Higher Order Nonlocal Models in Continuum Mechanics

小行星1716790 研究人员进行分析和计算的积分微分方程模型，出现在三个应用程序：动态断裂板，图像处理和相变的调查。 这些是非本地模型：该模型对系统中任何一点发生的情况的数学描述可以依赖于其他点发生的情况。 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不可用。 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个优势。 对于这些应用，研究人员研究解的正则性，能量泛函极小化的存在性和正则性（提供有关解的稳定性的信息），以及解的渐近行为。 这些问题在局部模型中也很重要，因此这里的进展有助于发展一种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探索，涉及系统各部分之间的非局部相互作用，如生物聚集，以及在制造，能源和基础设施中的应用，涉及断裂，图像或相变。 研究生参与该项目的工作。 研究人员进行了数学和计算的非局部模型，在动态断裂板，在图像处理中出现的调查，并在相变。 使用积分微分方程的低正则性解的优势被使用这些非平滑算子时数学工具的稀缺性（如紧致性参数）所抵消。 调查人员研究的问题的正规性的解决方案，整个损坏和未损坏的域和附近的边界，存在和非局部能量泛函极小的规律性，和渐近行为的解决方案。 他们开发预测模型的动态断裂板，他们研究的低规律性的解决方案，显示跳跃不连续性的相变所给予的双重非本地的卡恩-希利亚德系统。 为了解决这些问题，他们适应现有的技术（乘数方法，傅立叶变换方法，渐近展开，或DiGiorgi型参数）从本地理论的非本地框架，并开发新的方法，提供理论和方法的基础上研究非本地模型。 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.