We study the asymptotic behavior of the flow and extend the results obtained by Perthame [48] to arbitrary ML matrices. We then study the asymptotic behavior of the semigroup solution using Perron-Frobenius type theorems.
Notation
In this chapter we provide some background information necessary to develop the theory of partial differential equations on networks. We start with a few definitions and results in graph theory that will be important in our study of flows on networks.
Digraphs
However, since u has no input edge, u' in L(G) will have no input edge (and is therefore a source), which means that L(G) is not strongly connected, which is a contradiction. If there is at least one entering (leaving) edge point in G, then the matrix Φ+ (or Φ−) is surjective.
Perron-Frobenius Theorems
It is called irreducible if there exists a non-negative irreducible matrixAandη∈Rsuch thatA˜=A−ηI, otherwise we say it is reducible. Note that for an ML matrix A, the dominant eigenvalue is the number τ satisfying τ >ℜλ for any other eigenvalueλ of A.
Banach Lattices
So a lattice vector space X is an ordered vector space such that the infimum and supremum of any pair of elements in X are also contained in X. The space ℓ1 is an ideal of ℓ∞, but it is not a band: consider the sequence xn whose term is given by .
Positive Semigroups
Direct modelling with ML matrices
Then the change in the number of individuals in the state. Within a small time interval, ∆t can be expressed as. 3.1) Note that di in the above equation also includes the rate at which individuals migrate from state i to other states. In particular, if in the model described the number of births and the total number of individuals leaving (due to migration and death) each state are equal, the diagonal coefficients are the sum of the other terms in the respective columns, taken with a negative sign and thus , the sum of each column is zero. That is, processes in which the total number of individuals in the population remains constant [4].
In [48] Perthame showed the decay of the general relative entropy function for (3.2) in the case of strictly positive off-diagonal matrices A. In the second part of this chapter (Section 3.3) we reformulate these results in a more reduced form . case so that they can be applied to reducible matrices.
Irreducible matrices
This allows us to prove a number of classical estimates for the solution of (3.3) in a unified way. Furthermore, using H(u) = u2 and extending the Poincar´e inequality, we will show that the semigroup generated by A is strictly contractive in the subspace perpendicular to N with respect to the scalar product. In the next result, we show that Proposition 6.6 in [48] for solutions to the system (3.3) where A has strictly positive off-diagonal entries still holds when A is irreducible but not necessarily strictly positive off-diagonal.
If we continue with the same reasoning for all terms in the product, we find that mir−1. As a consequence of Theorem 3.2.1 together with Lemma 3.2.3, we have the following theorem on the solution of problem (3.3) (an extension of Proposition 6.5 in [48] to irreducible matrices).
Reducible matrices
As we saw in Example 3.3.1, Lemma 3.2.3 does not hold for reducible matrices, even if a positive-right eigenvector exists. Since the series is on the unit sphere in Y, it contains a convergent subseries whose limit ism˜˜ ∈Y. This limit also exists on the unit sphere (Bolzano-Weierstrass) and satisfies the assumptions of the lemma.
Equation (3.27) therefore holds if and only if each term on the left side of the equation is equal to zero. If A is a 3×3 triangular matrix in normal form, with g = 2, A1 and A2 are 1×1 blocks (ie, scalars), and N>0,v≥0, then the only vectors m for which equation 3.23 applies, vectors of the form (0 ,0,m3)T, where m3 is a vector whose dimension depends on the dimension of A3.
Relative Entropy Inequality for Reducible Matrices
Positive left eigenvector
In this section, we present a survey of some of the results on the transport equation on network structures obtained in and [15]. 4.1) where cj is the speed of the particles along the edge anduj(x, t) is the density of particles on edge ej at position x and time t. In [33] it was proved that the spectrum of (T(t))t≥0 not only depends on the structure of the cycles in G, but also on the rational dependence of the flow velocities on the edges that form a cycle.
After introducing the necessary notation, we provide a formal justification for the claim that the flow in the network described in (4.1) is related to the finite-dimensional system in (3.2). That is, the closed form of (4.8) is the system of equations (3.2) with A = B−I (note that we refer here to the matrix A in (3.2) and not to the differential expression introduced above).
Disconnected graphs
There are indications that (4.9) can be obtained from (4.8) as some asymptotic limit, but so far we have not been able to provide conclusive results in this direction. In general, if a network consists of a limited number of disjoint, strongly connected graphs, we solve the flow problem on each subgraph separately using the methods developed in [33] or in [14]. If Gis is a collection of gdisjoint strongly connected graphs, (A0, D(A0)) generates a C0 semigroup if and only if (Ai0, D(Ai0)) generates C0 semigroups for each i= 1,.
Connected graphs
Existence and uniqueness
Conversely, if Φ− is not surjective, then at least one row, say row k, is a linear combination of some other rows, that is: But this is not possible because each column must have at most one non-zero entry ( see note 2.2.5). So if Φ− is not surjective, then the only possibility is that it has a row of zeros, which implies that at least one of the vertices in the graph has no outgoing edges.
Since the graph is connected and we have assumed that there is no outgoing edge at vi, then there must be an incoming edge, i.e. at. Since we have assumed that there is an output edge at every vertex and wij 6= 0 if and only if φ−ij = 1, we have.
Spectral properties and asymptotic behaviour
- Same speed
- Same speed, with γ j = α j
- The primitive case
- Same speed, γ j 6 = α j
Since column B is stochastic, 1 is an eigenvalue of B, which means that some eigenvalues of A0 are of the form 2ıkπ. We denote by di the imprimitiveness index of the Bi matrix, which means that di is the number of different eigenvalues of the Bi module1. Then [Ti(t)u] is expanded to a periodic group in Xi with period di, the imprimitiveness index of the matrix Bi.
By comparing, we conclude that τ must be a multiple ofdi, therefore τi =di is the period of the semigroup(Ti(t))t≥0. The asymptotic behavior of the new matrix gives the asynchronous growth of the powers of P, and the qualitative behavior of the semigroup (˜T(t))t≥0 defined byT˜(t)f(x) = ˜Pnf(t+x) −n) is the same behavior as the original semigroup (T(t))t≥0.
Different speed along the vertices
According to Kirchoff's law for the artificial vertices and continuity of the test functions ξ on [0,c1. In the previous chapter we proved the existence of semigroup solutions to the flow problem in the graph without wells. We will show that the current will remain asymptotic in certain parts of the graph with cycles and that these subgraphs where the current remains asymptotic are those cycles with no outgoing current.
We also show that the flux at the edges in the acyclic part of the graph will exhaust in finite time while the flux in the cyclic parts of the graph with inflow and outflow will asymptotically exhaust. We start with some useful graph descriptions that will make the asymptotic flow description easier.
Graph Descriptions
If ǫ is generated by a vertex v∈V(G), then there is an incoming edge and an outgoing edge w such that the vertices u¯ and w¯ coincide with ǫ. Sidenej−1 andeij connect throughuij, there is an edge inΨG(uij)with head atΦG(eij) and an edge inΨG(uj+1)with tail atΦG(eij)soΨG(ui1)contains an edge connecting ΦG(eik− 1) )andΦG(ei1). If e is on a path connecting two cycles, then there is a vi1 and vik both on cycles such that vi1, ei1,.
Since vi1 has at least two outgoing edges (one edge is on the cycle and the other is on the path leading to ΦG(e), which is not on the cycle), ΨG(vi1) consists of at least two edges. Also, vik has at least two input edges, one edge on the cycle and the other on the containing path.
Asymptotic behaviour
Note that n2 is the number of vertices in Q1 that are not sources, or, if we use the original graph G, then 2 is the number of edges in G1 plus the cut set CGminus the number of edges with tails in the set V0(G) (see Eq. (5.3)). This result tells us that regardless of the initial distribution of the mass, after t = n2, all edges in the acyclic part of the graph will be exhausted. In fact, we can improve this result by noting that if brij is the (i, j)th entry in Br0, then by theorem brij gives the number of vi−vj paths of lengths.
Using the argument in [33], Theorem 4.5 together with Lemma 5.2.4, to ensure that the cycles in G correspond to cycles in Q in a one-to-one way, we see that the period of the semigroups (Ti( t) ))t≥0, described in the previous chapter, is equal to the greatest common divisor of the cycle lengths of edges in G which are among the states n1 +· · ·+ni−1 + 1,. In other words, the states corresponding to the subgraph whose adjacency matrix is Bi, fori=k,.
Different speeds
However, it increases the cycle lengths (and the length of anyu−vpath for each u, v ∈V(G)) and changes the time scale by N. However, their periods and the time required to exhaust the acyclic part of the graph changes. We further showed that the current in the strongly connected source components of the graph is depleted as t→.
In particular, we proved that the current in the acyclic part of the graph is exhausted in the finite time and that if cj = 1 for all j = 1,. How we define strong connectivity affects the reducibility of matrices A and B.
