Mathematical Sciences: Non-convex Energies and Dynamical Metastability

小行星9501060 主要研究者分析了涉及非凸能量泛函的数学模型，这些泛函包含了体积和梯度效应，以更好地理解动态亚稳性的性质，动态亚稳性通常以过渡结构的缓慢迁移的形式出现，例如陡峭的锋面或漩涡。 由Bronsard和Kohn开发的能量方法已被证明成功地建立了Allen-Cahn方程、Cahn-Hilliard方程、Cahn-Morral系统和用于模拟晶格相变的方程的动态亚稳态解的存在性。 研究者将把类似的技术应用于高维域中的方程，具有多个状态变量，具有多个（或连续的）低能相，具有涉及卷积的梯度能量，或具有复杂的动力学行为。 在物理系统的几个数学模型（特别是材料科学的模型）中，已经发现某些结构最终会发生重大变化，但需要非常长的时间才能完成。 这种现象，有时被称为动态亚稳性，对于识别和理解是重要的，因为材料的宏观性质（例如，脆性）可能敏感地取决于底层结构。 该项目将有助于这种理解，因此，新材料的开发和现有材料的改进使用。 表现出动态亚稳定性的模型是足够普遍的，这个项目的结果应该在其他情况下的影响，以及。 ***

9501060 Grant The principal investigator analyzes mathematical models involving nonconvex energy functionals that incorporate both bulk and gradient effects, in an effort to better understand the nature of dynamical metastability, which typically appears in the form of slow migration of transition structures, such as steep fronts or vortices. Energy methods of the type developed by Bronsard and Kohn have proven successful in establishing the existence of dynamically metastable solutions to the Allen-Cahn equation, the Cahn-Hilliard equation, Cahn-Morral systems, and equations for modeling phase transitions on lattices. The investigator will apply similar techniques to equations in higher-dimensional domains, with multiple state variables, with multiple (or a continuum of) low-energy phases, with gradient energies involving convolutions, or with complex dynamical behavior. %%% In several mathematical models of physical systems (in particular, models from materials science) certain structures have been discovered that eventually undergo significant changes but take an extraordinarily long time to do so. This phenomenon, sometimes called dynamical metastability, is important to identify and understand because macroscopic properties of a material (e.g., brittleness) may depend sensitively on the underlying structure. This project will contribute to this understanding and, therefore, to the development of new materials and the improved use of existing ones. The models exhibiting dynamical metastability are general enough that the results of this project should have implications in other contexts, as well. ***