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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 [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. / Thesis (Ph.D)-University of KwaZulu-Natal, Durban, 2013.

The relativistic Vlasov-Maxwell system (RVM for short) is a kinetic model that arises in plasma physics and describes the time evolution of an ensemble of charged particles that interact only through their self-induced electromagnetic field. Collisions among the particles are neglected and they are assumed to move at speeds comparable to the speed of light. If the particles are allowed to move in the three dimensional space, then the main open problem concerning this system is to prove (or disprove) that solutions with sufficiently smooth Cauchy data do not develop singularities in finite time. Since the RVM system is essential in the study of dilute hot plasmas, much effort has been directed to the solution of its Cauchy problem. The underlying hyperbolic nature of the Maxwell equations and their nonlinear coupling with the Vlasov equation amount for the challenges imposed by this system. In this thesis, we show that solutions of the RVM system with smooth, compactly supported Cauchy data develop singularities only if the charge density blows-up in finite time. In particular, solutions can not break-down due to shock formations, since in this case scenario the solution would remain bounded while its derivative blows-up. On the other hand, if the transversal component of the displacement current is neglected from the Maxwell equations, then the RVM system reduces to the socalled relativistic Vlasov-Darwin (RVD) system. The latter has useful applications in numeric simulations of collisionless plasma, since the hyperbolic RVM is now reduced to a more tractable elliptic system while preserving a fully coupled magnetic field. As for the RVM system, the main open problem for the RVD system is to prove whether classical solutions with unrestricted Cauchy data exist globally in time. In the second part of this thesis, we show that classical solutions of the RVD system exist provided the Cauchy datum satisfies some suitable smallness assumption. The proof presented here does not require estimates derived from the conservation of the total energy nor those on the transversal component of the electric field. These have been crucial in previous results concerning the RVD system. Instead, we exploit the potential formulation of the model equations. In particular, the Vlasov equation is rewritten in terms of the generalized variables and coupled with the equations satisfied by the scalar and vector Darwin potentials. This allows to use standard estimates for singular integrals and a recursive method to produce the existence of local in time classical solutions. Hence, by means of a bootstrap argument, we show that such solutions can be made global in time provided the Cauchy data is sufficiently small.

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Thesis (Ph.D.)--University of Adelaide, School of Mathematical Sciences, Discipline of Applied Mathematics, 2004. / "March, 2004" Includes bibliographical references (leaves 74-78).

Thesis (Ph. D.)--Ohio State University, 2004. / Title from first page of PDF file. Document formatted into pages; contains viii, 140 p.; also includes graphics. Includes bibliographical references (p. 139-140).

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Physics and Astronomy." Includes bibliographical references (p. 74-76).

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 131-137). Also available in electronic version. Access restricted to campus users.

Thesis (M.S.)--University of Wisconsin--Madison, 1965. / eContent provider-neutral record in process. Description based on print version record. Bibliography: l. 156-166.