Semigroup approach to diffusion and transport problems on networks

Models describing transport and diffusion processes occurring along the edges of a graph and interlinked by its vertices have been recently receiving a considerable attention. In this paper we generalize such models and consider a network of transport or diffusion operators defined on one dimensional domains and connected through boundary conditions linking the end-points of these domains in an arbitrary way (not necessarily as the edges of a graph are connected). We prove the existence of $C_0$-semigroups solving such problems and provide conditions fully characterizing when they are positive.


Introduction
Recently there has been an interest in dynamical problems on graphs, where some evolution operators, such as transport or diffusion, act on the edges of a graph and interact through its nodes. One can mention here quantum graphs, see e.g. [20,22,24,25], diffusion on graphs in probabilistic context, [10,16,22], transport problems, both linear and nonlinear, [3,11,13,14,23,26], migrations, [21], and several other applications discussed in e.g. [25,27]. In particular, the recent monograph [27] is a rich source of network models and methods. However, most of these works focus on a particular type of problems. For instance, in the quantum graph theory the main interest is to Research of J.B. and A.F was done during NRF/IIASA SA YSSP at the University of Free State and was partly supported by National Science Centre of Poland through the grant N N201605640. Research of P.N. was supported by TWOWS and the UKZN Research Fund. determine whether the operators defined on the edges of a graph are selfadjoint and the work is confined to the Hilbert space setting. Most papers on the linear transport theory on graphs focus on long term dynamics of the flow. Papers such as [10,16,22], motivated by probabilistic applications, look at Feller or Markov processes on graphs.
The present paper, which provides the theoretical foundation for [6], is similar in spirit to [10,16] in the sense that we prove the existence of strongly continuous semigroups in the space of continuous functions, as well as in the space of integrable functions, that solve the diffusion problem on a network. However, we extend the existing results of [10,16] by considering processes that are more general than the diffusion along edges of a geometric graph with Robin boundary conditions at its vertices in the sense that we allow for a communication between domains that are not necessarily physically connected. In fact, the models we analyse can be also interpreted as diffusion on a hypergraph, [7], but we shall not pursue this line of research in this paper. For completeness, we also present similar results for transport problems, generalizing [3] in a similar way.
To explain the idea of our extension, in the next section we consider two examples, see [5,10].

Motivation
First, let us introduce basic notation which will help to formulate the problems and results. We will work in a finite dimensional space, say, R m . The boldface characters will typically denote vectors in R m , e.g. u = (u 1 , . . . , u m ). We denote M = {1, . . . , m}. Further, for any Banach space X, we will use the notation X = X × . . . × X m times , e.g. for X = L 1 (I), I = [0, 1] we denote Let (A, D()) be an operator in X. If A is a generator, we will denote by {e tA } t≥0 the semigroup generated by A.
1.1.1. Diffusion. We consider a finite graph without loops and isolated edges G = (V, E) with, say, n vertices and m edges. On each edge there is a substance with density u j , j ∈ M, which diffuses along this edge according to where σ j > 0 are constant diffusion coefficients, and can also enter the adjacent edges. To simplify considerations, each edge is identified with the unit interval I. In the model of [10], the particles can permeate between the edges across the vertices that join them according to a version of the Fick law. To write down its analytical form, first we note that, since diffusion does not have a preferred direction, we can assign the tail, or the left endpoint, (that is, 0) and the head, or the right endpoint (that is, 1) to the endpoints of the edge in an arbitrary way. Let l i and r i be the rates at which the substance leaves e i through, respectively, the left and the right endpoints and l ik and r ik be the rates of at which it subsequently enters the edge e k . Then the Fick law at, respectively, the head and the tail of e i , gives where we have written u j (v) as v may be either the tail or the head of the incident edge e j . In particular, if there are no edges incident to the tail of e i , or there are no edges incident to the head of e i , then the Fick's laws take the form respectively, where either coefficient on the right hand side can be 0. It is clear that if r ij = 0, then l ij = 0 and if l ij = 0 then r ij = 0 and thus we can define an m × m matrix A by setting a ij = 1 if either r ij = 1 or l ij = 1 and zero otherwise. Then A is the adjacency matrix of the line graph L(G) of G, see e.g. [19]. However, it turns out that such a matrix is not easy to use and we see that introducing, for any i, j ∈ M, where v is either the tail or the head of the edge under consideration, the problem can be written as where u = (u 1 , . . . , u m ), D = diag{σ i } 1≤i≤m andů is the initial distribution. It turns out that there is no mathematical reason why the matrices K ω , ω ∈ Ω = {00, 01, 10, 11}, in (5) should be restricted to the matrices given by (4) which, indeed, form a strictly smaller class, see [5]. In this paper we study the well-posedness of (5) for arbitrary matrices K ω in spaces C(I) and L 1 (I), extending and simplifying the results of [10,16]. We also find necessary and sufficient conditions for the semigroup solving (5) to be positive.
1.1.2. Transport problems. We consider a digraph G = (V (G), E(G)) = ({v 1 , . . . , v n }, {e 1 , . . . , e m }) with n vertices and m edges. We suppose that none of the vertices is isolated. As before, each edge is normalized so as to be identified with I with the head at 1 and the tail at 0. Following [3,13,14,23,26], we consider a substance of density u j (x, t) on the edge e j , moving with speed c j along this edge. The conservation of mass at each vertex is expressed by the Kirchhoff law, where Φ − = (φ − ij ) 1≤i≤n,1≤j≤m and Φ + = (φ + ij ) 1≤i≤n,1≤j≤m are, respectively, the outgoing and incoming incidence matrices; that is, matrices with the entry φ − ij (resp. φ + ij ) equal to 1 if there is edge e j outgoing from (res. incoming to) the vertex v i , and zero otherwise. Note that due to definitions of the matrices Φ − and Φ + , the summation on the right hand side is over all incoming edges of the vertex v i and on the left hand side over all outgoing edges of v i .
In [3] we considered a slightly more general model where γ j > 0 and ξ j > 0 are the absorption/amplification coefficients at, respectively, the head and the tail of e j . Here the matrix {w ij } 1≤i≤n,1≤j≤m describes the distribution of the incoming flow at the vertex v i into the edges outgoing from it; it is a column stochastic matrix, [23]. We denote C = diag{c j } 1≤j≤m , Ξ = diag{ξ j } 1≤j≤m and Γ = diag{γ j } 1≤j≤m . It follows, [3], that if G has a sink, than there is no C 0 -semigroup solving (7). Hence we discard this case and then it can be proved, e.g. [13, Proposition 3.1], that (7) can be written as an abstract Cauchy problem where B is the (transposed) adjacency matrix of the line graph of G.
As with the diffusion problems, there is no mathematical reason to restrict our analysis to the matrices of the form Ξ −1 C −1 BΓC in the boundary conditions which, in fact, see [5], form a strict subset of the set of all matrices. Thus, we consider the following generalization of (8), (9), where K is an arbitrary matrix.
Example 1.1. The main difference between (10) with arbitrary K and the model with K given in (9) is that in the former, the exchange of the substance can occur instantaneously between any edges, while in the latter the edges must be physically connected by vertices for the exchange to take place. So, for instance, if there is a connection e 1 → e 2 and e 2 → e 3 , then there is no connection e 1 → e 3 . So, while in general (10) cannot model a flow in a physical network, it can describe e.g. a mutation process. Indeed, let a population of , is the density of cells of age x whose genotype belongs to a class j (for instance, having j copies of a gene of a particular type). We assume that cells of class j mature and divide upon reaching maturity at x = 1, with offspring, due to mutations, appearing in class i with probability k ij , i ∈ M. In such a case (k ij ) 1≤i,j≤m is a column stochastic matrix. We note that a particular case of this model is the discrete Rotenberg-Rubinov-Lebowitz model, [30], where the cells are divided into classes according to their maturation velocity. A similar interpretation can be given to (5) where the variable x, instead of the age, denotes the size of the organism, e.g. [18].

Well-posedness of the diffusion problem
We shall consider solvability of (5) in X = C(I) and X = L 1 (I). For technical reasons, we also shall need the solvability of (5) in W 1 1 (I). The norms in these spaces will be denoted, respectively, by · ∞ , · 0 , · 1 . However, if it does not lead to any misunderstanding, we will use X to denote any of these spaces and · or · X to denote the norm in X. Similarly, by ||| · ||| X we denote the operator norm in the space of bounded linear operators from X to X.
Our results are based on [17] and thus, introducing relevant spaces and operators, we try to keep notation consistent with op.cit. First, consider where γ i , i = 0, 1, is the trace operator at x = i (taking the value at x = i if X = C(I)). The domains of L are D(L) = C 1 (I) if X = C(I) and D(L) = W 2 1 (I) in two other cases. Then we define the operator Further, let us denote where K T denotes the transpose of K. Clearly (Φ * ) * = Φ. Let A denote the differential expression Au := D∂ xx u. Then we define the operators A α Φ , α = ∞, 0, 1 by the restriction of A to the domains respectively. As before, we drop the indices from the notation if it will not lead to any misunderstanding.

Basic estimates in the scalar case
Consider the general resolvent equation for (5) Since without the boundary conditions the system is uncoupled, its solution u is given by where and the vector constants C 1 and C 2 are determined by the boundary conditions. Further, denote The starting point are well-known estimates for the scalar Dirichlet and Neumann problems, e.g. [15]. We briefly recall them here in a slightly more precise form, similarly to [8]. In the scalar case we can use σ = 1 as will become clear when the estimate is derived. It follows that for the Dirichlet problem we have and for the Neumann problem Further, [15], and, for i = 0, 1, The following result plays an essential role in deriving precise estimates, Then, for either Dirichlet or Neumann problem in C(I), for λ ∈ Σ θ0 for any θ 0 < π, we have for ω = D, N Similarly, in L 1 (I), we have Consider now A 1 0 , see (13); that is, the operator corresponding to the Neumann boundary conditions in one dimension. We have Proposition 2.1. A 1 0 generates an analytic semigroup in W 1 1 (I) with the resolvent satisfying the estimate Proof. Consider the resolvent equation If f ∈ W 1 1 (I), then u ∈ W 3 1 (I) and we can differentiate the differential equation getting, for v : that is, v satisfies the resolvent equation for the Dirichlet problem. Hence Since λu − σ∂ xx u = f is equivalent to σ −1 λu − ∂ xx u = σ −1 f , we see that the estimates above are independent of σ.

Solvability of (5)
The ideas in this section are based on [10,16] but the analysis is simplified by using the analyticity of the semigroup and the application of [17,Theorem 2.4].

Since in the vector case and Neumann boundary conditions the resolvent equation decouples, we obtain
for any θ 0 < π and X = C(I), L 1 (I), W 1 1 (I). Hence, in particular A 0 generates an analytic semigroup in X.
We begin with a straightforward consequence of Ref. [17].
Theorem 2.2. Let X be either C(I), or W 1 1 (I). Then the operator A Φ generates an analytic semigroup in X with the resolvent satisfying the estimate for some α ≥ 0.
Proof. The result follows directly from [17, Theorem 2.4], since A 0 generates an analytic semigroup in X, L is an unbounded operator on X which is, however, bounded as an operator from D(A) to Y, and it is a surjection (the right inverse in each case is given by [L −1 r (y 0 , y 1 )](x) = 1 2 (y 1 − y 0 )x 2 − y 0 x. Furthermore, Φ is a bounded operator on X (if X = W 1 1 (I) this follows since In X = L 1 (I) the situation is more complicated as Φ is not bounded on X. First we show that R(λ, A 1 Φ ) extends to a resolvent on L 1 (I). Lemma 2.3. We have This shows that u n → u in W 2 1 (I) and thus v = Au. Since taking the trace of a W 2 1 (I) function and of its derivative is continuous in W 2 1 (I), the boundary values of u n and ∂ x u n are preserved in the limit and thus u ∈ D(A 0 Φ ). Hence On the other hand, let u ∈ D(A 0 Φ ).
. Thus there is a sequence (h n ) n∈N ⊂ C ∞ 0 (I) converging to v in W 2 1 (I) and hence u is the limit in W 2 1 (I) of functions u n = h n + f ∈ D(A 1 Φ ). Since the convergence in W 2 1 (I) implies the convergence of both u n and Au n in L 1 (I) we see that also Finally, we see that, as in the scalar case (36), the solvability of the resolvent problem for (5) is equivalent to solvability of the linear system (27) for the vectors C 1 , C 2 , where M = diag{µ}. Once the constants are found, the solution is given by the vector version of (15) and thus belongs to W 2 1 (I) ⊂ L 1 (I). The system above is solvable provided its determinant is different from zero. However, the determinant is clearly an entire function in µ and thus can have only isolated zeros in C. Hence, there must be positive values of λ ∈ ρ(A 0 Φ ).
Theorem 2.4. The operator A 0 Φ generates an analytic semigroup in L 1 (I). Proof. We use the formula from Lemma 1.4 of Ref. [17] stating that , see the proof of [17,Theorem 2.4]. However, R(λ, A 1 N ) extends by density to R(λ, A 0 N ) (the resolvent on L 1 (I)) which is a bounded linear operator from L 1 (I) to D(A 1 N ) ⊂ W 2 1 (I) which is continuously embedded in W 1 1 (I) and thus R(λ, A 1 Φ ) extends to a bounded linear operator, say R Φ (λ), on L 1 (I). Since R(λ, A 1 Φ ) is a resolvent on W 1 1 (I), we obtain, by density, that R Φ (λ) is a pseudoresolvent on L 1 (I). Furthermore, C ∞ 0 (]0, 1[) ⊂ D(A 1 Φ ), thus it is also a subset of the range of R Φ (λ) and therefore the range is dense in L 1 (I). Also, since the range of there is no need to extend (I − L λ Φ) −1 . Hence R Φ (λ) is a one-to-one operator as a composition of two injective operators. Thus, by Proposition III.4.6 in Ref. [15], R Φ (λ) is the resolvent of a densely defined operator, sayÂ Φ . Clearly,Â Φ is a closed extension of A 1 Φ and To prove that is a sectorial operator, we use the idea of [16] but in a somewhat simpler way. We consider the operator A ∞ Φ * . Let R # λ denote the adjoint to R(λ, A ∞ Φ * ) for λ ∈ ρ(A ∞ Φ * ), which acts in the space of signed (vector) Borel measures on I, [9]. R(λ, A ∞ Φ * ) is given by (15) with C 1 , C 2 given by (27) with the matrices K ω replaced by corresponding matrices in (12). Hence, apart from U µ (x), R(λ, A ∞ Φ * ) is a composition of an algebraic operator coming from inverting the matrix in (27) with a vector of functionals acting on f . Thus, if f ∈ L 1 (I) is the density of an absolutely continuous measure, a standard calculation shows that From the definition of the norm of a signed Borel measure, if the latter is absolutely continuous, its norm is equal to the L 1 norm of its density. Since taking the adjoint preserves the norm of the operator, we obtain Hence, by Theorem 2.2, A 0 Φ is sectorial and, being densely defined, it generates an analytic semigroup on L 1 (I).

Positivity of the semigroup
Let us recall that for an element u of a Banach lattice, we write u > 0 if 0 = u ≥ 0. We have the following result To prove the converse we introduce the following notation. For any α := (α 0 , α 1 ) ≥ 0 with α j i = 0 for some i ∈ M, j = 0, 1, we have Accordingly, we denote Let us assume that (30) is not satisfied. Then there is a non-diagonal element of Ξ which is strictly negative. Suppose (−1) r+1 k rs ij < 0 for some i = j and r, s = 0, 1, and consider a vector α with where t = 0 if r = 1 and t = 1 if r = 0, and for r = s for sufficiently small δ > 0. We shall prove that there exists a function 0 ≤ u ∈ C ∞ (I) satisfying u(r i ) = α r i and (−1) r+1 ∂ x u(r i ) = (Ξα) r i which additionally satisfies ∂ xx u(r i ) < 0.
For a given constants α r i , β r i = (Ξα) r i , r = 0, 1, we consider auxiliary and ∂ xx f r i = 2β r i . We observe that as long as α r i > 0, there is a one-sided interval ((0, ω i ) if r = 0 and (ω i , 1) if r = 1), where f r i ≥ 0 irrespective of the sign of β r i . On the other hand, if α r i = 0, for local nonnegativity we need β r i ≤ 0. Now let φ be a nonnegative C ∞ function which is 1 on [−a, a] and 0 outside [−2a, 2a] where 0 < 2a < min i∈M ω i and define Recalling that we assumed (−1) r+1 k rs ij < 0 for some i = j and r, s = 0, 1, we consider coefficients {α t j } t=0,1,j∈M satisfying (32). Then we have u(r i ) = 0. On the other hand, ∂ xx u(r i ) = 2β s i = 2(Ξα) r i < 0 by (33) or (34). Thus, there is a nonnegative element u ∈ D(A) for which ∂ xx u < 0 at a point where the global minimum of zero is attained.
We can use this result to prove an analogous result in L 1 (I).
Corollary 2.6. The operator A 0 Φ generates a positive semigroup if and only if the assumptions of Theorem 2.5 are satisfied.
Proof. In one direction the result immediately follows by density of C(I) in L 1 (I). Conversely, if {e tA 0 Φ } t≥0 ≥ 0 then, in particular, for any 0 ≤ů ∈ C(I) we have e tA 0 Φů = e tA ∞ Φů ≥ 0.

Solvability of (8)
We return to the problem (8) where, we emphasize, K is an arbitrary matrix.
In this section we restrict our attention to X = L 1 (I) as in C(I) the operator A is not densely defined. Let us recall that A is the realization of A = diag{−c j ∂ x } 1≤j≤m on the domain D(A) = {u ∈ W 1 1 (I); u(0) = Ku(1)}. The following theorem for (7) has been proved in [3] (see also [4]) but the proof for any nonnegative matrix K is practically the same. Here we extend this proof to an arbitrary K. However, for the proof in the general case we need to provide basic steps of the proof for nonnegative matrices.
Using the fact that taking the modulus does not change the norm, we find with M and ω following from the Hille-Yosida estimates for A |K| . The fact that K ≥ 0 yields the positivity of the semigroup follows from the first part of the proof. To prove the converse, let k ij < 0 for some i, j and consider the initial condition f (x) = (f 1 (x), . . . , f m (x)) with f k = 0 for k = j and f j ∈ C 1 ([0, 1]) with f j (0) = f j (1) = 0, so that f ∈ D(A), and f j (x) > 0 for 0 < x < 1. Then, at least for t < min 1≤j≤m {1/c j }, u i satisfies t ≥ x c i and we see that the solution is negative for such t. This ends the proof.
We conclude the paper by noting an interesting corollary which also uses monotonicity properties of the problem and which is important in asymptotic analysis, see [6]. For given velocities C = (c 1 , . . . , c m ), let c = min j∈M {c j } and C min = {c, . . . , c} m times . For a particular velocity matrix C, let A K,C denote the generator of the semigroup solving (10).
Corollary 3.1. There holds e tA K,C ≤ e tA |K|,C min .