Coupled local/nonlocal models in thin domains

Remark 1.

We can also consider kernels that are not in convolution form, that is, J ( x , y ) and G ( x , y ) with J ∈ C ( R N × R N , R ) nonnegative, with J ( x , x ) > 0 , symmetric J ( x , y ) = J ( y , x ) and integrable, and G ∈ C ( R N × R N , R ) nonnegative, nontrivial and integrable. To simplify the presentation we will deal with convolution type kernels in the proofs.

Observe that it is common to assume that the integral of J and G is equal to one. This assumption is related to the probabilistic interpretation of the model given in [24] and [18]. For example, in this interpretation, G ( ( x 1 , x 2 ) , y ) is the probability of a particle (or an individual of a biological species) that is at ( x 1 , x 2 ) jumps to y in a time step). So, in this case, we have ∫ R 1 G ( x 1 , x 2 , y ) d y = 1 . To obtain our results we only need the integrability of the kernels, hence we do not assume that they are normalized to have integral equal to one.

Associated with the energy (1.3) we have the evolution problem given by its gradient flow (with respect to L 2 ( Ω ∪ R ) ). This gives rise to an diffusion problem. Take ( u , v ) as the solution of the abstract O D E problem ( u , v ) ′ ( t ) = − ∂ E [ ( u , v ) ( t ) ] , t ⩾ 0 , with u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) and, ∂ E [ ( u , v ) ] the subdifferential of E. Then, it turns out (see [24] and [18]) that ( u , v ) solves a system composed by a heat equation (local diffusion) of the form (1.1) in Ω and a nonlocal diffusion equation in R, (1.2), coupled via source terms in the equations (when A = Ω in (1.2)) or via a boundary flux on Γ ⊂ ∂ Ω (when A = Γ in (1.2)). See Sections 1.1 and 1.2 below.

Also from [24] and [18] we know that the associated evolution problem is well-posed in the sense that there are existence and uniqueness of solutions. There are two alternative proofs of this fact. The first one uses a fixed point argument while the second relies on semigroup theory. Besides, a comparison principle holds. Also, the total mass of the initial condition is preserved along the evolution and the solutions converge exponentially fast to the mean value of the initial condition. Notice that, according to [24] and [18], we do not impose any continuity of the densities throughout the interface between the local and nonlocal domain, but we can guarantee continuity of the densities u and v inside the local and nonlocal subdomains Ω and R, respectively, by assuming continuity of the initial conditions. Also there is a probabilistic interpretation of this model (we refer one more time to [24] and [18]). In this interpretation individuals cannot diffuse neither jump from the exterior R N ∖ Ω into Ω or the other way around (the integrals accounting for jumps do not consider the complement of Ω). There is no interchange of mass between Ω ∪ R and its complement. Therefore, the total mass is preserved and we can call our problem as being of Neumann type.

The study of nonlocal problems with smooth kernels has been widely considered recently, see [6,8–10,12,14,21–23] and the book [1]. This kind of equation is getting attention due to its potential applications in ecology, physics, and engineering, and to its flexibility to accurately capture effects that are not easily obtained from classical local models. Biological mobility models of animals and plants are examples of how distinct patterns of mobility can affect the success of invasions [8,34]. In epidemiology, the effects of long-range interactions are responsible for the spreading of diseases around the world [35]. Nonlocal patterns also play an important role in molecular interactions in dissimilar interfaces, continuum mechanics, [25,30], and peridynamics (a model of elasticity and mechanics), [32,33].

There are different strategies for couplings between local and nonlocal models. Let us briefly summarize previous results in [16,18,19,23,24,27], see also the review [15]. In [16], local and nonlocal problems are coupled trough a prescribed solid region in which both kinds of equations overlap (the value of the solution in the nonlocal part of the domain is used as a Dirichlet boundary condition for the local part and vice-versa). This kind of coupling gives continuity of the solution in the overlapping region but does not preserve the total mass. Here we follow [24] and [18] (see also [23,27]). In probabilistic terms, in the model described in [24], particles may jump across the interface between the two regions but can not pass coming from the local side unless they jump. Finally, in [18], the authors studied local and nonlocal diffusion models in different zones coupled via the fluxes across the surface that separates the two regions.

Here, we take as the nonlocal region a thin domain, that is, we consider R ε ⊂ R N ( R ε is assumed to be open and bounded), depending on a small parameter ε ∈ ( 0 , 1 ] that will go to zero and that measures the thickness of the domain. Therefore, in our model problem we have two full dimensional domains, the local domain Ω ⊂ R N (that is fixed) and the nonlocal domain R ε ⊂ R N = R N 1 × R N 2 . We denote x = ( x 1 , x 2 ) a point in R N = R N 1 × R N 2 . The domain R ε is assumed to be a general thin domain defined as R ε = { ( x 1 , ε x 2 ) ∈ R N 1 × R N 2 : ( x 1 , x 2 ) ∈ R } , with R ⊂ R N = R N 1 × R N 2 . Notice that R ε is a domain that is thin in the x 2 -variable. See Fig. 1.

OPEN IN VIEWER

Fig. 1.

Perturbed domain.

Here we want to pass to the limit as ε → 0 in the previous setting and obtain a nontrivial diffusion model in which we couple local and nonlocal diffusion equations, (1.1) and (1.2) that take place in domains of different dimension (we deal here with local diffusion in the full-dimensional domain and nonlocal diffusion in the lower-dimensional one).

For simplicity, we will concentrate in the product case and take R ε as R ε = R 1 × ε R 2 = { ( x 1 , ε x 2 ) : x 1 ∈ R 1 , x 2 ∈ R 2 } . Our results are valid in a more general setting (see Remark 2 below) but we prefer to avoid extra notations and simplify the changes of variables that are needed in the proofs. The typical configuration under study is depicted in Fig. 1.

Remark 2.

Instead of a thin domain like R ε = { ( x 1 , ε x 2 ) : x 1 ∈ R 1 , x 2 ∈ R 2 } , we could have a more complex domain, which could be described by some function g related to the geometry of the channel R ε , more exactly, on the way the channel R ε collapses to a general manifold R 1 . If we want to construct a more general geometry of the channel we could, for instance, in two dimensions, consider the channel R ε = { ( x , y ) : 0 < x 1 < 1 , 0 < x 2 < ε g ( x 1 ) } , although more general and complicated geometries are allowed, see [2].

OPEN IN VIEWER

Fig. 2.

Limit domain.

Main goal. Let Ω ε = Ω ∪ R ε ⊂ R N and consider a local/nonlocal coupling in this domain (see Sections 1.1 and 1.2 for a precise statement of the involved equations and the obtained results). As we have mentioned, our main goal is to study the limit as the nonlocal region, R ε , gets thinner, that is, to study the limit as ε → 0 . When passing the limit as ε → 0 , the “limit” domain, Ω 0 (see Fig. 2) will be the union of Ω and the lower dimensional domain R 1 . In the limit of the solutions to our coupled models, we will obtain solutions to a local equation in the domain Ω (with a nonlocal source) and a nonlocal equation in a domain of smaller dimension, R 1 . After obtaining the limit equations, we will also prove some qualitative properties of this limit problem (like conservation of the total mass and study the asymptotic behaviour of the solutions).

Concerning references for equations in thin domains we refer to [2–5,28,31] that develop some techniques and methods to understand the effects of the geometry of the thin domain on the solutions of elliptic and parabolic singular problems. We can find some applications in elastic beam theories (as torsion and warping functions) [29], lubrification [7], fluid flows as ocean dynamics, geophysical fluid dynamics, and fluid flows in cell membranes, see for instance [26].

Our results can be viewed as an extension of [2] and [28]. In [2], the authors investigate the dynamics of a local reaction-diffusion equation with homogeneous boundary condition in a dumbbell domain. The dumbbell domain is composed by two disconnected regions joined by a thin channel, that depends on a thickness parameter ε and degenerates to a line segment as the parameter ε → 0 . As part of a series of articles (see [3,4]) the authors also prove some properties about the continuity of the set of equilibria. On the other hand, in [28] the authors deal with nonlocal evolution problems with non singular kernels in thin domains obtaining a limit problem when the thickness of the domain goes to zero, but without considering any coupling with a local part of the problem. Passing to the limit in these coupling terms is the main contribution of this work.

1.1. Coupling using source terms

We need to compare the solutions of the problem posed in the perturbed domain Ω ε = Ω ∪ R ε ⊂ R N and the solutions to the limit problem in the limit domain Ω 0 . Since the solutions live in different spaces, to obtain convergence we need some care, not only in the choice of the functional space, but also with the metric chosen in this space. Decomposing a function w ∈ L 2 ( Ω ε ) as w = u + v , with u = w χ Ω and v = w χ R ε , we define the metric in L 2 ( Ω ε ) as (1.4) ‖ w ‖ L 2 ( Ω ε ) 2 = ∫ Ω | u | 2 + 1 ε N 2 ∫ R ε | v | 2 . Remark that we multiply the norm of the involved functions in the thin part of the domain R ε by a factor ε − ( N 2 ) . Now, we can define the energy functional E ε ( u , v ) : = 1 2 ∫ Ω | ∇ u | 2 d x + 1 4 ε 2 N 2 ∫ R ε ∫ R ε J ( x − y ) | v ( y ) − v ( x ) | 2 d y d x (1.5) + 1 2 ε N 2 ∫ R ε ∫ Ω G ( x − y ) | v ( x ) − u ( y ) | 2 d y d x , which is finite in B : = { ( u , v ) ∈ L 2 ( Ω ε ) : u ∈ H 1 ( Ω ) , v ∈ L 2 ( R ε ) } . Notice that in this energy functional we have two terms, 1 2 ∫ Ω | ∇ u | 2 d x and 1 4 ε 2 N 2 ∫ R ε ∫ R ε J ( x − y ) ( v ( y ) − v ( x ) ) 2 d y d x , that are naturally associated with the equations (1.1) and (1.2) plus a coupling term given by 1 2 ε N 2 ∫ R ε ∫ Ω G ( x − y ) ( v ( x ) − u ( y ) ) 2 d y d x .

Now, let us consider the evolution problem obtained as the gradient flow associated with this energy with respect to the norm previously defined in (1.4), that is, ( u ( t ) , v ( t ) ) will be the solution of the abstract O D E problem ( u , v ) ′ ( t ) = − ∂ E ε [ ( u , v ) ( t ) ] , t ⩾ 0 , with initial data u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ε ( x ) . Here ∂ E [ ( u , v ) ] denotes the subdifferential of E at the point ( u , v ) . To see what kind of equations we are solving here, let us compute the derivative of E at ( u , v ) in the direction of φ ∈ C 0 ∞ ( Ω ε ) , ∂ φ E ε ( u , v ) = lim h → 0 E ε ( u + h φ , v + h φ ) − E ε ( u , v ) h = ∫ Ω ∇ u ∇ φ d x + 1 ε N 2 ∫ R ε 1 2 ε N 2 ∫ R ε J ( x − y ) ( v ( y ) − v ( x ) ) ( φ ( y ) − φ ( x ) ) d y d x + 1 ε N 2 ∫ R ε ∫ Ω G ( x − y ) ( v ( x ) − u ( y ) ) d y φ ( x ) d x − ∫ Ω 1 ε N 2 ∫ R ε G ( x − y ) ( v ( y ) − u ( x ) ) d y φ ( x ) d x . Since ⟨ ∂ E [ u , v ] , φ ⟩ = ∂ φ E ( u , v ) , we can derive the local/nonlocal problem associated to this gradient flow that is given by the following system of equations: (1.6) ∂ u ∂ t ( x , t ) = △ u ( x , t ) + 1 ε N 2 ∫ R ε G ( x − y ) ( v ( y , t ) − u ( x , t ) ) d y , ( x , t ) ∈ Ω × ( 0 , + ∞ ) , ∂ u ∂ η ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , + ∞ ) , ∂ v ∂ t ( x , t ) = 1 ε N 2 ∫ R ε J ( x − y ) ( v ( y , t ) − v ( x , t ) ) d y ∂ v ∂ t ( x , t ) = − ∫ Ω G ( x − y ) ( v ( x , t ) − u ( y , t ) ) d y , ( x , t ) ∈ R ε × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω , v ( x , 0 ) = v 0 ε ( x ) , x ∈ R ε .

To clarify, let us state precisely what we will understand by a solution to (1.6): ( u , v ) u ∈ C ( [ 0 , T ] , H 1 ( Ω ) ) and v ∈ C ( [ 0 , T ] , L 2 ( R ε ) ) is a solution to (1.6) if it solves the weak form of the gradient flow ( u , v ) ′ ( t ) = − ∂ E ε [ ( u , v ) ( t ) ] integrated for t ∈ [ 0 , T ] , that is, ∫ Ω u ( x , T ) φ ( x , T ) d x − ∫ Ω u 0 ( x ) φ ( x , 0 ) d x + ∫ R ε v ( x , T ) φ ( x , T ) d x − ∫ R ε v 0 ε ( x ) φ ( x , 0 ) d x = ∫ 0 T ∫ Ω ∇ u ( x , t ) ∇ φ ( x , t ) d x d t + ∫ 0 T 1 ε N 2 ∫ R ε 1 2 ε N 2 ∫ R ε J ( x − y ) ( v ( y , t ) − v ( x , t ) ) ( φ ( y , t ) − φ ( x , t ) ) d y d x d t + ∫ 0 T 1 ε N 2 ∫ R ε ∫ Ω G ( x − y ) ( v ( x , t ) − u ( y , t ) ) d y φ ( x , t ) d x d t (1.7) − ∫ 0 T ∫ Ω 1 ε N 2 ∫ R ε G ( x − y ) ( v ( y , t ) − u ( x , t ) ) d y φ ( x , t ) d x d t , for every test function φ such that φ | Ω ∈ C ( [ 0 , T ] , H 1 ( Ω ) ) and φ | R ε ∈ C ( [ 0 , T ] , L 2 ( R ε ) ) . Existence and uniqueness of such solutions were proved in [24].

As we have mentioned previously, our aim is to pass to the limit as ε → 0 in this evolution problem (1.6). To introduce a candidate to be a limit problem for (1.6), defined in the domain Ω 0 (see Fig. 2) we will perform a change of variables (as described in [28]) in the thin domain, R ε , in order to fix it. The change of variables is given by R = R 1 × R 2 ∋ ( x 1 , w 2 ) ⟼ ( x 1 , ε w 2 ) ∈ R 1 × ε R 2 = R ε . That is, we take w 2 = x 2 ε and z 2 = y 2 ε . One remark concerning notation is needed, we choose to call w 2 = x 2 ε and then we have that ( x 1 , w 2 ) ∈ R 1 × R 2 .

With these variables we can fix the domain which allows us to analyze the asymptotic behavior as ε → 0 in a fixed space of functions. To fix the initial condition for v after the change of variables, we take v ( x , 0 ) = v 0 ε ( x 1 , x 2 ) = v 0 ( x 1 , w 2 ) for some fixed function v 0 . The problem (1.6) becomes after this change of variables the following equations in the fixed domain Ω ˆ = Ω ∪ R : (1.8) ∂ u ε ∂ t ( x , t ) = Δ u ε ( x , t ) + ∫ R G ε ( x − y ) ( v ε ( z , t ) − u ( x , t ) ) d z , ( x , t ) ∈ Ω × ( 0 , + ∞ ) , ∂ u ε ∂ η ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , + ∞ ) , ∂ v ε ∂ t ( w , t ) = ∫ R J ε ( x − y ) ( v ε ( z , t ) − v ε ( w , t ) ) d z ∂ v ε ∂ t ( w , t ) = − ∫ Ω G ε ( x − y ) ( v ε ( w , t ) − u ε ( y , t ) ) d y , ( w , t ) ∈ R × ( 0 , + ∞ ) , u ε ( x , 0 ) = u 0 ( x ) , x ∈ Ω , v ε ( w , 0 ) = v 0 ( w ) , w ∈ R , where J ε ( x − y ) = J ( x 1 − y 1 , ε ( w 2 − z 2 ) ) , G ε ( x − y ) = G ( x 1 − y 1 , ε w 2 − z 2 ) and v ε ( x 1 , w 2 , t ) = v ( x 1 , ε w 2 , t ) . Notice that the problem (1.8) is similar to the ones obtained previously in thin domains (see for instance [2,28]).

Now, we are ready to state our main result for this coupling.