Metastable Dynamics of Convection-diffusion-reaction Equations

Metastable dynamics, which qualitatively refers to physical processes that involve an extremely slow approach to their nal equilibrium states, is often associated with singularly perturbed convection-di usion-reaction equations. A problem exhibits metastable behavior when the approach to equilibrium occurs on a time-scale of order O(eC" ), where C > 0 and " is the singular perturbation parameter. The studies of these mathematical models are not only signi cant in their own right, but also useful in simulating and explaining observed physical phenomena and exploring possibly certain unknown ones. A typical common characteristic associated with these convection-di usion-reaction equations is that the linearized operator is exponentially ill-conditioned. By exponential ill-conditioning we mean that the linearized operator has an exponentially small eigenvalue. As a result, conventional analytical methods and numerical schemes may fail to provide accurate information about the metastable behavior. This thesis is concerned with developing a systematic and robust approach based on asymptotic and numericalmethods to quantify the dynamic metastability associated with various problems. Using the asymptotic method called the projection method which was originated by Ward in [108], we have succeeded in deriving ordinary di erential equations (ODEs) or di erential algebraic equations (DAEs) which characterize the metastable patterns for several problems, including, the phase separation of a binary alloy modeled by the viscous Cahn-Hilliard equation, the upward propagation of a ame front in a vertical channel modeled by the Mikishev-Rakib-Sivashinsky equation, and two problems in slowly varying geometries. The main role of our numericalmethod called the transverse method of lines is to give a numerical justi cation of these ODEs/DAEs and to provide ii useful information about the metastable solutions in their transient phases and collapse phases during which our asymptotic method fails. From the numerical point of view, little is known of the nature concerning the convergence and stability of any numerical scheme that computes metastable behavior, as a result of the exponential ill-conditioning of the linearized operator. In this thesis, several nite di erence schemes and their convergence are analyzed rigorously for a boundary layer resonance problem. Our results from this problem are shown numerically to be also valid for other nonlinear metastable problems and some guidelines in designing e ective numerical schemes are provided. The analytical and numerical results show that our approach is a powerful and general tool to quantitatively study the metastable patterns in various physical problems. In addition, the metastable behavior revealed by our analysis appears to be also rather interesting from the viewpoint of physical applications. iii Table of