Pattern Formation in Thin-Film Ferromagnets and in Spinodal Decomposition of Polymer Solutions

计划中的工作研究了凝聚态物理学中两个简单连续模型的预测能力。这些模型是否预测了在一个有序参数下，在特定区域中实验观察到的某种空间模式的形成？ 所考虑的两个模型是a）微磁模型和B）两种不同粘度的不可压缩流体的混合物的准定常流动模型，该流动由作用在每个组分上的缓慢变化的力驱动。 在模型a）的情况下，目标是识别物理参数的范围，其中模型再现在具有薄膜几何形状的铁磁体中观察到的磁化的特征域和壁图案。 在模型B）的情况下，目的是询问它是否解释了海绵状图案的产生和自相似粗化，海绵状图案是由聚合物和水的混合物的旋节分解期间聚合物相的浓缩形成的。 从数学上讲，模型a）以磁化中的非凸、非局部最小化问题的形式出现。 模型B）以梯度通量类型的偏微分方程的形式出现，其随时间演化聚合物浓度。 对于模型a），数学目标是在目标参数极限中表征极小值的变分极限。将给出严格的证明。模型B）实际上是对实验主义者提出的模型的修正。我们的目标是表明，修改是必要的，至少在质量上重现他们的观察。渐近分析和数值模拟相结合将被采用。建模某些现象是控制它们的关键-并利用它们。一个好的模型既简单又有很高的预测能力。科学和工程中的模型通常是用数学语言表达的，用模型进行预测就相当于解决一个数学问题。本文的目的是将现代数学应用于两个具体的模型，以评估模型预测的质量并改进模型。 第一个模型描述了磁性薄膜如磁记录介质中的涂层的行为，目的是更好地理解微磁化图案的大小和形状。 这是这种介质的存储容量的主要因素。 第二个模型描述了聚合物-水混合物随时间的行为。这种混合物用于聚合物加工，在生物化学中普遍存在。 目的是了解聚合物浓缩过程中经常形成的模式。我们所使用的新工具是一台功能强大的计算机，它可以用数值方法解决模型背后的数学问题。在计算机上实现一个数学问题是微妙的，需要对问题有很好的理论理解。现代数学分析工具已成为这项任务不可或缺的，并将与计划工作的数值计算结合使用。

The planned work investigates the predictive power of two simple continuum models in condensed matter physics. Do themodels predict the formation of certain spatial pattern in anorder 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 ofdifferent viscosity, a flow driven by slowly changing forces on eachcomponent. In case of model a), the objective is to identify aregime 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 decompositionof 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 bythe experimentalists. The goal is to show that the modificationis necessary to reproduce their observations at least qualitatively.A combination of asymptotic analysis and numerical simulation willbe employed.Modeling certain phenomena is the key to control them - and to makeuse 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 specificmodels in order to assess the quality of a model's predictions andto 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.