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vii, 159 leaves : ill., graphs, tables ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1979

The classical existence theorem for the steady Navier-Stokes equations, based on a bound for the solution's Dirichlet integral, provides little qualitative information about the solution. In particular, if a domain is unbounded, it is not evident that the solution will be unique even when the data are small. Inspired by the works of Odqvist for the interior problem and of Finn for the problem of flow past an obstacle, we give a potential theoretic construction of a solution of the steady Navier-Stokes equations in several domains with noncompact boundaries. We begin by studying a scalar quasilinear elliptic problem in a half space, which serves as a model problem for the development of some of the methods which are later applied to the Navier-Stokes equations. Then, we consider Navier-Stokes flow in a half space, modeling such phenomena as a jet emanating from a wall, with prescribed boundary values. The solution which is obtained decays like |x|⁻² at infinity and has a finite Dirichlet integral. Finally, we solve the problem of flow through an aperture in a wall between two half spaces, with a prescribed net flux through the aperture, or with a prescribed pressure drop between the two half spaces. A steady solution is constructed which decays like |x|⁻² at infinity. For small data, uniqueness is proven within the class of functions which decay like |x|⁻¹ at infinity and have finite Dirichlet integrals. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis investigates the viability of two boundary element methods for solving steady state problems, the continuous least squares method and the Galerkin minimization technique. In conventional boundary element methods, the singularities of the fundamental solution involved are usually located at fixed points on the boundary of the problem's domain or on an auxiliary boundary. This leads to some difficulties: when the singularities are located on the problem domain's boundary, it is not easy to evaluate the solution for points on or near that boundary whereas if the singularities are placed on an auxiliary boundary, this auxiliary boundary would have to be carefully chosen. Hence the methods studied here allow the singularities, initially located at some auxiliary boundary, to move until the best positions are found. These positions are determined by attempting to minimize the error via the least squares or the Galerkin technique. This results in a highly accurate, adaptive, but nonlinear method. We study various methods for solving systems of nonlinear equations resulting from the Galerkin technique. A hybrid method has been implemented, which involves the objective function from the least squares method while the gradient is due to the Galerkin method. Numerical examples involving Laplace's equation in two dimensions are presented and results using the discrete least squares method, the continuous least squares method and the Galerkin method are compared and discussed. The continuous least squares method appears to give the best results for the sample problems tried. / Science, Faculty of / Computer Science, Department of / Graduate

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We consider bounded weak solutions u of a degenerate parabolic-hyperbolic equation defined in a subset [mathematical symbols]. We define strong notion of trace at the boundary [mathematical symbols] reached by L¹ convergence for a large class of functionals of u. Such functionals depend on the flux function of the degenerate parabolic-hyperbolic equation and on the boundary. We also prove the well-posedness of the entropy solution for scalar conservation laws with a strong boundary condition with the above trace result as applications. / text

LD2668 .T4 1942 F3 / Master of Science