A slowly-varying or thin-layer multiscale assumption empowers macroscale understanding of many physical scenarios from dispersion in pipes and rivers, including beams, shells, and the modulation of nonlinear waves, to homogenisation of micro-structures. Here we begin a new exploration of the scenario where the given physics has non-local microscale interactions. We rigorously analyse the dynamics of a basic example of shear dispersion. Near each cross-section, the dynamics is expressed in the local moments of the microscale non-local effects. Centre manifold theory then supports the local modelling of the system's dynamics with coupling to neighbouring cross-sections as a non-autonomous forcing. The union over all cross-sections then provides powerful new support for the existence and emergence of a macroscale model advection-diffusion PDE global in the large, finite-sized, domain. The approach quantifies the accuracy of macroscale advection-diffusion approximations, and has the potential to open previously intractable multiscale issues to new insights.

中文翻译：

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非局部相互作用的严格建模决定了宏观对流扩散 PDE

缓慢变化或薄层的多尺度假设有助于宏观理解许多物理场景，从管道和河流中的扩散，包括梁、壳和非线性波的调制，到微观结构的均质化。在这里，我们开始对给定物理具有非局部微尺度相互作用的场景进行新的探索。我们严格分析了剪切分散的基本示例的动力学。在每个横截面附近，动力学以微观非局部效应的局部矩表示。然后，中心流形理论支持系统动力学的局部建模，并将耦合到相邻横截面作为非自主强迫。然后，所有横截面的联合为大型有限尺寸域中宏观模型对流扩散 PDE 全局的存在和出现提供了强大的新支持。该方法量化了宏观对流扩散近似的准确性，并有可能将以前棘手的多尺度问题打开为新的见解。