In this dissertation, we employ variational methods to obtain a new existence result for solutions of a Ginzburg-Landau type equation on a Riemannian manifold. We prove that if $N$ is a compact, orientable 3-dimensional Riemannian manifold without boundary and $\gamma$ is a simple, smooth, connected, closed geodesic in $N$ satisfying a natural nondegeneracy condition, then for every $\ep>0$ sufficiently small, $\exists$ a critical point $u^\ep\in H^1(N;\mathbb{C})$ of the Ginzburg-Landau functional \bd\ds E^\ep(u):=\frac{1}{2\pi |\ln\ep|}\int_N |\nabla u|^2+\frac{(|u|^2-1)^2}{2\ep^2}\ed and these critical points have the property that $E^\ep(u^\ep)\rightarrow\tx{length}(\gamma)$ as $\ep\rightarrow 0$. To accomplish this, we appeal to a recent general asymptotic minmax theorem which basically says that if $E^\ep$ $\Gamma$-converges to $E$ (not necessarily defined on the same Banach space as $E^\ep$), $v$ is a saddle point of $E$ and some additional mild hypotheses are met, then there exists $\ep_0>0$ such that for every $\ep\in(0,\ep_0),E^\ep$ possesses a critical point $u^\ep$ and $\lim_{\ep\rightarrow 0}E^\ep(u^\ep)=E(v)$. Typically, $E$ is only lower semicontinuous, therefore a suitable notion of saddle point is needed. Using known results on $\mathbb{R}^3$, we show the Ginzburg-Landau functional $E^\ep$ defined above $\Gamma$-converges to a functional $E$ which can be thought of as measuring the arclength of a limiting singular set. Also, we verify using regularity theory for almost-minimal currents that $\gamma$ is a saddle point of $E$ in an appropriate sense.

In this thesis, we consider the two-space homogenization method, which produces macroscopic expressions out of descriptions of the behaviour of the microstructure. Specifically, we focus on its application to poroelastic media. After describing the method, we provide examples to demonstrate that the resultant expressions are equivalent to an explicit derivation, which might not always be possible, and to outline the method for proving that the expressions converge to their macroscopic equivalents. Upon providing the basis for this method, we follow Burridge and Keller’s work for using this to prove the existence of Biot’s consolidation equations for poroelastic media and to provide expressions for the derivation of the parameters of these equations from the microstructure [5]. We then discuss the benefits and challenges that arise from this formulation of Biot’s consolidation equations.

In this dissertation we develop, analyze and implement effective numerical methods for multiscale phenomena arising from flows in heterogeneous porous media. The main purpose is to develop innovative numerical and analytical methods that can capture the effect of small scales on the large scales without resolving the small scale details on a coarse computational grid. This research activity is strongly motivated by many important practical applications arising in contaminant transport in heterogeneous porous media, oil reservoir simulations and subsurface characterization. In the work, we investigate three main multiscale numerical methods, i.e., multiscale finite element method, partition of unity method and mixed multiscale finite element method. These methods employ limited single or multiple global information. We apply these numerical methods to partial differential equations (elliptic, parabolic and wave equations) with continuum scales. To compute the solution of partial differential equations on a coarse grid, we define global fields such that the solution smoothly depends on these fields. The global fields typically contain non-local information required for achieving a convergence independent of small scales. We present a rigorous analysis and show that the proposed global multiscale numerical methods converge independent of small scales. In particular, a global mixed multiscale finite element method is extensively studied and applied to two-phase flows. We present some numerical results for two-phase simulations on coarse grids. The numerical results demonstrate that the global multiscale numerical methods achieve high accuracy.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 167-180). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--Kent State University, 2008. / Title from PDF t.p. (viewed Nov. 10, 2009). Advisor: Hassan Allouba. Keywords: stochastic partial differential equations; compact support property. Includes bibliographical references (p. 30).

No description available.

In this thesis we derive and solve equations governing the flow of slender threads and sheets of viscous fluid. Our method is to solve the Navier-Stokes equations and free surface conditions in the form of asymptotic expansions in powers of the inverse aspect ratio of the fluid, i.e. the ratio of a typical thickness to a typical length. In the first chapter we describe some of the many industrial processes in which such flows are important, and summarise some of the related work which has been carried out by other authors. We introduce the basic asymptotic methods which are employed throughout this thesis in the second chapter, while deriving models for two-dimensional viscous sheets and axisymmetric viscous fibres. In chapter 3 we show that when these equations govern the straightening or buckling of a curved viscous sheet, simplification may be made via the use of a suitable short timescale. In the following four chapters, we derive models for nonaxisymmetric viscous fibres and fully three-dimensional viscous sheets; for each we consider separately the cases where the dimensionless curvature is small and where the dimensionless curvature is of order one. We find that the models which result bear a marked similarity to the theories of elastic rods, plates and shells. In chapter 8 we explain in some detail why the Trouton ratio - the ratio between the extensional viscosity and the shear viscosity - is 3 for a slender viscous fibre and 4 for a slender viscous sheet. We draw our conclusions and suggest further work in the final chapter.

In this thesis, we consider the two-space homogenization method, which produces macroscopic expressions out of descriptions of the behaviour of the microstructure. Specifically, we focus on its application to poroelastic media. After describing the method, we provide examples to demonstrate that the resultant expressions are equivalent to an explicit derivation, which might not always be possible, and to outline the method for proving that the expressions converge to their macroscopic equivalents. Upon providing the basis for this method, we follow Burridge and Keller’s work for using this to prove the existence of Biot’s consolidation equations for poroelastic media and to provide expressions for the derivation of the parameters of these equations from the microstructure [5]. We then discuss the benefits and challenges that arise from this formulation of Biot’s consolidation equations.

No description available.

In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have solutions, global supersolutions which guarantee solutions to the conformal constraint equations for near-constant-mean-curvature (near-CMC) data as well as for far-from-CMC data, a proof of the limit equation criterion in the near-CMC case, as well as a model problem on the relationship between the asymptotic constants of solutions and the ADM mass. We also prove a characterization of the Yamabe classes on asymptotically Euclidean manifolds and resolve the (conformally) prescribed scalar curvature problem on asymptotically Euclidean manifolds for the case of nonpositive scalar curvatures. This dissertation includes previously published coauthored material.