\tilde{ρ} \in L^{1} ((0, T) \times ω)$ the limit of $\tilde{ρ^{ε}}$ and ρ the weak limit of $ρ^{ε}$ . We have $\begin{matrix} e^{- ϕ_{H}^{ε}} ⇀ e^{- ϕ_{e} (x)} : = \int_{Y} e^{- ϕ_{H} (x, y)} d y weakly- ⋆ in L^{\infty} . \end{matrix}$ This implies that $ρ^{ε} : = \tilde{ρ^{ε}} e^{- ϕ_{H}^{ε}} ⇀ ρ : = \tilde{ρ} e^{- ϕ_{e} (x)}$ . Moreover, we have $\begin{matrix} ρ^{ε} - ρ e^{ϕ_{e} - ϕ_{H}^{ε}} = (\tilde{ρ^{ε}} - ρ e^{- ϕ_{e} (x)}) e^{- ϕ_{H}^{ε}} \end{matrix}$ and hence $\begin{matrix} ρ^{ε} - ρ e^{ϕ_{e} - ϕ_{H}^{ε}} \to 0 in L_{t, x}^{1} and a.e. \end{matrix}$ The strong convergence (in $L^{2}$ ) of $\sqrt{\tilde{f^{ε}}}$ and $\sqrt{\tilde{ρ^{ε}} M}$ towards $\sqrt{ρ e^{ϕ_{e}}}$ and the two-scale convergence of the sequence $e^{- ϕ_{H}^{ε}}$ towards $e^{- ϕ_{e}}$ leads to: $\begin{matrix} \sqrt{f^{ε}} \to \sqrt{ρ M_{ϕ_{H}}} in L^{2} ((0, T) \times Ω) two-scale, \end{matrix}$ and $\begin{matrix} \sqrt{ρ^{ε} M} \to \sqrt{ρ M_{ϕ_{H}}} in L^{2} ((0, T) \times Ω) two-scale . \end{matrix}$ Using Proposition 4.4, there exists a potential $ϕ_{P} \in L^{\infty} (0, T; W^{1, 2})$ such that $- Δ ϕ_{P} = ρ$ , $ϕ_{P} = 0$ on $\partial ω$ and $\begin{matrix} \nabla_{x} ϕ_{P}^{ε} \to \nabla_{x} ϕ_{P} in L^{2} (0, T; L^{p} (ω)) for 1 ⩽ p < 2 and a.e. \end{matrix}$ To finish the passage to limit $(ε \to 0)$ we need to identify the weak limit J of the spectral current density $J^{ε}$ . Let us first define $ξ_{ε, λ}$ for all $λ > 0$ by $\begin{matrix} (44) & ξ_{ε, λ} (t, x, v) : = \frac{1}{ε \sqrt{M}} (\sqrt{f^{ε} + λ M e^{- ϕ_{H}^{ε}}} - \sqrt{(ρ^{ε} + λ e^{- ϕ_{H}^{ε}}) M}) \end{matrix}$ which is bounded in $L^{2} (M d t d x d v)$ by using that $| ξ_{ε, λ} | ⩽ | r_{ε} |$ . Let us denote by $\tilde{ξ_{λ}}$ the two scale limit of $ξ_{ε, λ}$ when ε goes to zero, we note that $\tilde{ξ_{λ}}$ is bounded in $L^{2} ((0, T) \times Ω \times Y, d t d x d y M d v)$ . So we denote the subsequence $\tilde{ξ}$ is the weak limit of $\tilde{ξ_{λ}}$ when λ goes to zero. Then we can establish the following proposition:

Proposition 6.1.
$\begin{matrix} J^{ε} ⇀^{D^{'}} J (t, x) : = - 2 \sqrt{ρ} D (x) \cdot (\nabla_{x} \sqrt{ρ} + \frac{1}{2} \sqrt{ρ} \nabla_{x} (ϕ_{P} + ϕ_{e})), \end{matrix}$

where $D$ is the diffusion matrix defined by ( 29 ).
Proof.
Using (44), we can write $\begin{matrix} f^{ε} = ρ^{ε} M + 2 ε ξ_{ε, λ} \sqrt{ρ^{ε} + λ e^{- ϕ_{H}^{ε}}} M + ε^{2} ξ_{ε, λ}^{2} M . \end{matrix}$ Then, we obtain $\begin{matrix} (45) & J^{ε} = 2 \sqrt{ρ^{ε} + λ e^{- ϕ_{H}^{ε}}} \int_{R^{d}} ξ_{ε, λ} v M d v + ε \int_{R^{d}} ξ_{ε, λ}^{2} v M d v . \end{matrix}$ Thus, using that $| ξ_{ε, λ} | ⩽ | r_{ε} |$ , Proposition 4.5 leads to $\begin{matrix} J = 2 \int_{Y} \sqrt{(\tilde{ρ} + λ) e^{- ϕ_{H}}} [\int_{R^{d}} \tilde{ξ_{λ}} v M d v] d y . \end{matrix}$ Sending λ to 0, we infer that $\begin{matrix} (46) & J = 2 \sqrt{\tilde{ρ}} \int_{Y} \int_{R^{d}} \tilde{ξ} v M e^{- ϕ_{H} (x, y) / 2} d v d y . \end{matrix}$ Now, we have to compute $\tilde{ξ}$ to get the expression of J.

First, we remark that $\begin{array}{rcl} \frac{L_{FP} (f^{ε})}{2 ε θ_{ε, λ}} & = & \frac{L_{FP} (ξ_{ε, λ} \sqrt{ρ^{ε} + λ e^{- ϕ_{H}^{ε}}} M + ε ξ_{ε, λ}^{2})}{\sqrt{f^{ε} + λ M e^{- ϕ_{H}^{ε}}}} \\ = & \frac{\sqrt{\tilde{ρ^{ε}} + λ}}{\sqrt{\tilde{f^{ε}} + λ M}} L_{FP} (ξ_{ε, λ} M) + ε \frac{L_{F P} (ξ_{ε, λ}^{2})}{\sqrt{f^{ε} + λ M e^{- ϕ_{H}^{ε}}}} . \end{array}$ Since for $λ > 0$ , $\sqrt{(\tilde{ρ^{ε}} + λ) M}$ and $\sqrt{\tilde{f^{ε}} + λ M}$ converge to $\sqrt{(\tilde{ρ} + λ) M}$ . Then, if we take an oscillating test function $ψ = ψ (t, x, \frac{x}{ε}, v)$ , where $ψ \in D_{#} : = D ((0, T) \times Ω; C_{#}^{\infty} (Y))$ , we get $\begin{matrix} lim_{λ \to 0} lim_{ε \to 0} \int \frac{L_{FP} (f^{ε})}{2 ε θ_{ε, λ}} ψ d v d x d t = \int \int_{Y} \frac{L_{FP} (\tilde{ξ} M)}{\sqrt{M}} ψ d y d v d x d t . \end{matrix}$ Next, let φ be a test function belonging to $D ((0, T) \times Ω)$ . Using that $\begin{matrix} θ_{ε, λ} : = \sqrt{f^{ε} + λ M e^{- ϕ_{H}^{ε}}} = ε \sqrt{M} ξ_{ε, λ} + \sqrt{(\tilde{ρ^{ε}} + λ) M e^{- ϕ_{H}^{ε}}} \end{matrix}$ satisfies (32), we get $\begin{array}{l} \int_{0}^{T} \int_{Ω} (ε \sqrt{M} ξ_{ε, λ} + \sqrt{(\tilde{ρ^{ε}} + λ) e^{- ϕ_{H}^{ε}} M}) [ε \partial_{t} φ + v \cdot \nabla_{x} φ - \nabla_{x} ϕ_{T}^{ε} \cdot \nabla_{v} φ] \\ + \int_{0}^{T} \int_{Ω} [\frac{L_{FP} ({\tilde{f}}^{ε})}{2 ε θ_{ε, λ}} φ + \frac{λ M e^{- ϕ_{H}^{ε}} v \cdot \nabla_{x} ϕ_{P}^{ε}}{2 θ_{ε, λ}} φ] = 0 . \end{array}$

Choosing $φ = ψ (t, x, \frac{x}{ε}, v) = ψ$ , where $ψ \in D_{#}$ , the above formulation becomes $\begin{array}{rcl} B_{ε, λ} & : = & \int_{0}^{T} \int_{Ω} ε \sqrt{M} ξ_{ε, λ} [ε \partial_{t} ψ + v \cdot \nabla_{x} ψ - \nabla_{x} (ϕ_{T}^{ε}) \cdot \nabla_{v} ψ] \\ + \int_{0}^{T} \int_{Ω} \sqrt{(\tilde{ρ^{ε}} + λ) e^{- ϕ_{H}^{ε}} M} [ε \partial_{t} ψ + v \cdot \nabla_{x} ψ - \nabla_{x} (ϕ_{T}^{ε}) \cdot \nabla_{v} ψ] \\ + \int_{0}^{T} \int_{Ω} \sqrt{M} ξ_{ε, λ} [v \cdot \nabla_{y} ψ - \nabla_{y} ϕ_{H} \cdot \nabla_{v} ψ] \\ + \frac{1}{ε} \int_{0}^{T} \int_{ω} \sqrt{(\tilde{ρ^{ε}} + λ)} \nabla_{y} \cdot [\int_{R^{d}} v \sqrt{M e^{- ϕ_{H}^{ε}}} ψ d v] d x d t \\ + \int_{0}^{T} \int_{Ω} [\frac{L_{FP} ({\tilde{f}}^{ε})}{2 ε θ_{ε, λ}} ψ + \frac{λ M v \cdot \nabla_{x} ϕ_{P}^{ε} ψ e^{- ϕ_{H}^{ε}}}{2 θ_{ε, λ}}] \\ = & 0 . \end{array}$ Let us denote $E_{#}$ by $\begin{matrix} E_{#} : = {ψ \in D_{#} / \nabla_{y} \cdot (\int_{R^{d}} v ψ d v) = 0} \end{matrix}$ and pass to the limit only for $\sqrt{M e^{- ϕ_{H}}} ψ \in E_{#}$ in order to overcome the difficulty due to the singular term in $B_{ε, λ}$ . So that using the property $[v \cdot \nabla_{y} - \nabla_{y} ϕ_{H} \cdot \nabla_{v}] \sqrt{M e^{- ϕ_{H}}} = 0$ . Then, we obtain $\begin{array}{rcl} lim_{λ \to 0} lim_{ε \to 0} B_{ε, λ} & = & \int \tilde{ξ} M e^{- ϕ_{H} / 2} [v \cdot \nabla_{y} - \nabla_{y} ϕ_{H} \cdot \nabla_{v}] (\frac{ψ}{\sqrt{M e^{- ϕ_{H}}}}) \\ + \int \frac{ψ}{\sqrt{M e^{- ϕ_{H}}}} L_{FP} (\tilde{ξ} M e^{- ϕ_{H} / 2}) + \int \sqrt{ρ M_{ϕ_{H}}} [v \cdot \nabla_{x} ψ - \nabla_{x} (ϕ_{H} + ϕ_{P}) \cdot \nabla_{v} ψ] \\ = & 0 . \end{array}$ Integrating by parts, we get $\begin{array}{rcl} \int ψ \sqrt{M e^{- ϕ_{H}}} [\frac{L_{ϕ_{H}} (\tilde{ξ} M e^{- ϕ_{H} / 2})}{M e^{- ϕ_{H}}} + \frac{[v \cdot \nabla_{x} - \nabla_{x} (ϕ_{H} + ϕ_{P}) \cdot \nabla_{v}] (\sqrt{ρ M_{ϕ_{H}}})}{\sqrt{M e^{- ϕ_{H}}}}] d v d x d y d t = 0 \end{array}$ which means that $\begin{matrix} \frac{L_{ϕ_{H}} (\tilde{ξ} M e^{- ϕ_{H} / 2})}{M e^{- ϕ_{H}}} + \frac{[v \cdot \nabla_{x} - \nabla_{x} (ϕ_{H} + ϕ_{P}) \cdot \nabla_{v}] (\sqrt{ρ M_{ϕ_{H}}})}{\sqrt{M e^{- ϕ_{H}}}} \in E_{#}^{⊥} . \end{matrix}$

The following lemma characterizes the orthogonal of $E_{#}$ . Lemma 6.2 ([14]).

Let $T \in D_{#}^{'}$ . Then, T belongs to $E_{#}^{⊥}$ if and only if there exists $S \in D_{#}^{'}$ such that $\nabla_{v} S = 0$ and $T = v \cdot \nabla_{y} S$ .

According to this lemma, there exists $S \in D_{#}^{'}$ satisfying $\nabla_{v} S = 0$ and $\begin{matrix} \frac{L_{ϕ_{H}} (\tilde{ξ} M e^{- ϕ_{H} / 2})}{M e^{- ϕ_{H}}} + \frac{[v \cdot \nabla_{x} - \nabla_{x} (ϕ_{H} + ϕ_{P}) \cdot \nabla_{v}] (\sqrt{ρ M_{ϕ_{H}}})}{\sqrt{M e^{- ϕ_{H}}}} = v \cdot \nabla_{y} S . \end{matrix}$ Multiplying this equation by $M e^{- ϕ_{H}}$ , combined with (30), implies $\begin{matrix} L_{ϕ_{H}} (\tilde{ξ} M e^{- ϕ_{H} / 2} - M e^{- ϕ_{H}} S + (\nabla_{x} \sqrt{ρ} + \frac{1}{2} \nabla_{x} (ϕ_{e} + ϕ_{P}) \sqrt{ρ}) χ e^{ϕ_{e} / 2}) = 0 \end{matrix}$ and then $\begin{array}{rcl} (47) & \tilde{ξ} M e^{- ϕ_{H} / 2} & = & M e^{- ϕ_{H}} S - (\nabla_{x} \sqrt{ρ} + \frac{1}{2} \nabla_{x} (ϕ_{e} + ϕ_{P}) \sqrt{ρ}) χ e^{ϕ_{e} / 2} . \end{array}$ Let us go back to the expression of the current density. Using (47), we obtain $\begin{array}{rcl} J & = & 2 \sqrt{\tilde{ρ}} \int_{Y} \int_{R^{d}} \tilde{ξ} v M e^{- ϕ_{H} / 2} d y d v \\ = & 2 \sqrt{ρ e^{ϕ_{e}}} \int_{Y} \int_{R^{d}} [v M e^{- ϕ_{H}} S - v (\nabla_{x} \sqrt{ρ} + \frac{1}{2} \nabla_{x} (ϕ_{e} + ϕ_{P}) \sqrt{ρ}) χ e^{ϕ_{e} / 2}] \\ = & - 2 \sqrt{ρ} (\int_{Y} \int_{R^{d}} v \otimes χ) \cdot [\nabla_{x} \sqrt{ρ} + \frac{1}{2} \nabla_{x} (ϕ_{e} + ϕ_{P}) \sqrt{ρ}] . \end{array}$ Now, we would like to explain how we can pass to the limit in the boundary condition and how we can rewrite the current J by using the following lemma: Lemma 6.3.

Let ω be a regular and bounded open subset of $R^{d}$ and ρ be a positive function of $L^{\infty} (0, T; L^{1} (ω))$ satisfying $\begin{matrix} \{\begin{matrix} \nabla_{x} \sqrt{ρ} + \frac{1}{2} \nabla_{x} (ϕ_{P} + ϕ_{e}) \sqrt{ρ} = G \in L^{2} (0, T; L^{2} (ω)), \\ - Δ_{x} ϕ_{P} = ρ, \\ \nabla_{x} ϕ_{P} \in L^{\infty} (0, T; L^{2} (ω)), \\ ρ = ρ_{b} on \partial ω . \end{matrix} \end{matrix}$ Then, $\begin{matrix} ρ \in L^{2} (0, T; L^{2} (ω)), \sqrt{ρ} \in L^{2} (0, T; H^{1} (ω)) and \nabla_{x} (ϕ_{P} + ϕ_{e}) \sqrt{ρ} \in L^{2} (0, T; L^{2} (ω)) . \end{matrix}$

Proof.

The proof of this lemma can be found in [16]. □

Finally, using the previous lemma, we can see easily that we can rewrite the current $\begin{array}{rcl} J & = & - 2 \sqrt{ρ} D (x) \cdot (\nabla_{x} \sqrt{ρ} + \frac{1}{2} \sqrt{ρ} \nabla_{x} (ϕ_{P} + ϕ_{e})) \\ = & - D (x) (\nabla_{x} ρ + ρ \nabla_{x} (ϕ_{P} + ϕ_{e})) . \end{array}$ This ends the proof of Theorem 1.2. □

References

Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), 1482–1518. doi:10.1137/0523084.

Ben Abdallah and

M.L.

Tayeb, Diffusion limit for the one dimensional Boltzmann–Poisson system, Discrete Contin. Dyn. Syst. Ser. B 4 (2004), 1129–1142.

Ben Abdallah and

M.L.

Tayeb, Diffusion approximation and homogenization of the semiconductor Boltzmann equation, Multiscale Model Simul. 4 (2005), 896–914. doi:10.1137/040611227.

J.-F.

Coulombel,

Golse and

Goudon, Diffusion approximation and entropy-based moment closure for kinetic equations, Asymptot. Anal. 45 (2005), 1–39.

Degond and

Zhang, Diffusion approximation of a scattering matrix model of a semiconductor superlattice, SIAM J. Appl. Math. 63 (2002), 279–298. doi:10.1137/S0036139999360015.

DiPerna and

P.L.

Lions, Solutions globales d’équations du type Vlasov–Poisson, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 655–658.

Dolbeault, Free energy and solutions of the Vlasov–Poisson–Fokker–Planck system: External potential and confinement (large time behavior and steady states), J. Math. Pures Appl. 78 (1999), 121–157. doi:10.1016/S0021-7824(01)80006-4.

El Ghani, Diffusion limit for the Vlasov–Maxwell–Fokker–Planck system, IAENG Int. J. Appl. Math. 40 (2010), 159–166.

El Ghani and

Masmoudi, Diffusion limit for the Vlasov–Poisson–Fokker–Planck system, Comm. Math. Sci. 8 (2010), 463–479. doi:10.4310/CMS.2010.v8.n2.a9.

10.

Frenod,

P.A.

Raviart and

Sonnendrücker, Two scale expansion of a singularly perturbed convection equation, J. Math. Pures Appl. 80 (2001), 815–843. doi:10.1016/S0021-7824(01)01215-6.

11.

Golse,

P.-L.

Lions,

Perthame and

Sentis, Regularity of the moments of solution of a transport equation, J. Funct. Anal. 76 (1988), 110–125. doi:10.1016/0022-1236(88)90051-1.

12.

Goudon, Hydrodynamic limit for the Vlasov–Poisson–Fokker–Planck system: Analysis of the two-dimentional case, Math. Models Methods Appl. Sci. 15 (2005), 737–752. doi:10.1142/S021820250500056X.

13.

Goudon and

Mellet, Homogenization and diffusion asymptotics of the linear Boltzmann, ESAIM Control Optim. Calc. 9 (2003), 371–398. doi:10.1051/cocv:2003018.

14.

Goudon and

Poupaud, Approximation by homogenization and diffusion of kinetic equations, Commun. Partial Differ. Equ. 26 (2001), 537–569. doi:10.1081/PDE-100002237.

15.

Herda, On massless electron limit for a multispecies kinetic system with external magnetic field, J. Differential Equations 260 (2016), 7861–7891. doi:10.1016/j.jde.2016.02.005.

16.

Masmoudi and

M.L.

Tayeb, Diffusion limit of a semiconductor Boltzmann–Poisson system, SIAM J. Math. Anal. 36 (2007), 1788–1807. doi:10.1137/050630763.

17.

Masmoudi and

M.L.

Tayeb, Diffusion and homogenization approximation for semiconductor Boltzmann–Poisson system, J. Hyp. Diff. Equ. 5 (2008), 1–20. doi:10.1142/S0219891608001465.

18.

Masmoudi and

M.L.

Tayeb, Homogenization and hydrodynamic limit for Fermi–Dirac statistics coupled to a Poisson equation, Commun. Pure Appl. Math. 68 (2015), 1742–1773. doi:10.1002/cpa.21593.

19.

Masmoudi,

M.L.

Tayeb and

Tlili, Homogenization of a spherical harmonics expansion model, Math. Models Methods Appl. Sci. 28 (2018), 453–485. doi:10.1142/S0218202518500124.

20.

Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), 608–623. doi:10.1137/0520043.

21.

Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asymptot. Anal. 4 (1991), 293–317. doi:10.3233/ASY-1991-4402.

22.

Poupaud and

Soler, Parabolic limit and stability of the Vlasov–Fokker–Planck system, Math. Models Methods Appl. Sci. 10 (2000), 1027–1045. doi:10.1142/S0218202500000525.

23.

Sentis, Approximation and homogenization of transport process, SIAM J. Appl. Math. 93 (1980), 134–141. doi:10.1137/0139011.

24.

M.L.

Tayeb, Homogenized diffusion limit of a Vlasov–Poisson–Fokker–Planck model, Ann. Henri Poincaré 17 (2016), 2529–2553. doi:10.1007/s00023-016-0484-7.

25.

Weinan, Homogenization of linear and nonlinear transport equations, Commun. Pure Appl. Math. 45 (1992), 301–326. doi:10.1002/cpa.3160450304.

26.

Wu,

T.C.

Lin and

Liu, Diffusion limit of kinetic equations for multiple species charged particles, Arch. Rational Mech. Anal. 215 (2015), 419–441. doi:10.1007/s00205-014-0784-3.