Transport on network structures.

Abstract

This thesis is dedicated to the study of flows on a network. In the first part of the work, we

describe notation and give the necessary results from graph theory and operator theory that will

be used in the rest of the thesis. Next, we consider the flow of particles between vertices along an

edge, which occurs instantaneously, and this flow is described by a system of first order ordinary

differential equations. For this system, we extend the results of Perthame [48] to arbitrary

nonnegative off-diagonal matrices (ML matrices). In particular, we show that the results that

were obtained in [48] for positive off diagonal matrices hold for irreducible ML matrices. For

reducible matrices, the results in [48], presented in the same form are only satisfied in certain

invariant subspaces and do not hold for the whole matrix space in general.

Next, we consider a system of transport equations on a network with Kirchoff-type conditions

which allow for amplification and/or absorption at the boundary, and extend the results obtained

in [33] to connected graphs with no sinks. We prove that the abstract Cauchy problem associated

with the flow problem generates a strongly continuous semigroup provided the network has no

sinks. We also prove that the acyclic part of the graph will be depleted in finite time, explicitly

given by the length of the longest path in the acyclic part.