Mathematical and computational problems in energy driven pattern formation

任何复杂物理系统的构型都可以用关联能来描述。当系统耗散能量时，动力学跟随能量的梯度流动。研究这些系统的第一步自然是分析能量的临界点。对于本文所关注的系统，这些临界状态及其能量取决于多个参数。在被建模的原始物理系统中通常存在一些噪声，因此系统通常会进化到能量的全局最小值。在应用科学中，大多数基于能量的模型涉及的参数往往不能绝对确定。因此，在整个参数范围内研究这种模型，特别是检测系统行为发生定性变化的参数值，是很重要的。实现这一目标的第一步是理解模型的平衡集及其各自的能量。我的研究计划的长期目标是开发计算工具，可以模拟具有精确计算误差界限的复杂数学模型，其中每个计算都是存在性证明。为了实现这一目标，我将在未来五年内与HQP合作，开发方法来解决与大田-川崎能源和材料科学相关模型相关的问题。该项目将在三个相互关联的主题上结合各种研究目标：相图的严格计算；大田-川崎能量的扩展、约束和几何效应。

The configuration of any complex physical systems can be described by an associated energy. The dynamics then follow the gradient flow of this energy as the system dissipates energy. The first natural step in studying these systems is to analyze the critical points of the energy. For the systems of interest in this proposal these critical states and their energies depend on multiple parameters. There is generally some noise in the original physical systems being modeled thus systems generically evolve to global minimizers of the energy.? Most energy based models in the applied sciences involve parameters which often cannot be determined with absolute certainty. It is therefore important to study such models for whole parameter ranges, and in particular, to detect parameter values at which the system behaviour changes qualitatively. A first step towards accomplishing this goal is the understanding of the set of equilibria of the model and their respective energies. The long-term goal of my research program is the development of computational tools which can simulate complex mathematical models with accurate computed error bounds where every computation is an existence proof. Towards this goal I will work with HQP over the next five years on developing methods to address questions related to the Ohta-Kawasaki energy and related models from material science. The project will tie together various research objectives over three interconnected themes: Rigorous computation of phase diagrams; Extensions of the Ohta-Kawasaki energy and Confinement and geometric effects.