On miscible fluids in porous media and absorbing boundary conditions for electromagnetic wave propagation and on elastic and nearly elastic waves in the frequency domain

Xiaobing Feng, Purdue University

Abstract

The mathematical models for the incompressible and compressible, single-phase miscible displacement of one fluid by another in a porous medium, under the physical assumptions that no volume change results from the mixing of the components and that a pressure-density relation exists for each component in a form that is independent of mixing, are given by nonlinear partial differential systems of two equations. The primary objective of the first part of this thesis is to study the initial-boundary value problems for both models. For the incompressible model, we prove existence of a global in time weak solution and uniqueness for the semiclassical solution in the two spatial variables case. For the compressible model, existence and uniqueness of the local in time strong solution is established for the case of one spatial variable. In the second part we study electromagnetic wave transmission through an infinite medium. Because of the restrictions imposed by present-day computers, we must solve these problems in a finite domain with an artificial boundary. In this part, we first find the theoretical perfectly absorbing boundary condition on the boundary of a half-space domain for the Maxwell system by considering the system in whole instead of considering each component of electromagnetic fields individually. It is proved, together with each of these local absorbing boundary conditions and the initial conditions, that the Maxwell system is a well-posed initial-boundary value problem in the sense of Kreiss. The reflection coefficients are also computed for local absorbing boundary conditions as a plane wave strikes the artificial boundary. In the last part of this thesis we consider the propagation of waves in either an elastic or a nearly elastic solid. In the frequency domain, both the elastic waves and nearly elastic waves are described by a sequence of noncoercive elliptic systems. In this part of the thesis, we first establish existence and uniqueness of the solution to the systems with absorbing boundary conditions. Then we derive an asymptotic elliptic regularity coefficient estimate as the frequency tends to zero and infinity, which is essential to understanding the solution in the space-time domain. Finally, an iterative domain decomposition procedure is proposed for both the standard weak formulations and the mixed hybrid weak formulations of the systems and the convergence of the domain decomposition procedure is also established for both cases. (Abstract shortened by UMI.)

Degree

Ph.D.

Advisors

Douglas, Purdue University.

Subject Area

Mathematics|Petroleum production|Geophysics

COinS