In this thesis, we establish a general and type independent theory for the existence and regularity of generalized solutions of large classes of systems of nonlinear partial differential equations (PDEs). In this regard, our point of departure is the Order Completion Method. The spaces of generalized functions to which the solutions of such systems of PDEs belong are constructed as the completions of suitable uniform convergence spaces of normal lower semi-continuous functions. It is shown that large classes of systems of nonlinear PDEs admit generalized solutions in the mentioned spaces of generalized functions. Furthermore, the generalized solutions that we construct satisfy a blanket regularity property, in the sense that such solutions may be assimilated with usual normal lower semi-continuous functions. These fundamental existence and regularity results are obtain as applications of basic topological processes, namely, the completion of uniform convergence spaces, and elementary properties of real valued continuous functions. In particular, those techniques from functional analysis which are customary in the study of nonlinear PDEs are not used at all. The mentioned sophisticated methods of functional analysis are used only to obtain additional regularity properties of the generalized solutions of systems of nonlinear PDEs, and are thus relegated to a secondary role. Over and above the mentioned blanket regularity of the solutions, it is shown that for a large class of equations, the generalized solutions are in fact usual classical solutions of the respective system of equations everywhere except on a closed, nowhere dense subset of the domain of definition of the system of equations. This result is obtained under minimal assumptions on the smoothness of the equations, and is an application of convenient compactness theorems for sets of sufficiently smooth functions with respect to suitable topologies on spaces of such functions. As an application of the existence and regularity results presented here, we obtain for the first time in the literature an extension of the celebrated Cauchy-Kovalevskaia Theorem, on its own general and type independent grounds, to equations that are not analytic. / Thesis (PhD)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

It is well known that a classical solution of the initial value problem for a scalar conservation law may fail to exist on the whole domain of definition of the problem. For this reason, suitable generalized solutions of such problems, known as weak solutions, have been considered and studied extensively. However, weak solutions are not unique. In order to obtain a unique solution that is physically relevant, the vanishing viscosity method, amongst others, has been employed to single out a unique solution known as the entropy solution. In this thesis we present an alternative approach to the study of the entropy solution of conservation laws. The main novelty of our approach is that the theory of entropy solution of conservation law is presented in an operator theoretic setting. In this regard, the Order Completion Method for nonlinear PDEs, in the context of convergence vector spaces, is modified to obtain an operator equation which generalizes the initial value problem. This equation admits at most one solution, which may be represented as a Hausdorff continuous function. As a particular case, we apply our method to obtain the entropy solution of the Burger's equation. Copyright / Dissertation (MSc)--University of Pretoria, 2011. / Mathematics and Applied Mathematics / Unrestricted