Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales

We derive quantitative error estimates for coupled reaction-diffusion systems, whose coefficient functions are quasi-periodically oscillating modeling the microstructure of the underlying macroscopic domain. The coupling arises via nonlinear reaction terms and we allow for different diffusion length scales, i.e. whereas some species have characteristic diffusion length of order 1 other species may diffuse with the order of the characteristic microstructure-length scale. We consider an effective system, which is rigorously obtained via two-scale convergence, and we derive quantitative error estimates.


Introduction
Many mathematical models arising from biological, physical or engineering problems involve effects on microscopic scales, e.g. spatial inhomogeneities of the underlying material. In view of numerical simulations as well as more profound structural insight, we are interested in finding effective, or homogenized, models. From the analytical perspective, we ask for a rigorous justification of the effective model and, if available, error estimates describing the difference to the original macroscopic model.
We refer to the books [1,2,3,4] for a general survey of homogenization theory. An important step in the theory of periodic homogenization was the introduction of two-scale convergence in [5,6], which allows to rigorously treat systems involving different diffusion length scales, see e.g. [7,8]. So far, the notion of two-scale convergence is a weak convergence. The periodic unfolding technique, introduced in [9], allows for a natural definition of strong two-scale convergence and, hence, the treatment of nonlinear problems, cf. [10,11,12,13,14,15,16]. Based on this strong notion of convergence, one can ask for quantitative error estimates, see e.g. [17,18,19,20,21], as well as for numerical simulations, see e.g. [22,23,24,25,26,27]. For applications of periodic homogenization in physics and engineering, we refer to e.g. [28,29,30,31] for systems of reactiondiffusion type in heterogeneous media as well as to e.g. [32,33] for two-scale models on the evolution of damage.
The objective of this contribution are coupled reaction-diffusion systems of the following type in Ω (1.1) supplemented with homogeneous Neumann boundary conditions and initial conditions. Here, (u ε , v ε ) : [0, T ] × Ω → R m 1 +m 2 denote the concentrations of m 1 "classically" diffusing species with characteristic diffusion length of order O(1) and m 2 slowly diffusing species of order O(ε). Moreover, D i : Ω × Y → R (m i ×d)×(m i ×d) denotes the diffusion coefficients and F i : Ω × Y × R m 1 +m 2 → R m i the nonlinear reaction terms and both, D i and F i , are assumed to be periodic in y = x/ε w.r.t. a prescribed microstructure. It was shown in [34] that the solutions (u ε , v ε ) converge for ε → 0 to a limit (u, V ) that decomposes into a one-scale function u(t, x) and a two-scale function V (t, x, y), which solve the effective system u t = div(D eff (x)∇u) + − Y F 1 (x, y, u(x), V (x, y)) dy in Ω, V t = div y (D 2 (x, y)∇ y V ) + F 2 (x, y, u, V ) in Ω × Y. (1.2) In order to install the limit passage (1.1) → (1.2), we employ the technique of two-scale convergence via periodic unfolding, cf. (2.6). This involves the periodic unfolding operator T ε : L 1 (Ω) → L 1 (Ω × Y), the folding operator F ε : L 1 (Ω × Y) → L 1 (Ω) and the gradient folding operators G 0 ε respective G 1 ε , cf. Section 2.1. With this method, the strong two-scale convergence of the slowly diffusing species v ε , i.e. max 0≤t≤T T ε v ε (t)−V (t) L 2 (Ω×Y) → 0, was proved in [34], cf. Section 3.1, whereas the strong convergence u ε → u follows immediately from the compact embedding H 1 (Ω) ⊂ L 2 (Ω). This result was obtained under the assumption of L ∞ -regularity of the coefficients and global Lipschitz continuity of the reaction terms, cf. (3.6.A1)-(3.6.A4). One major analytical difficulty to overcome is the periodicity defect [17] or T ε -property of recovered periodicity [34], i.e.
The aim of this paper is to derive in Theorem 3.2 the error estimate In the interior of the domain Ω, the convergence rate in (1.4) can be improved to ε 1/2 , see Theorem 3.3. We assume additional spatial regularity w.r.t. the macroscopic scale x ∈ Ω of the given data (3.6.A5), i.e. ∇ x D i , ∇ x F i ∈ L ∞ (Ω × Y), and the effective solution (u, V ) (3.6.A6), i.e. u ∈ H 2 (Ω), V ∈ H 1 (Ω; H 1 (Y)). We assume neither additional spatial regularity of the original solutions (u ε , v ε ) nor of the corrector functions. In [20], a reaction-diffusion system predicting concrete corrosion is considered, but the system does not include slowly diffusing species v ε . Nevertheless, for the classically diffusing species u ε and its gradient ∇u ε the convergence rate ε 1/2 and ε 1/4 , respectively, is rigorously proved by the method of periodic unfolding. For systems involving slowly diffusing species v ε , convergence rates of order ε 1/2 are derived in [24,21] via the method of asymptotic expansion assuming continuous given data and limit solutions.
The distinctive feature of this contribution is the nonlinear coupling of the classically and slowly diffusing species combined with the periodic unfolding method, which allows to avoid any assumption of spatial continuity. Our proof to (1.4), in the first part, follows along the lines of [34] and we derive the Gronwall-type estimate unfolding operators, see the lemmas 3.5, 3.6, and 3.7 in Section 3.3, which heavily rely on the improved regularity w.r.t. x ∈ Ω and ideas from [17]. Moreover, we use a quantification result for the periodicity defect (1.3.PD) from [18], see Lemma 3.8.
The structure of the paper is as follows: In Section 2, we introduce basic notations, definitions, and results concerning periodic unfolding (Sec. 2.1) and two-scale convergence (Sec. 2.2). In Section 3, we consider the coupled systems (1.1)-(1.2) and derive the error estimate (1.4). Therefore, we list our assumptions and recall the existing convergence result (Sec. 3.1), state our Main Theorem (Thm. 3.2 & 3.3), explain the structure of its proof (Sec. 3.2), and we derive preparatory error estimates (Sec. 3.3). Finally, we give the proof of Theorem 3.2 (Sec. 3.4) and we discuss the obtained results (Sec. 3.5).

Two-scale convergence
Here, and throughout this paper, x denotes the macroscopic variable and the microscopic variable y captures periodic oscillations in x/ε. In order to describe the convergence from (1.1) to (1.2), we introduce the concept of two-scale convergence, which is designed for problems with underlying periodic microstructure. The definition of two-scale convergence (2.6), introduced in Section 2.2, is based on the periodic unfolding technique, described in Section 2.1, and with this it reduces to the notion of classical weak and strong convergence in the two-scale space L 2 (Ω × Y).
2.1. Periodic unfolding, folding, and gradient folding operators Throughout this paper, let Ω ⊂ R d be a bounded domain with Lipschitz boundary ∂Ω. Following [9,35,13], Y = [0, 1) d denotes the unit cell so that R d is the disjoint union of translated cells λ + Y , where λ ∈ Z d . Identifying opposite faces of Y gives the periodicity cell Y, i.e. the torus But, in notation, we will not distinguish between elements of the unit cell y ∈ Y and the ones of the periodicity cell y ∈ Y. Using the mappings x ∈ R d . Then, we can identify every periodic function f with a function f on Y. Introducing the small length scale parameter ε > 0, we define the sets With this definition of the subset Ω ε ⊂ Ω, we sort out microscopic cells ε[x/ε] Y +Y which overlap the boundary ∂Ω. Moreover, we have vol(Ω\ Ω ε ) = O(ε) for those cells which are only partially contained in Ω. Based in these notations, the periodic unfolding operator T ε : L 1 (Ω) → L 1 (Ω×Y) is defined via, cf. [9,35], Moreover, we have the crucial properties, cf. [35], where ω F (ε) = Ω\ Ωε F dx. It holds ω F (ε) → 0 as ε → 0 for all F ∈ L p (Ω) with p > 1, due to vol(Ω\ Ω ε ) → 0. The rate of ω(ε) depends on the norm of the function F . For the reverse operation, we define the folding operator F ε : for all x ∈ Ω ε and (F ε U )(x) = 0 otherwise. Even for smooth functions U : Ω × Y → R the folded function F ε U is only piecewise constant in x, hence ∇(F ε U ) cannot be determined in the classical sense. Therefore, we define the socalled gradient folding operator G 0 ε , respective G 1 ε , which suitably regularizes the folded function F ε U . The definition of the above mentioned gradient folding operator is taken from [ For ε > 0 fixed, the Lax-Milgram lemma yields the existence of a unique weak solution u ε ∈ H 1 (Ω) of (2.4)/(2.5), so that the gradient folding operators are indeed well-defined.

Weak and strong two-scale convergence
We are now in the position to give the definition of weak and strong two-scale convergence following again [9,35,13]. The notion of two-scale convergence was first introduced in [5] and coincides for bounded sequences with Definition (2.6a), here below. For a more detailed comparison of the different definitions see [13,Sec. 2.3].
The unfolding operator T ε is defined for the class of Lebesgue-integrable functions, where boundary values play no role, so that in particular L 2 (Ω × Y) = L 2 (Ω × Y ). In view of the periodicity defect (1.3.PD), we carefully distinguish the spaces H 1 (Y ) and H 1 (Y) = H 1 per (Y ), where the latter one is a closed subspace of H 1 (Y ). For brevity, we set (2.7) implies the existence of a weakly two-scale convergent subsequence. However, for given U ∈ X the gradient folding operator guarantees even strong two-scale convergence. So, (G 1 ε U ) ε ⊂ X recovers any function U ∈ X via strong two-scale convergence and it is shown in [16,Prop. 2.11] that

Convenient commutation relations, such as
Instead, we have that the different folding operators are comparable in the sense that their difference vanishes, see [34,Prop. 3.9], (2.8)

Error estimates for reaction-diffusion systems
We consider a system of two coupled reaction-diffusion systems, where the coupling arises via the nonlinear reaction term (f ε 1 , f ε 2 ), whereas the diffusion tensor has block structure We supplement (3.1.P ε ) with homogenous Neumann boundary conditions on ∂Ω and prescribed initial values u ε (0) = u ε 0 respective v ε (0) = v ε 0 . In [34] (see Theorem 3.1 below) it was proven that (u ε , v ε ) converges for ε → 0 to a limit (u, V ) that decomposes into a one-scale function u(t, x) and a two-scale function V (t, x, y) which solve the effective system Here, the effective diffusion tensor D eff and the effective u-reaction f eff only depend on the macroscopic variable x ∈ Ω, while the diffusion tensor D 2 and the V -reaction F 2 depend on the two-scale variables (x, y) ∈ Ω × Y, see (3.6.A1)-(3.6.A2) and (3.3)-(3.5), below. The functionto-function map f eff : The effective diffusion tensor D eff : Ω → R (m 1 ×d)×(m 1 ×d) is given componentwise via the classical homogenization formula, see e.g. [1,6,37], for i, k = 1, ..., m 1 , j, l = 1, ..., d, where the so-called correctors z ij ∈ H 1 av (Y) solve the local problem in the weak sense: (3.5)

Convergence rates for the initial values
(3.6.A7) We obtain the two evolution triples X ⊂ H ⊂ X * and X ⊂ H ⊂ X * . The assumptions (3.6.A1)-(3.6.A4) guarantee the existence of unique weak solutions (u ε , v ε ) of (3.1.P ε ) and (u, V ) of (3.2.P 0 ). Further, the differentiability of the reaction terms and the additional regularity of the initial values (3.6.A4) ensure improved time-regularity of the solutions and the following a priori bounds: there exists C b ≥ 0 independent of ε so that, cf. [  (3.7) Moreover, we have the following convergence result. to the weak solution (u, V ) of (3.2.P 0 ) in the following sense: (3.8b) One may drop the additional assumptions div(D ε 1 ∇u ε 0 ), div(ε 2 D ε 2 ∇v ε 0 ) ∈ H on the initial values, see [38]. Therein, it is shown that any solution with u ε 0 , v ε 0 ∈ H can be approximated by a solution satisfying improved time-regularity as in (3.7).

Main Theorem and outline of the proof
Under the assumption of additional spatial regularity (3.6.A5)-(3.6.A7), we derive the following error estimates for the strong convergences in (3.8). We emphasize that we do not assume improved spatial regularity for the original macroscopic solutions (u ε , v ε ).
Theorem 3.2. Let (u ε , v ε ) and (u, V ) denote the solutions of (3.1.P ε ) and (3.2.P 0 ), respectively, and let the assumptions in (3.6) hold true. Then there exists a constant C ≥ 0 independent of ε such that Moreover, we find the improved convergence rate in the interior of the domain Ω.
Here, we focus on Theorem 3.2 and for the proof of Theorem 3.3, we refer to [38]. Therein, it shown that away from the boundary ∂Ω, the error √ ε of lower order does not need to be considered, cf. Lemma 3.4, and the periodicity defect error is of improved order ε using [17, Thanks to (3.6.A5), we can equally choose where the correctors z ij ∈ H 1 av (Y) solve the local problem (3.5). Since u ∈ H 2 (Ω) by (3.6.A6), we obtain immediately U ∈ H 1 (Ω; H 1 av (Y)) and in particular we do not assume any improved regularity for the correctors z ij . Note, (3.9b) implies the strong two-scale convergence ∇u ε 2s − → ∇u + ∇ y U in L 2 (0, T ; H), which also holds in (3.8b) under the assumptions of Theorem 3.1.

Preparatory error estimates
We recall that Ω is a bounded domain with Lipschitz boundary such that we have in general Ω ε Ω. With this, the treatment of cells ε(λ i + Y ) intersecting the boundary ∂Ω is crucial. Therefore, we begin with a rather classical result for the error on Ω\Ω , where Ω = {x ∈ Ω | dist(x, ∂Ω) > }. The following lemma will be applied to the estimation of the boundary terms ω(ε) in (2.2) by choosing = ε √ d. Lemma 3.4 ([17,18,38]). For u ∈ X and U ∈ H 1 (Ω; L 2 (Y)), it holds for all > 0 where the constant C ≥ 0 only depends on the properties of the domain Ω.
The most important observation in deriving the error estimates (3.9a)-(3.9b) is the quantification of the well-known two-scale property, cf. [13,Prop. 2.4(e)], for every U ∈ L 2 (Ω×Y) exists a sequence (u ε ) ε ⊂ L 2 (Ω) such that u ε 2s − → U in L 2 (Ω × Y). For example, such a sequence is given by u ε = F ε U . More precisely, based in the explicit definitions of T ε and F ε , it holds: Lemma 3.5. For all U ∈ H 1 (Ω; L 2 (Y)), there exists a constant C ≥ 0, only depending on Ω and Y , such that Proof. We use the unfolding criterion (2.2) and we apply the Poincaré-Wirtinger inequality on each cell int(ε(λ i + εY )) ⊂ Ω ε so that Hence, we have the desired estimate.
As a direct consequence of Lemma 3.5, we have, e.g. [17,Eq. (3.4)], for u ∈ X : For possibly discontinuous functions U ∈ H 1 (Ω; L 2 (Y)), the "naive folding" x → U (x, x/ε) is not well-defined. But, in the proof of Lemma 3.7 below, exactly such a "naive folding" is employed. Therefore, we need a suitable regularization U ε of U so that ϑ ε (x) = U ε (x, x/ε) is well-defined and the difference F ε U − ϑ ε H is of order O(ε+ √ ε). Therefore, we use in addition to G 0 ε respective G 1 ε another regularization of the folding operator F ε , namely, the so-called scale-splitting operator Q ε , cf. [9,35,17].
The next Lemma quantifies the convergence (2.8) and relies on Lemma 3.6. It is applied to the estimation of the folding mismatch ∆ u ε 1 respective ∆ v ε 1 . Lemma 3.7. For all (u, U ) ∈ H 1 (Ω) × H 1 (Ω; H 1 av (Y)) respective U ∈ H 1 (Ω; H 1 (Y)), there exists a constant C ≥ 0 such that  (3.19b) and it utilizes the gradient folding operator G 1 ε in the case γ = 1. In the case γ = 0, i.e. (3.19a), we resort to G 0 ε and we only point out the differences afterwards. The case γ = 1 : By an orthogonality argument, cf. [38], we may assume that U (x, y) = w(x)z(y) with w ∈ X and z ∈ H 1 (Y).
We use Lemma 3.8 below to estimate the periodicity defect error ∆ u ε 2 respective ∆ v ε 2 . formal asymptotic expansion is used and then, the convergence rate O(ε 1/2 ) is proved under the assumption of significantly more spatial regularity of the limit solution. In Theorem 3.3, our method reproduces the rate O(ε 1/2 ) as in [24,Thm. 4.5] and [21,Thm. 3.1] under significantly weaker assumptions on the given data and limit solutions.