## 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.