SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 5792-5839, January 2020.
We introduce a new class of models for emergent dynamics. It is based on a new communication protocol which incorporates two main features: short-range kernels which restrict the communication to local metric balls, and anisotropic communication kernels, adapted to the local density in these balls, which form topological neighborhoods. We prove flocking behavior---the emergence of global alignment for regular, nonvacuous solutions of the $n$-dimensional models based on short-range topological communication. Moreover, global regularity (and hence unconditional flocking) of the one-dimensional model is proved via an application of a De Giorgi-type method. To handle the nonsymmetric singular kernels that arise with our topological communication, we develop a new analysis for local fractional elliptic operators (interesting in its own right) encountered in the construction of our class of models.

中文翻译：

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基于拓扑的分数扩散和具有短程相互作用的新兴动力学

SIAM数学分析杂志，第52卷，第6期，第5792-5839页，2020年1月。
我们介绍了用于突发动力学的一类新模型。它基于一种新的通信协议，该协议包含两个主要功能：将通信限制为本地度量标准球的短距离内核，以及适应这些球中局部密度（形成拓扑邻域）的各向异性通信内核。我们证明了植群行为---基于短程拓扑通信的$ n $维模型的常规，非空解的全局对齐的出现。此外，通过应用De Giorgi型方法证明了一维模型的整体规则性（因此实现了无条件的植绒）。为了处理拓扑通信中出现的非对称奇异内核，