Global well-posedness of solutions for the epitaxy thin film growth model

In this paper, we consider the global well-posedness of solutions for the initial-boundary value problems of the epitaxy growth model. We first construct the local smooth solution, then by combining some a priori estimates, continuity argument, the local smooth solutions are extended step by step to all t > 0, provided that the initial datums sufficiently small and the smooth nonlinear functions satisfy certain local growth conditions

Local existence and uniqueness in the largest critical space for a surface growth model

We show the existence and uniqueness of solutions (either local or global for small data) for an equation arising in different aspects of surface growth. Following the work of Koch and Tataru we consider spaces critical with respect to scaling and we prove our results in the largest possible critical space such that weak solutions are defined. The uniqueness of global weak solutions remains unfortunately open, unless the initial conditions are sufficiently small.Comment: 17 page

Coarsening in High Order, Discrete, Ill-Posed Diffusion Equations

We study the discrete version of a family of ill-posed, nonlinear diffusion equations of order 2n. The fourth order (n=2) version of these equations constitutes our main motivation, as it appears prominently in image processing and computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The second order equation (n=1) corresponds to another famous model from image processing, namely Perona and Malik\u27s anisotropic diffusion, and was studied in earlier papers. The equations studied in this paper are high order analogues of the Perona-Malik equation, and like the second order model, their continuum versions violate parabolicity and hence lack well-posedness theory. We follow a recent technique from Kohn and Otto, and prove a weak upper bound on the coarsening rate of the discrete in space version of these high order equations in any space dimension, for a large class of diffusivities. Numerical experiments indicate that the bounds are close to being optimal, and are typically observed

Topics in PDE-based Image Processing.

The content of this dissertation lies at the intersection of analysis and applications of PDE to image processing and computer vision applications. In the first part of this thesis, we propose efficient and accurate algorithms for computing certain area preserving geometric motions of curves in the plane, such as area preserving motion by curvature. These schemes are based on a new class of diffusion generated motion algorithms using signed distance functions. In particular, they alternate two very simple and fast operations, namely convolution with the Gaussian kernel and construction of the distance function, to generate the desired geometric flow in an unconditionally stable manner. We present applications of these area preserving flows to large scale simulations of coarsening, and inverse problems. In the second part of this dissertation, we study the discrete version of a family of ill-posed, nonlinear diffusion equations of order $2n$. The fourth order ($n=2$) version of these equations constitutes our main motivation, as it appears prominently in image processing and computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The second order equation ($n=1$) corresponds to another famous model from image processing, namely Perona and Malik's anisotropic diffusion, and was studied in earlier papers. The equations studied in this paper are high order analogues of the Perona-Malik equation, and like the second order model, their continuum versions violate parabolicity and hence lack well-posedness theory. We follow a recent technique from Kohn and Otto, and prove a weak upper bound on the coarsening rate of the discrete in space version of these high order equations in any space dimension, for a large class of diffusivities. Numerical experiments indicate that the bounds are close to being optimal, and are typically observed.Ph.D.Applied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/78774/1/mareva_1.pd

Non-Classical Crystallization Pathways in Eutectic-Forming Systems

Crystallization is the central process of synthesizing materials across length scales, with ubiquitous examples in synthetic, biogenic, and geologic environments. During crystallization a continuum of patterns could emerge due to the interplay of growth kinetics, material or solution chemistry, and crystallographic defects. In particular, solidification of eutectic alloys, characterized by the proximity of their compositions to a nonvariant point in the phase diagram, produces multi-phased micro- and nanostructures with diverse morphologies. This spontaneous pattern formation lies among the broader self-organization strategies that can be easily scaled to large areas, potentially enabling higher throughput and lower cost than serial processes. This dissertation sheds new light on non-classical pathways for eutectic crystallization, perplexing characteristics that cannot be satisfactorily explained nor predicted by classical nucleation and growth models. The scope of this work entails a platform combining advanced experimental techniques – precise synthesis along with multiscale, three-dimensional, and time-resolved measurements – and computational methods – computer vision and machine learning – for tracking eutectic formation at temperature and their structural evolution under external stimuli. The first thrust of this dissertation focuses on crystallization in the presence of chemical modifiers, and the second thrust on the emergence of two-phase metastable spirals and their response in extreme environments. Thrust one demonstrates cases in which the interaction of the modifier with the growing crystal is either synergistic, illustrated in a case study of Al-Si and Al-Ge eutectics, or antagonistic, shown in the growth of primary Si crystals. Thrust two focuses on spiral growth in the Zn-Mg system, and their behavior at elevated temperatures. These spirals are thermodynamically metastable, so their successful synthesis requires steering the system down certain kinetic pathways on intermediate time-scales. Collectively, our multi-modal characterization studies provide the necessary benchmark data for simulations of complex self‐organization patterns, thus expanding the horizon for the design of next‑generation alloys with superior properties.PHDChemical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155123/1/moniri_1.pd