On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption

Motivated by the applications from chemical engineering, in this paper we present a rigorous derivation of the effective model describing the convection-diffusion-reaction process in a thin domain. The problem is described by a nonlinear elliptic problem with nonlinearity appearing both in the governing equation as well in the boundary condition. Using rigorous analysis in appropriate functional setting, we show that the starting singular problem posed in a two-dimensional region can be approximated with one which is regular, one-dimensional and captures the effects of all physical processes which are relevant for the original problem.


1.
Introduction. Convection-diffusion-reaction problems naturally appear if physical processes in chemical engineering are modeled and, thus, are of great interest both from the theoretical and practical point of view. Depending on the problem we want to model, different types of equations and boundary conditions can be considered. In the present paper, we study a stationary convection-diffusion equation in a thin (or long) channel with nonlinear reaction term concentrated in a narrow (oscillating) strip near one part of the channel wall. On the opposite part of the channel boundary, a nonlinear condition is prescribed modeling the reaction catalyzed by the wall. This type of elliptic boundary value problem describes, for instance, a transport of the solute by convection and diffusion where the solute particles undergo an irreversible chemical reaction on the one part of the boundary 1 and react among themselves in the vicinity of the other one. Our goal is to rigorously derive the effective model described by the one-dimensional boundary-value problem and 580 I. PAŽANIN AND M. C. PEREIRA providing a good approximation of the governing problem when the ratio between channel's thickness and its length is small.
The study of the solute transport problem goes back to the celebrated work of Taylor [20] who first discussed the dispersion of a passive solute in a laminar flow. Extending Taylor's analysis, Aris [4] formally derived the effective equations describing the problem in the absence of the chemical reaction. Rigorous derivation of the asymptotic model for a solute transport in the presence of the first-order (linear) chemical reaction on the channel wall was given in [16]. With the same type of boundary condition, a general model of convection-diffusion with reaction was treated in [1] via homogenization. The effects of the curved geometry and fluids microstructure on solute dispersion in pipe-like domains were investigated in [15,17]. Last but not least, let us also mention some contributions in the engineering literature as [7,19,22].
In the above mentioned papers, the problems under consideration were linear. In the present paper, we deal with a nonlinear elliptic problem with nonlinearity appearing both in the governing equation as well in the boundary condition. Diffusion problems with reaction terms concentrating in the neighborhood of the boundary were successfully addressed in recent papers by the second author of this paper. Combining techniques from geometric theory of parabolic problems, perturbation of linear operators and concentrated integrals, the continuity of the dynamics given by dissipative reaction-diffusion equations posed in a fixed domain have been discussed in [3,18]. Indeed, potential applications including management and control of aquatic ecological systems where one finds localized concentrations in connection with boundary complexity are quoted in [18].
In thin channel and with homogeneous Neumann boundary condition, a nonlinear diffusion elliptic problem has been considered in [8]. It is important to emphasize that here we consider a more general situation. On one hand, we allow convection, diffusion and nonlinear reactions concentrated close to a portion of the boundary. On the other hand, we combine homogeneous and nonlinear Neumann boundary conditions on the domain boundary. Our main result provides a way how to replace a singular nonlinear elliptic problem posed in a two-dimensional region with one which is regular, one-dimensional and captures the effects of all physical processes which took place in the original problem. As far as we know, this is the first attempt to carry out such rigorous analysis and we believe that the result could be instrumental for creating more efficient numerical algorithms for approximating the solution of the convection-diffusion-reaction problems.
2. Formulation of the problem and the statement of the main result. We study the asymptotic behavior of a family of solutions given by the nonlinear elliptic equation with the following boundary conditions D ∂w ∂ν = g(w ) on Γ and ∂w ∂ν = 0 on ∂R \ Γ.
The domain R is a simple thin (or long) channel given by We denote by Γ ⊂ ∂R the lower wall of the channel, namely In the governing equation (2.1), the velocity field is assumed to be incompressible and given. Since we are studying the process in a thin domain, it is reasonable to take the velocity to be unidirectional implying Q = Q (y), due to the incompressibility condition. We set where Q ∈ L ∞ (0, H) is a non-negative function. D > 0 is the molecular diffusion, c is the reaction coefficient, the vector ν = (ν 1 , ν 2 ) is the unit outward normal to ∂R and ∂ ∂ν is the outside normal derivative. Observe that the reaction mechanism on the boundary Γ is assumed to be weak and we model that by taking that the wall absorption parameter is of order O( ) (see first equation from (2.2)). In case of the weak wall absorption, the loss of the solute at Γ is not considerable and, consequently, the effects of the reaction at the boundary remain in the limit problem. 2 Nonlinearities f, g : R → R are supposed to be C 2 -functions with bounded derivatives. From the point of view of investigating the asymptotic behavior of problems such as (2.1)-(2.2), to assume f and g bounded with bounded derivatives does not imply any restriction since we are interested here in solutions uniformly bounded in L ∞ -norms. In fact, under the assumption that the solutions are uniformly bounded in L ∞ , we may perform a cut-off in f and g outside of an appropriate region |u| ≤ R without modifying any of these solutions.
We also assume that G is l(x)-periodic in y for each x ∈ (0, 1), namely G(x, y + l(x)) = G(x, y), for all y, with the period function l positive and uniformly bounded, Clearly the open set θ is a neighborhood for the upper boundary of R whose thickness and oscillatory behavior depend on the positive parameters α and β respectively. Note that α and β set the thickness and oscillating order when goes to zero. Also, if G only depends on the first variable x, then the function G is independent of and the narrow strip θ does not possess oscillatory behavior.
In order to model the concentration of reactions in the small region θ ⊂ R , we will proceed as in [2,5]. We will combine the characteristic function χ θ , the parameter and the nonlinear reaction f by the term Moreover, since R ⊂ (0, 1)×(0, H) is thin and degenerates into the unit interval as goes to zero, it is reasonable to expect that the family of solutions w converges to a solution of a one-dimensional equation capturing the variable profile of the oscillatory behavior of the narrow strip θ as well as the effect of the nonlinear boundary condition.
We will show that the limit problem for (2.1)-(2.2) is the following one: where the constant q and the function µ : (0, 1) → (0, ∞) are given by Notice that the positive constant q is the average of the velocity Q and the nonnegative coefficient µ ∈ L ∞ (0, 1) is related to the oscillating strip θ set by the function G . In view of that, we conclude that the limit problem (2.6) captures all the effects we seek for: the effects of convection, the reactions on the boundary and inside the oscillating strip and also the effect of the geometry of the region where those reactions take place.
In our analysis we combine the approaches from [8,2,11]. More precisely, we apply methods from [11] to deal with the thin channel, and we use the concentrated integrals discussed in [8,2,5,12] in order to obtain µ(x), which is the mean value of G(x, ·) for each x ∈ (0, 1). The coefficient µ captures the oscillatory behavior and the geometry of the narrow strip where the reactions are concentrated. If G does not depend on the second variable y, then the narrow neighborhood does not have oscillatory behavior, and so, µ(x) = G(x) in (0, 1).
In order to study problem (2.1) in the thin domain R , we perform a convenient change of variable leading to the following problem where the function χ o : R 2 → R is the characteristic function of the narrow strip o given by The vector N = (N 1 , N 2 ) is the outward unit normal to ∂Ω and Ω ⊂ R 2 is the set The equivalence between problems (2.1)-(2.2) and (2.8) can be observed by changing the scale of the channel R and the narrow strip θ through the isomorphism (x 1 , x 2 ) → (x 1 , −1 x 2 ) which consists in stretching the x 2 -direction by a factor of −1 . The factor −2 establishes a very fast diffusion in the x 2 -direction. Indeed, we have rescaled the neighborhood θ into the strip o ⊂ Ω and substituted the thin domain R for a domain Ω independent on , at a cost of introducing a very strong diffusion mechanism in the x 2 -direction.
Due to this strong diffusion mechanism it is expected that solutions of (2.8) will become more and more homogeneous in the x 2 -direction when decreases, such that the limit solution will not depend on x 2 and therefore the limit problem will be one dimensional. This is in fully agreement with the intuitive idea that an equation in a thin domain should approach one in a line segment. Now we are in position to state our main result: is a family of solutions of problem (2.8), there exists a subsequence, still defined by u , and a function u ∈ H 1 (Ω) with u(x 1 , x 2 ) = u(x 1 ), solution of the problem (2.6), such that On the other hand, if a solution u of (2.6) is hyperbolic, then there exists a sequence u of solutions of problem (2.8) satisfying Recall that a steady state solution u of a nonlinear differential equation is called hyperbolic if λ = 0 is not an eigenvalue of the linearized problem around u. For instance, if u satisfies equation (2.6) and is hyperbolic, then λ = 0 is not an eigenvalue of the eigenvalue problem Remark 2. Let us denote E = {u ∈ H 1 (Ω) : u is a solution of (2.8)} for each > 0. In view of that, assertions a) and b) from Theorem 2.1 respectively mean upper and lower semicontinuity of the equilibria set of the parabolic problem associated to (2.8) at = 0. In this sense we are proving the continuity of the steady state solutions given by (2.8) at = 0 which reach the limit equation (2.6) as → 0.
Remark 3. Finally, we notice that a standard nonlinear reaction term h(w ) acting in the whole thin domain and defined by a function h can be added to the right hand side of equation (2.1). This term will appear in the first line of the limit equation (2.6) as an extra term in the right hand side. The proof of this statement is exactly the same and we mention it here in order to give a more complete picture of the problem.
3. Basic facts. In this section we state basic results, introducing notations and writing our problem in an abstract setting. We also discuss existence of solutions and how concentrating integrals converge to boundary integrals employing the results from [8,2,5,12]. Throughout this work we denote by H 1 (U ) the Hilbert space set by H 1 (U ) with the equivalent norm where U is an arbitrary open set of R 2 . Note that · H 1 (U ) ≥ · H 1 (U ) wherever ∈ [0, 1]. As we will see, the strong diffusion mechanism in front of the second derivative makes this space a suitable one to deal with the thin domain problems. Consequently, we get ∂u ∂x 2 L 2 (Ω) → 0, as → 0. 3.1. Abstract settings and existence of solutions. In order to write problems (2.6) and (2.8) in an abstract form, we introduce the bilinear forms a : H 1 (Ω) × H 1 (Ω) → R and a 0 : where the constant q is given by (2.7) and the constant H comes from the domain Ω.
It is not difficult to see that a is continuous for all > 0. Moreover, for each > 0, we can define the linear operators A : Hence, we can write the problem (2.8) in the abstract form A u = F (u), for > 0, where F : H 1 (Ω) → H −s (Ω), 1/2 < s < 1, is given by Recall that f and g are C 2 -nonlinearities, bounded with bounded derivatives, and o is the narrow strip defined in (2.9). Here γ : H t (Ω) → L 2 (Γ) is the trace operator with 1/2 < t.
In a similar way we can write the limit problem (2.6) in an abstract form where µ ∈ L ∞ (0, 1) is the coefficient introduced in (2.7).
Remark 5. Under our assumptions, it is known that functions F and F 0 are Fréchet differentiable. The proof can be seen for example in [3, Lemma 3.6 and 3.7].

Remark 6.
Since · H 1 (Ω) ≥ · H 1 (Ω) for all ∈ [0, 1], we also get from Lemma 3.2 that a (u, u) ≥ k u 2 H 1 (Ω) , ∀ ∈ [0, 1] and u ∈ H 1 (Ω). As a consequence of Lemma 3.2 we have that the unbounded operator A is invertible, and then, we have that u ∈ H 1 (Ω) is a solution of (2.8), if and only if satisfies u = A −1 F (u ). That is, u must be a fixed point of the nonlinear map wherever c > Q 2 L ∞ (0,H) /4D. Analogously, we deduce that the solutions of the limit problem (2.6) can be obtained as fixed points of the map Moreover, as a consequence of Theorem 2.1, some solutions of (2.6) also can be obtained as limits of the solutions of the perturbed problem (2.8).
is a bounded set. But, for λ = 1 and any ϕ ∈ O , it follows from Lemma 3.2 and 3.3 (shown in the next section) that where k and C are positive constants independent of ∈ [0, 1], andĈ is the imbedding constant. Therefore, since · H 1 (Ω) ≤ · H 1 (Ω) , wherever ∈ (0, 1), there exists K > 0 such that ϕ H 1 (Ω) ≤ K for all ϕ ∈ O , proving our statement. In a similar way, we can also show the existence of solutions for problem (2.6).

3.2.
Concentrating integrals. Next we colect some results that we need in order to prove the main result.
Then, for small 0 , there exists a constant C > 0 independent of and v, such that for any 0 < ≤ 0 , we have 1 Proof. We show the first case, the proof of the last one is similar. Performing a simple change of variable we get Now from [2, Lemma 2.1] we know that there exist 0 and C > 0 independent of such that 1 where w = v •τ and τ : R 2 → R 2 is given by τ (x 1 , x 2 ) = (x 1 , G(x 1 )−x 2 ). Hence, we can conclude the proof of the result using that norms w H s (τ −1 (Ω)) and v H s (Ω) are equivalents (for instance, see [5,Section 2]). Now we evaluate the convergence of the integrals with nonlinear terms.
Proof. Since g is bounded with bounded derivatives, it is clear that g is globally Lipschitz. Hence it is not difficult to see that g(u ) → g(u) in L 2 (Ω) as → 0. Moreover, Thus, we obtain (3.8) combine continuity of the trace with u u and ϕ ϕ, weakly in H 1 (Ω). Now let us evaluate 1 where I i sets the integrals in an obvious way. We will get I i → 0 as → 0 for each i = 1, 2, 3 proving (3.9). Due to Lemma 3.1, we have I 3 → 0. On the other hand, Lemma 3.3 implies for any s ∈ (1/2, 1] that Thus, since from ϕ ϕ, weakly in H 1 (Ω), we have ϕ → ϕ, strongly in H s (Ω), for all 0 ≤ s < 1, then we obtain I 1 → 0.
Finally, we use that f is globally Lipschitz, as well as Lemma 3.3 to get Hence, since u → u, strongly in H s (Ω) for all 0 ≤ s < 1, we also obtain I 3 → 0 as → 0.
Remark 9. From now on, we will omit the trace operator symbol aiming to simplify the notation.
We next discuss some properties of the maps A −1 F . They will be necessary in order to prove the lower semicontinuity of the solutions at = 0.
Thus A 0 w, ϕ = F 0 (u), ϕ for all ϕ ∈ H 1 (0, 1), and so, w = A −1 0 F 0 (u), which implies On the other hand, due to (3.13) and Lemma 3.4, we have F (u ), w → F 0 (u), w , in such a way that, Consequently, we complete the proof of ii) using (3.13) and (3.14). Finally let us prove iii). Since we are supposing u − u H 1 (Ω) → 0, we have u H 1 (Ω) ≤ C for some C > 0 independent of . Hence, arguing as in the proof of item ii), for any subsequence, we still can extract another subsequence such that Thus, since this has been shown for any arbitrary sequence, the proof of assertion iii) is completed.  Then there exist a subsequence, still denoted by u , and a function u ∈ H 1 (Ω), depending only on the first variable, that is, u(x 1 , x 2 ) = u(x 1 ), solution of the problem (2.6), such that Moreover, we have that the family u is uniformly bounded in L ∞ (Ω).
Proof. First we note that the the first statement is a direct consequence of inequality · H 1 (Ω) ≥ · H 1 (Ω) for ∈ [0, 1], and Lemma 3.5 since u = A −1 F (u ). Then, let us show u is uniformly bounded in L ∞ (Ω). For all ϕ ∈ H 1 (Ω), we have Let us take ϕ = U = (u − k) + in (4.1) for some k > 0 where φ + denotes the positive part of a function φ. Thus, adding and subtracting k in a suitable way, we Also, since g is bounded in R, it follows from Lemma 3.3 that 1 where C > 0 is a constant, independent of , and A k = {x ∈ Ω : u (x) > k}.
Hence, if we take k big enough satisfying k > sup x∈R f (x), we get On the other hand, due to Sobolev embeddings and (4.2), we have Hence, we conclude from [14, Lemma 5.1] that U is uniformly bonded in L ∞ , and then, u + . Notice we can proceed in a similar way for u − , the negative part of u , which implies the desired result.
Finally we will show the lower semicontinuity of the steady state solutions u at = 0. As we will see, under the assumption of the solutions are hyperbolic, it is a direct consequence of Lemma 3.5 and [21, Theorem 3]. Then there exists a sequence of solutions u of problem (2.8) such that u − u H 1 (Ω) → 0, as → 0.