Higher Order Nonlocal Models in Continuum Mechanics

1716790Radu研究人员对三种应用中出现的积分-微分方程模型进行了分析和计算研究：板中的动态断裂、图像处理和相变。这些都是非局部模型：模型对系统任何点发生的事情的数学描述可以依赖于在其他点发生的事情。这种非局部相关性使模型的分析变得复杂，而且局部模型(通常是偏微分方程组，通常具有高于二阶的导数)的标准分析工具和结果是不可用的。作为补偿，非局部模型不需要像微分方程模型那样要求解的光滑性。这在研究断裂和其他固有的不连续或奇异现象时可能是一种优势。对于这些应用，研究人员研究解的正则性，能量泛函极小值的存在性和正则性(它提供关于解的稳定性的信息)，以及解的渐近行为。这些问题在局部模型中也很重要，所以这里的进展有助于发展一种非局部模型的理论，与经典局部模型的理论平行。这样的理论将产生更广泛的影响，因为非局部模型正在许多其他涉及系统各部分之间非局部相互作用的应用中被探索，例如生物聚集，以及在制造、能源和基础设施中涉及断裂、图像或相变的应用。研究生参与该项目的工作。研究人员对在平板动态断裂、图像处理和相变中出现的非局部模型进行了数学和计算研究。在处理这些非光滑算子时，数学工具(例如紧致性论据)的稀缺性抵消了处理积分-微分方程式的低正则解的优势。研究人员研究了损伤区域和非损伤区域以及边界附近解的正则性，非局部能量泛函极小元的存在性和正则性，以及解的渐近行为。他们开发了平板动态断裂的预测模型，并研究了由双非局域Cahn-Hilliard系统给出的显示相变中跳跃不连续性的低正则解。为了解决这些问题，他们将现有的技术(乘子方法、傅里叶变换方法、渐近展开或DiGiorgi型变元)从局域理论引入非局域框架，并发展出新的方法，为非局域模型的研究提供了理论和方法论基础。研究生参与该项目的工作。

1716790Radu The investigators conduct an analytical and computational investigation of integro-differential equation models that arise in three applications: dynamic fracture in plates, image processing, and phase transitions. These are nonlocal models: the model's mathematical description of what happens at any point of the system can depend on what is happening at other points. This nonlocal dependence complicates the analysis of the models, and the standard analytical tools and results for local models (which are usually systems of partial differential equations, often with derivatives of higher than second order) are not available. In compensation, the nonlocal models need not require such smoothness of their solutions as differential equation models do. This can be an advantage in studying fracture and other inherently discontinuous or singular phenomena. For these applications, the investigators study regularity of solutions, existence and regularity for minimizers of energy functionals (which provide information about the stability of solutions), and asymptotic behavior of the solutions. These issues are important in local models as well, so advances here help develop a theory for nonlocal models that parallels the theory for classical local models. Such a theory would have wider consequences, because nonlocal models are being explored in many other applications that involve nonlocal interactions between parts of a system, such as biological aggregation, as well as in applications in manufacturing, energy, and infrastructure that involve fracture, images, or phase transitions. Graduate students participate in the work of the project. The investigators conduct a mathematical and computational investigation of nonlocal models that arise in dynamic fracture in plates, in image processing, and in phase transitions. The advantage of working with low-regularity solutions to integro-differential equations is offset by the scarcity of mathematical tools (such as compactness arguments) when working with these non-smoothing operators. The investigators study the issues of regularity of solutions throughout the damaged and undamaged domain and near the boundary, existence and regularity of minimizers for nonlocal energy functionals, and asymptotic behavior of solutions. They develop predictive models for dynamic fracture in plates, and they study low-regularity solutions that show jump discontinuities in phase transitions as given by doubly-nonlocal Cahn-Hilliard systems. To tackle these problems, they adapt existing techniques (multiplier methods, Fourier transform methods, asymptotic expansions, or DiGiorgi-type arguments) from the local theory to the nonlocal framework, and develop new methods that provide theoretical and methodological foundation for the study of nonlocal models. Graduate students participate in the work of the project.