Transport on network structures.
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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  to arbitrary nonnegative off-diagonal matrices (ML matrices). In particular, we show that the results that were obtained in  for positive off diagonal matrices hold for irreducible ML matrices. For reducible matrices, the results in , 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  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.