The planned work investigates the predictive power of two simple continuum models in condensed matter physics. Do the models predict the formation of certain spatial pattern in an order parameter, patterns which are experimentally observed in special regimes? The two considered models are a) the micromagnetic model and b) a model for quasi-stationary flow of a mixture of two incompressible fluids of different viscosity, a flow driven by slowly changing forces on each component. In case of model a), the objective is to identify a regime of the physical parameters in which the model reproduces the characteristic domain and wall pattern of the magnetization which is observed in a ferromagnet with a thin film geometry. In case of model b), the objective is to inquire if it explains the creation and self-similar coarsening of sponge-like pattern, which are formed by the concentration of the polymer phase during spinodal decomposition of a mixture of polymer and water. Mathematically speaking, model a) comes in the form of a non-convex, non-local minimization problem in the magnetization. Model b) comes in the form of a partial differential equation of gradient flux type, which evolves the polymer concentration in time. For model a), the mathematical goal is to characterize variational limits of minimizers in the targeted parameter limit. Rigorous proofs will be given. Model b) actually is a modification of the model proposed by the experimentalists. The goal is to show that the modification is necessary to reproduce their observations at least qualitatively. A combination of asymptotic analysis and numerical simulation will be employed.

Modeling certain phenomena is the key to control them - and to make use of them. A good model is both simple and of high predictive power. Models in sciences and engineering usually are cast in mathematical language, and making prediction with the model then amounts to solving a mathematical problem. The goal of this work is to use modern mathematics for two specific models in order to assess the quality of a model's predictions and to improve the models. The first model describes the behavior of magnetic thin films such as coatings in magnetic recording media. The goal is a better understanding of the size and shape of micromagnetization patterns. This is a major factor in the storage capacity of such media. The second model describes the behavior of polymer-water mixtures over time. Such mixtures are used in polymer processing and are ubiquitous in biochemistry. The goal is to understand the patterns that often form during the polymer concentration process. The new tool at our disposal is a powerful computer, which allows to solve the mathematical problem behind the model numerically. Implementing a mathematical problem on the computer is delicate and requires a good theoretical understanding of the problem. Modern tools in mathematical analysis have become indispensible for this task and will be used in conjunction with numerical calculations for the planned work.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Deborah Lockhart
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University of California Santa Barbara
Santa Barbara
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