A mathematical model of stationary charging processes in polar dielectrics: theoretical analysis

We address some global solvability issues for non-linear stationary convection-reaction-diffusion problems. Global solvability of the boundary value problem for the stationary model of a charging process of polar dielectrics under non-equilibrium external conditions is proved. The maximum principle for volume charge density is established.


Introduction
In recent decades, among interdisciplinary researches and applications a special place is occupied by "advection-reaction-diffusion" or "convection-reaction-diffusion" processes. This is due to the increasing role of mathematical modeling in the description of behavior of complex formalized systems as well as wide possibilities of predicting the characteristics of analyzed phenomena using techniques of computational experiment and computer simulation [1]. The deterministic approach leads to a fundamental description of convection-reactiondiffusion processes models in forms of initial-boundary value problems for parabolic partial differential equations in stationary modes and boundary value problems for elliptic equations in the case of stationary states.
"Convection-reaction-diffusion" equations can be characterized by "polyfunctionality" and employed in numerous applied fields for formalization of phenomena and processes of various nature: chemistry, biology, theory of heat and mass transfer, condensed matter physics, hydrodynamics, etc. [2][3][4][5].
This class of mathematical models also includes a drift-diffusion model of charging processes of dielectric materials under the influence of nonequilibrium external conditions. Applying the framework of this scientific area, the "drift" of charge carriers (electrons and holes) is considered, that represents an analogue of convection or advection. One of the most important particular problems is the development of a drift-diffusion approach for modeling the process of charging polar dielectrics induced by an electron irradiation [6]. This problem has been the focus of practical interests due to the need to predict the state of functional dielectrics at diagnostics and modification of their properties with scanning electron microscopy techniques. A wide range of modern studies is devoted to the development of fundamental foundations and mathematical models, the creation mathematical and software to examine charging processes in dielectrics stimulated by electron bunches [7][8][9][10][11].
In a series of papers (for example, [11]), techniques and methods for physical and mathematical modeling of dynamic charging processes in ferroelectric materials are presented.  2 We proposed a modification of the classical model of charging process for ferroelectrics taking into account the radiation-induced conductivity. Mathematically the model is described by a system of equations, including nonlinear time-dependent reaction-drift-diffusion equation to define the spatial distribution of the volume charge density, the local-instantaneous Poisson's equation to calculate the potential distribution and the dependence between a field intensity and potential which are induced by injected charges. Since analytical solutions can be found only for certain classes of such problems, numerical methods play a significant role in the practice of computer simulation [12].
However, as the analysis suggests, if a dielectric sample is being under electron irradiation for a typical exposed time, this time is much longer, than the time of transition of the dynamic system to a stationary state (less than a microsecond). In this terms, a detailed study of mathematical models describing stationary modes becomes relevant for practice of electron microscopy.
It should be noted that a lot of attention was paid to the development mathematical models of charging effects as well as to the implementation of computational algorithms and application the special software. However, in spite of analysis of correctness of models takes a special place in the process of creating a fundamental theory of this complex phenomenon, nowadays results of theoretical studies have not been reported in the literature.
In this study, we carry out a justification of solvability of the stationary convection-reactiondiffusion problem using the mathematical apparatus described in [13]. According to this approach, the boundary value problem is represented as an operator equation. It is further proved that the equation operator is continuous, bounded, monotonous and coercive, which implies the solvability of the operator equation, and therefore of the original problem. This technique has been also applied in [14,15].

Statement of the boundary value problem
The mathematical model of charging processes in polar dielectrics induced by sufficiently long electron irradiation can be described by a boundary value problem for a stationary drift-reactiondiffusion equation. In a bounded domain Ω ⊂ R 3 with a boundary Γ the following boundary value problem is considered: Here ρ is the volume charge density, E is the electric field intensity vector, ϕ is the potential, d is the diffusion coefficient of electrons, µ n is the drift mobility of electrons, ε is the dielectric permittivity, ε 0 is the dielectric constant, f is the generating term responsible for the action of a volume charge source in an object. Applying the operator div to the second equation in (2), taking into account the first equation in (2) we arrive to relation div E = 1 εε 0 |ρ| in Ω.
Below we will refer to the problem (1), (3), (4) for given function f as Problem 1.
In current paper we will prove the global solvability of the Problem 1 and the nonlocal uniqueness of its solution.
Using the Green's formula (7), we prove the following lemma. Lemma 3.2. Let under conditions (i) E ∈ Z, ρ ∈ H 1 0 (Ω) and (4) holds. Then the following relation holds: As h = ρ the relation (10) takes the form: Proof. Using the Green's formula (7), we obtain Taking into account the following relation: div (hE) = ∇h · E + h div E in L 3/2 (Ω) (13) and the equality (4), from (12) we arrive at (10). Setting h = ρ in (10), we obtain (11). Let us multiply the equation (1) by a function h ∈ H 1 0 (Ω) and integrate over Ω using the Green's formula (6). Then we obtain the weak formulation of the Problem 1 The function ρ ∈ H 1 0 (Ω), which satisfies (14), will be called a weak solution of Problem 1. To prove the solvability of problem (14) we will use the following theorem (see [13, p. 182]). Theorem 3.1. Let V is the reflexive separable Banach space. Let the operator A : V → V * has the following properties: 1) the operator A is bounded and semi-continuous, that is, for all u, v, w ∈ V the form (A(u + λv), w) is continuous for λ ∈ R; 2) the operator A is monotonous, that is, Then the mapping A : V → V * is surjective, that is, for any l ∈ V * there exists such u ∈ V that A(u) = l.
The inequality (20) means that the operator A is coercive and condition 3) is met. Therefore, the Theorem 3.1 yields the solvability of the operator equation and, as a consequence, the existence of a solution ρ ∈ T of the problem (14). For this solution, by virtue of the relation (20) the estimate holds Arguing as [15], we prove that the weak solution ρ ∈ H 1 0 (Ω) of Problem 1 is unique. The following theorem holds. Theorem 3.2. Assume that the assumptions (i), (ii) hold. Then there exists a unique weak solution ρ ∈ H 1 0 (Ω) of the Problem 1, and the estimate (21) holds. Using some concepts of [17], let us prove maximum principle for the volume charge density ρ.
Let f max be a positive number and let, in addition to (i), (ii), the following condition holds: (iii) 0 ≤ f ≤ f max a.e. in Ω.
Arguing as [18], we prove the following lemma. Further studies are required to examine inverse problems for the model (1), (3), (4). According to the framework of the optimization approach, inverse problems will be reduced to control problems (see [19][20][21]). For control problems, optimal solutions can be obtained similar to the results reported in studies [21,22].
Additional studies are necessary to demonstrate the possibilities of model applications by means of numerical simulations of stationary charging characteristics obtained for typical polar dielectrics exposed to electron irradiation.