Existence and two-scale convergence of the generalised Poisson–Nernst–Planck problem with non-linear interface conditions

The paper is devoted to the existence and rigorous homogenisation of the generalised Poisson– Nernst–Planck problem describing the transport of charged species in a two-phase domain. By this, inhomogeneous conditions are supposed at the interface between the pore and solid phases. The solution of the doubly non-linear cross-diffusion model is discontinuous and allows a jump across the phase interface. To prove an averaged problem, the two-scale convergence method over periodic cells is applied and formulated simultaneously in the two phases and at the interface. In the limit, we obtain a non-linear system of equations with averaged matrices of the coefﬁcients, which are based on cell problems due to diffusivity, permittivity and interface electric ﬂux. The ﬁrst-order corrector due to the inhomogeneous interface condition is derived as the solution to a non-local problem.


Introduction
In this paper, we study a generalised Poisson-Nernst-Planck (PNP) problem formulated in a twophase domain composed of periodic cells. Non-linear interface conditions are strongly motivated by electro-chemical interfacial reactions, which are of primary importance for electrokinetic applications modelling, for example, electrolyte, Li-Ion batteries and fuel cells. The existence result is stated by means of the fixed point approach. In order to derive an averaged problem with respect to a decreasing cell size, the two-scale convergence method is applied. The PNP problem describes cross-diffusion of multiple charged species, which are expressed in terms 684 V. A. Kovtunenko and A. V. Zubkova of species concentrations and overall electrostatic potential. For the reason of thermodynamic consistency, the concentrations of charged species should satisfy the total mass balance. The consistent generalisation of the multi-component PNP model is made following [14,18] and the authors themselves [21,28] based on consideration of the pressure in the mixture with respect to a flow model (Darcy, Stokes).
From a geometric viewpoint, we consider a two-phase space with a microstructure which describes the solid and pores separated by the interface. In the two-phase domain, the field variables are discontinuous since allowing a jump across the interface. For further homogenisation, we will assume that the geometry can be filled periodically with repeating cells and a thin boundary layer complement to the periodic domain. The homogenisation parameter ε ∈ (0, 1) describes the size of a periodic cell. The cell consists of the unit solid particle surrounded by the unit pore and separated by an interface.
From the viewpoint of partial differential equations, the parabolic-elliptic equations constituting the PNP system are non-linear, coupled, and differ on the two phases. The double non-linearity appears, first, in the diffusion fluxes and, second, in the interface fluxes. In the classic formulation, the L ∞ -estimation of the species concentrations is needed to provide wellposedness of the problem. The solvability of classic PNP systems was studied, for example, in [6,36] based on the Tikhonov-Schauder fixed point theorem, and in [22] Moser's iteration technique was applied. Following a general approach [36], in the present work, we prove wellposedness of the variational formulation of the generalised PNP problem. The uniform bound is provided by the non-negativity and the total mass balance that hold in the generalised formulation. The a priori estimates are obtained involving reasonable assumptions on the diffusivity and permittivity matrices and the boundary fluxes.
For basics of homogenisation techniques, we refer to the two-scale convergence in [1,33], and to formal two-scale asymptotic expansion in [4,37,42]. There is a lot of literature going far beyond linear diffusion, homogeneous Neumann boundary conditions and perforated (one-phase) domains. More complex transport in porous media was considered, for example, in [2,15,38]. For an oscillating third boundary condition, we cite [3]. The two-scale convergence was applied to the classic non-linear PNP equations in [24,34,39], and to steady-state nonlinear Poisson-Boltzmann equations in [16]. Homogenisation of the PNP system in a one-phase perforated domain with the homogeneous Neumann boundary condition and with a jump of the electrical flux was studied in [19]. It is worth remarking that the asymptotic appearance on the one-phase and two-phase domains is different.
In the homogenisation context, very few results are available for inhomogeneous transmission conditions, which are usually assumed to be linear. The works [12,13] were devoted to the homogenisation of stationary one-component linear diffusion equations in the two-phase domain under linear transmission conditions with a continuous flux. The limit in the linear diffusion system with non-linear transmission conditions was obtained in [20] in the sense of the two-scale convergence. The corrector residual estimates were derived in [29,35]. In [5], a thermal transfer was considered in a two-phase domain with an imperfect interface, where both the temperature and the flux are discontinuous across the interface. Coupled multi-component reaction-diffusion systems were treated with respect to non-linear reaction terms over the domain in [10] and examined for degenerate asymptotic behaviour in [32] using the two-scale convergence. Finally, in [11], a non-linear transmission condition was treated with respect to the homogenisation procedure with the help of Minty's argument.  For a small-scale parameter ε ∈ (0, 1), every spatial point x ∈ R d can be decomposed into the floor part x ε ∈ Z d and the fractional part x ε ∈ Y . There exists a bijection C : Z d → N implying a natural ordering, and its inverse is C −1 : N → Z d . Based on (2.1), we can determine a local cell Y l ε with the index l = C x ε , such that x ∈ Y l ε , and x ε ∈ Y are the local coordinates with respect to the cell Y l ε . Let be a domain in R d with the Lipschitz continuous boundary ∂ and the unit normal vector ν, which is outward to . Let I ε := {l ∈ N : Y l ε ⊂ } be the set of indexes of all periodic cells contained in , and ε := int l∈I ε Y l ε be the union of these cells. For every index l ∈ I ε , after rescaling y = x ε , the local coordinate y ∈ ω determines the solid particle such that x ε ∈ ω l ε with the boundary ∂ω l ε . Its complement composes the pore l ε := Y l ε \ ω l ε by analogy with = Y \ ω. Gathering over all local cells, we define the multi-component disconnected domain of periodic particles (the solid phase) denoted by ω ε := l∈I ε ω l ε with the union of boundaries ∂ω ε := l∈I ε ∂ω l ε .
In the homogenisation theory, usually, x refers to as a macro-variable, y as a micro-variable, and (x, y) as the two-scale variables. For fixed ε > 0, the two-phase domain Q ε ∪ ω ε with the external boundary ∂ and the interface ∂ω ε is illustrated in Figure 1.
Arbitrary functions u(y) ∈ H 1 ( ∪ ω) and f (x) ∈ H 1 (Q ε ∪ ω ε ) given with respect to the micro y and macro x variables allow discontinuity across the interfaces ∂ω and ∂ω ε , respectively. In the unit cell Y , we distinguish the negative face ∂ω − as the boundary of the particle ω and the positive face ∂ω + as the opposite part of the boundary of the pore . Similarly, in each local cell Y l ε , we distinguish (∂ω l ε ) − and (∂ω l ε ) + . Gathering over all local cells establishes the positive and negative faces of the interface as ∂ω ± ε = l∈I ε (∂ω l ε ) ± . We set the interface jump of u across ∂ω and of f across ∂ω ε by Existence and two-scale convergence of generalised PNP problem where the corresponding traces of u at ∂ω ± and f at ∂ω ± ε are well defined, see [23,Section 1.4].

Problem formulation
In the two-phase domain Q ε ∪ ω ε , we consider the number n 2 of charged species with specific charges z i and unknown concentrations c ε i , i = 1, . . . , n, together with the overall electrostatic potential ϕ ε . They solve the system of Poisson and Nernst-Planck equations for i = 1, . . . , n: where the indicator function 1 Q ε is equal to 1 in Q ε ,and 0 in ω ε . Here, the Nernst-Planck implies drift-diffusion equations in Q ε , while linear diffusion together with simple Ohm's law are suggested in ω ε (see [14,18,28] and references therein for the modelling aspects in solid electrolyte). The mixed type non-linear interface conditions are stated on ∂ω ε : for fixed parameters α > 0, κ > 0, γ 0. The scaling of g i with ε as γ = 0 in (2.3c) is natural since it just compensates the interface length |∂ω ε | = O(ε −1 ). Then, the uniform a priori estimate (see (2.23)) forces conditions γ 0 and κ 0 in the scaling of the non-linearity in (2.3a). Whereas κ > 0 is assumed for the averaging procedure (see (3.9)). The factor 1/ε in (2.3d) will be explained later in (2.14). Below we explain the constitutive relations (2.3) in more details. The non-linear convection terms ϒ 0 and ϒ j , j = 1, . . . , n, and given by where k B and N A are the Boltzmann and Avogadro constants, is the temperature, and the notation c + k := max(0, c k ) was used. The expressions (2.4) imply a generalisation of the diffusion fluxes J ε i that preserves the non-negativity c ε 0 and the total mass balance n k=1 c ε k = C > 0 following the approach of [14,36].
In (2.3d), the periodic at the interface ∂ω ε function is set g ε (t, x) := (U ε g)(t, x) = g t, x ε . Here, g ∈ L ∞ (0, T; L 2 (∂ω)) denotes the electric flux through the interface in the unit cell. The family of matrices A ε (x) : . . , n, in (2.3a) is determined in and periodic in ε . The averaging operator U ε is introduced in Appendix A.
In the two-phase unit cell ∪ ω, we employ the d-by-d matrix of permittivity A and the twoparameter family of d-by-d diffusivity matrices D ij , i, j = 1, . . . , n, which satisfy the following assumptions. • A(y) ∈ L ∞ ( ∪ ω) d×d is uniformly bounded and symmetric positive definite (spd): there exist 0 < a ā such that a|ξ | 2 ξ A(y) ξ ā|ξ | 2 for ξ ∈ R d ; (2.5) • D ij (y) ∈ L ∞ ( ∪ ω) d×d are uniformly bounded and elliptic: • as ε → 0 the asymptotic expansion of the diffusivity matrices holds: stands for the pair of traces at the phase interface ∂ω ε . We assume that the functions (ĉ ε ,φ ε ) → g i , R 2n × R 2 → R, i = 1, . . . , n, describing the interface fluxes of species, are strong-to-strong continuous in L 2topology (e.g., Lipschitz-continuous), and satisfy balance of the mass: Motivated by bounded statistics, an example verifying assumptions (2.9) (see [28]) is Moreover, it satisfies the positive production rate condition (see [36]): By multiplying (2.3a), (2.3b) with smooth test functions and integrating by parts due to (2.3c), (2.3d), we set a variational formulation of the inhomogeneous PNP problem in the two-phase domain. Given final time T > 0, find discontinuous over interface functions satisfying the following variational equations for i = 1, . . . , n:

10a)
Existence and two-scale convergence of generalised PNP problem In fact, since ϒ j and g i in (2.10a) are uniformly bounded, the terms build a linear and continuous functional and The system of parabolic (2.10a) and elliptic (2.10b) equations is supported by the standard initial and Dirichlet boundary conditions: The given functions Based on the properties (2.5)-(2.7) and (2.9), we prove well-posedness of the generalised PNP problem (2.10).

Well-posedness
In the following, we use the trace theorem for functions f ∈ H 1 (Q ε ∪ ω ε ) with K 0 > 0: (2.13) and the Poincaré inequalities that hold when f = 0 on ∂ (see [16]): 14) It is worth noting that the discontinuous Poincaré inequality (the second line in (2.14)) prescribes the scaling of the interface term in the left-hand side of the Poisson equation (2.10b) and justifies the equivalent to L ∞ (0, T; H 1 (Q ε ∪ ω ε )) norm in (2.16b).

Theorem 2.1 (Well-posedness)
A solution (c ε , ϕ ε ) ∈ W to the inhomogeneous PNP problem (2.10) exists and satisfies the total mass balance

V. A. Kovtunenko and A. V. Zubkova
The following a priori estimates hold in the norm of W with K c , K φ > 0: Proof To prove the assertion, we apply the Schauder-Tikhonov fixed point theorem [40].
Starting with a smooth initialisation c for m > m 0 , m ∈ N, we find the solution (c m , ϕ m ) ∈ W, which satisfy the initial and Dirichlet boundary conditions (2.12) and the linearised equations: for all test functions (c,φ) ∈ W such thatc =φ = 0 on ∂ . In (2.17b), the notation g m−1 We show that M is continuous and has compact image. We start with uniform a priori estimates.
Estimation for ϕ m . Let us choose in (2.17a) the test functionφ =φ m := ϕ m − ϕ D , which is zero at (0, T) × ∂ due to the Dirichlet boundary condition, and rearrange the terms using ϕ D = 0 on ∂ω ε such that: Applying Young's inequality with a weight δ > 0, we obtain the following upper bounds of the terms in the right-hand side of (2.18). First, estimating from above ϒ 0 (c m−1 ) CZ in (2.4), where Z := n k=1 |z k | > 0, and using the Poincaré inequality from (2.14), we get Third, using the uniform boundedness |g ε | |g| and |∂ω ε | = O(ε −1 ), this gets Summarising the three above estimates of the right-hand side of (2.18), we infer the asymptotic relation: On the left-hand side of the equation (2.18), using the spd-property of the matrix A ε in (2.5), the term I m ϕ is estimated from below: Gathering together (2.19) and (2.20), for δ chosen sufficiently small such that δ < min{a, α}, it follows the uniform with respect to ε estimate for all m: Applying to the differenceφ m = ϕ m − ϕ D , the triangle inequality, using ϕ D = 0 on ∂ω ε , and taking the supremum over t ∈ (0, T), from (2.21), it follows the uniform in m and ε estimate: where the terms in the right-hand side of (2.23) are: We estimate these four integrals using Young's inequality with a weight δ > 0. Applying the trace theorem (2.13) and the growth condition (2.9b), the integral I m 1 is estimated as Similarly, using the uniform estimate ϒ j (c m−1 ) Thus, I m c in (2.23) is estimated from above as follows with 0 < K 5 = O 1 δ : |I m l | δ c m 2 L 2 (0,τ ;L 2 (Q ε ∪ω ε )) n + ∇c m 2 L 2 (0,τ ;L 2 (Q ε ∪ω ε )) d×n + K 5 . (2.24) Due to the compatibility conditions, we havec m i (0) = 0 and integrate by parts: In view of the uniform ellipticity of D ij ε in (2.6), we estimate I m c in (2.23) from below and combine it with the upper bound (2.24) to obtain: For δ < d, applying the Grönwall inequality leads to the estimate c m (τ ) 2 2δτ e 2δτ ). Therefore, taking in (2.25) the supremum over τ ∈ (0, T), we get c m 2 L ∞ (0,T;L 2 (Q ε ∪ω ε )) n + ∇c m 2 L 2 (0,T;L 2 (Q ε ∪ω ε )) d×n K 6 , K 6 > 0. (2.28) We show that M is continuous with respect to the two non-linear terms in (2.17).
The first non-linearity occurs at the interface. Due to the continuous dependence of g i onĉ and ϕ, it holds the limit g m i → g i (ĉ ε ,φ ε ) provided by the componentwise convergenceĉ m →ĉ ε and ϕ m →φ ε in the strong topology of L 2 (∂ω ε ) as m → ∞.
The second non-linearity is due to non-linear terms ϒ 0 and (ϒ 1 , . . . , ϒ n ). To prove their continuity, it needs to establish the total mass balance for c m and c ε . For this task, we sum up the equations (2.17b) over i = 1, . . . , n skipping the trivial terms n j=1 ϒ j (c m ) = 0 according to (2.4) and n i=1 g m i = 0 due to the assumption (2.9a).
We estimate σ m analogously to (2.25) with K = 0, from which it follows σ m ≡ 0 and Passing m → ∞ in virtue of (2.28), the total mass balance (2.15) holds for the limit function c ε from (2.28).
For any c such that n k=1 c k = C, hence C n k=1 c + k 1, we will show that ϒ 0 (c) and ϒ j (c), j = 1, . . . , n, are Lipschitz continuous. Due to (2.15) and (2.29), we can take c m and c ε from (2.28) as the argument for ϒ 0 and estimate the difference:

V. A. Kovtunenko and A. V. Zubkova
Therefore, the Lipschitz continuity of ϒ j justifies the limit in the non-linear term in (2.17b). Applying the Cauchy-Schwartz inequality, the convergences (2.28), the compact embedding H 1 (Q ε ) → L 2 (Q ε ), and the boundedness of ϒ j , it follows that with smooth test functionsc i ∈ C ∞ (Q ε ) and K > 0. The limit in ϒ 0 in (2.17a) is analogous. Since We note that the non-negativity c ε 0 under the positive production rate assumption (2.9c) on g i and for a stronger than (2.7) decoupling assumption D ij = δ ijD on the diffusivity matrices D ij is proved in [26,27].

Homogenisation procedure
For homogenisation of the PNP problem (2.10), we start with auxiliary cell problems, which are due to periodic matrices of permittivity and diffusivity and periodic electric flux at the interface. A two-scale convergence to an averaged PNP problem is established. After that we proceed with corollaries and state the corrector term due to the non-periodic interface reactions, which refines the two-scale convergence. Respective homogenisation tools that we employ are technical and deduced separately in the Appendix.

Auxiliary cell problems
Later on we will use the following auxiliary cell problems in the space of periodic through ∂Y , discontinuous across ∂ω functions for all test functions u ∈ H 1 # ( ∪ ω). In (3.1), the notation ∂ y N(y) ∈ R d×d for y ∈ ∪ ω stands for the matrix of derivatives with entries (∂ y N) ij = ∂N i ∂y j for i, j = 1, . . . , d, and I ∈ R d×d is the identity matrix. A solution of (3.1) exists, and it is defined up to a piecewise constant in ∪ ω, see [42,Chapter 1.2]. Moreover, since the strict inclusion ω ⊂ Y is assumed, this fact follows that N = −y and ∂ y N = −I in ω.
• The cell problem due to the periodic permittivity matrix: find a vector function = ( 1 , . . . , d )(y) ∈ (H 1 # ( ∪ ω)) d such that ∪ω (I + ∂ y )A∇ y u dy + ∂ω α y u y dS y = 0, (3.2) for all u ∈ H 1 # ( ∪ ω). Compared to (3.1), the integral over the interface ∂ω appears in (3.2) due to the interface term in (2.10b). Similarly, a solution exists, and it is defined up to a constant in the cell Y .
It is worth noting that, by the standard variational analysis, from (3.4a) and (3.4b), we derive the strong formulation of the limit diffusion problem in (0, T) × : Proof For the proof, we apply the homogenisation tools deduced in the Appendix.
Since the a priori estimates (2.16) hold for the solution, the cases (ii) and (iii) in Lemma A.5 from Appendix A provide existence of functions yielding the two-scale limit according to the convergences (A.12) and (A.14): The case (iiia) of Lemma A.5 ensures the Dirichlet condition following from (2.12): In addition, we will use the auxiliary asymptotic results below. For periodic functions U ε g := g ε , since T ε g ε = (T ε U ε )g = g converges to itself in L 2 ( ) × L 2 (∂ω), according to Lemma A.3 of the two-scale convergence this implies: We apply Lemma A.5 to pass to the limit in the inhomogeneous problem (2.10). First, we consider the equation (2.10b). For arbitrary v ∈ H 1 0 ( ) in the domain, we takeφ = v as a test function in (2.10b) such that v = 0 on ∂ω ε and the boundary terms disappear: (3.10) Existence and two-scale convergence of generalised PNP problem 697 Adding and subtracting ϒ 0 (c 0 ) yield the decomposition Since ϒ 0 (c 0 )v ∈ H 1 0 ( ), its integral over the pore part Q ε can be rewritten over with the help of the asymptotic formula from [16, Lemma 2]: Provided by the total mass balance (2.15) and the non-negativity assumption c ε 0, the function c ε → ϒ 0 in (2.4) is linear. Henceforth, from the composition rule (A.3c) of the unfolding operator T ε (see Definition A.1) implying T ε ϒ 0 (c ε ) = ϒ 0 (T ε c ε ) and the convergence (3.7), it follows and together with (3.11), Since T ε A ε = A in ε due to the periodicity of A as stated in (A.3b) and | \ ε | → 0, based on cases (ib) and (iiib) in Lemma A.4 with q = v, we get the limit in (3.10): In order to represent ϕ 1 in (3.13), we employ the cell problems (3.2) and (3.3) as follows. For arbitrary w(y) ∈ H 1 # ( ∪ ω) in the cell, we can take in (2.10b) another test functionφ = εη ε U ε w with a cut-off function η ε supported in ε and equals one outside an ε-neighborhood of ∂ ε (see [4, Chapter 1, Section 5]). Indeed, the periodicity of w on ∂Y guarantees continuity of U εφ across the local cells, then (2.10b) turns into where η ε = 1 at the interface ∂ω ε . Passing here to the limit as ε → 0 due to (A.7) and (A.8) in cases (ii) and (iiia) of Lemma A.4, by virtue of ε Q ε κϒ 0 (c ε )U ε w dx → 0 due to (3.12), this leads to the following variational equality: We take the test function u = w in the cell problems Then, subtracting the result from the equation (3.14), after gathering the same terms (∇ϕ 0 ) A∇ y w and g w y were shortened and this gives: where we used the identity (∇ϕ 0 ) ∂ y = ∇ y [(∇ϕ 0 ) ] . The linear homogeneous equation (3.15) has a solution ψ(t, x, y) ). We substitute w = ψ into (3.15) and due to (2.5) get the lower bound 1 |Y | a ∪ω |∇ y ψ| 2 dy + α ∂ω ψ 2 y dS y dx 0.
Therefore, ψ(t, x) is independent on y, which implies the representation of the gradient and of the jump of ϕ 1 with respect to y as Now, we substitute the expressions (3.16) into the limit equation (3.13) and obtain Moving the terms independent on y outside the integral over ∪ ω in (3.17) and introducing the notation of the averaged matrix A 0 ∈ R d×d and vector G 0 ∈ R d by we rewrite (3.17) in the form (3.19) which implies a homogeneous problem for the averaged solution ϕ 0 ∈ L ∞ (0, T; H 1 ( )) supported by the Dirichlet boundary condition (3.7c). We cite [1, Section 3] for the homogenisation procedure of non-linear operators using a two-scale convergence. Second, we proceed with the equation (2.10a). For functions v i (t, x) ∈ H 1 (0, T; L 2 ( )) ∩ L 2 (0, T; H 1 0 ( )) in the domain such that v i = 0, i = 1, . . . , n, we test the equations (2.10a) withc i = v i using the expansion (3.9) and the asymptotic decoupling assumption (2.8), then pass to the limit as ε → 0 analogously to (3.13) to get where we understand the time derivative in the weak sense similarly to (2.11).
Due to the convergences (3.7a), (3.7b), and the fact that the time derivative, non-linear and the boundary terms vanish, similarly to (3.14) gives us the limit equation as ε → 0: We multiply the cell problem (3.1) with (∇c i ) |Y | and take the test function u = w i : Integrating it over (0, T) and and then subtracting from the equation (3.21), after gathering the same terms and shortening (∇c 0 i ) D∇ y w i , we get From the homogeneous equation (3.22), it follows that The matrix D 0 ∈ R d×d of the averaged diffusivity is defined by:  We integrate by parts with respect to time the first term in (3.26) and take into account the initial condition σ 0 = 0 at t = 0 since n i=1 c in i = C is assumed, such that σ 0 (T) 2 dx. Taking into account the ellipticity condition for the averaged matrix D 0 , this gives the lower estimate: from which it follows σ 0 2 c := σ 0 2 L ∞ (0,T;L 2 ( )) + σ 0 2 L 2 (0,T;H 1 ( )) = 0, hence σ 0 ≡ 0, and the total mass balance n i=1 c 0 i = C holds. Next, decomposing on ∂ , and derive due to the orthogonality of (c 0 i ) + and (c 0 Henceforth, it follows thus providing that (c 0 i ) − 2 c = 0 and c 0 0 component wisely. This completes the proof. The two following corollaries suggest a next asymptotic term as ε → 0. Corollary 3.3 (Corrector due to interface fluxes) For a weak-to-weak continuous function g i (e.g., linear one), a corrector due to the interface fluxes is given by functions χ ε i ∈ L ∞ (0, T; L 2 (Q ε ∪ ω ε )) ∩ L 2 (0, T; H 1 (Q ε ∪ ω ε )), i = 1, . . . , n, such that satisfying the variational equation for all test functionsc i ∈ H 1 (0, T; L 2 (Q ε ∪ ω ε )) ∩ L 2 (0, T; H 1 (Q ε ∪ ω ε )), where g 0 i := g i ([c 0 , c 0 ], [ϕ 0 , ϕ 0 ]) and D ε = U ε D with the matrix D from (2.8). As ε → 0, the corrector obeys the convergence

31a)
and the two-scale convergences where o(1) expresses the lower-order asymptotic terms due to (2.8).
The two-scale convergences (3.31b) hold after applying the homogenisation result of Theorem 3.1 to the problem (3.30) for χ ε . In doing so, we conclude with a trivial solution of the averaged problem corresponding to (3.30) because of the homogeneous boundary and initial conditions (3.29).

Discussion
The averaged PNP problem (3.4) is coupled and non-linear. However, solving first the system of linear diffusion equations (3.4a) with respect to c 0 and substituting it into (3.4b) give the linear elliptic equation with respect to ϕ 0 . The scaling of non-linearities is crucial. The non-linearity ϒ j (c ε ) in the cross-diffusion equations (2.3a) is scaled with ε κ . The non-linear fluxes g i (ĉ ε ,φ ε ) at the phase interface on the right-hand side in the formula (2.10a) are multiplied by ε 1+γ . The negative values κ < 0 and γ < 0 are not admissible within the uniform a priori estimate of the solution (c ε , ϕ ε ) established in Theorem 2.1. The case κ = 0, which describes strongly non-linear equations, is not allowed within the asymptotic method used for assertion in Theorem 3.1. In this sense, the values κ > 0 and γ 0 taken in the paper are sharp ones.
In order to express an asymptotic order of the convergence (3.5) as ε → 0, it needs to derive residual error estimates. For the larger value of γ = 1, thus avoiding the non-linearity at the interface from the homogenisation procedure, the corrector estimates are derived recently in [29].
We note that the scaling with γ 0 yields no contribution of the interface reaction term in the macroscopic model (3.4a). For a possible remedy, in [25], we investigate the bi-domain setting of a non-linear transmission problem for the linear diffusion equation in connected domains. We denote by θ (x) := ∪ωq dy. The left-hand side of (A.22) builds a linear continuous form, and the right-hand side implies the duality ∇f 0 , θ (C ∞ 0 ( ) * ,C ∞ 0 ( )) for arbitrary θ ∈ L 2 ( ) according to [1,Lemma 2.10].