Topologically Based Fractional Diffusion and Emergent Dynamics with Short-Range Interactions
Roman Shvydkoy, Eitan Tadmor · SIAM Journal on Mathematical Analysis · 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.