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：板的动态断裂，图像处理和相变。这些是非局部模型：模型对系统任何一点发生的事情的数学描述可以依赖于其他点发生的事情。这种非局部依赖性使模型的分析变得复杂，并且局部模型（通常是偏微分方程系统，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平滑性。这在研究断裂和其他固有的不连续或奇异现象时具有优势。对于这些应用，研究者研究了解的正则性，能量泛函的极小值的存在性和正则性（它提供了解的稳定性信息），以及解的渐近行为。这些问题在局部模型中也很重要，因此这里的进展有助于发展与经典局部模型理论平行的非局部模型理论。这样的理论将有更广泛的影响，因为非局部模型正在许多其他应用中被探索，这些应用涉及系统各部分之间的非局部相互作用，例如生物聚集，以及在涉及断裂、图像或相变的制造业、能源和基础设施中的应用。研究生参与项目工作。研究人员进行了非局部模型的数学和计算研究，这些模型出现在板的动态断裂、图像处理和相变中。在处理这些非平滑算子时，处理积分-微分方程的低正则解的优势被数学工具（如紧性参数）的稀缺性所抵消。研究了整个损伤区域和边界附近解的正则性、非局部能量泛函的极小值的存在性和正则性以及解的渐近性等问题。他们开发了板块动态断裂的预测模型，并研究了双非局部Cahn-Hilliard系统给出的相变中显示跳跃不连续的低规则解。为了解决这些问题，他们将现有的技术（乘数方法、傅立叶变换方法、渐近展开或digiorgi型参数）从局部理论应用到非局部框架，并开发出新的方法，为非局部模型的研究提供理论和方法基础。研究生参与项目工作。

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.