Topologically-based fractional diffusion and emergent dynamics with short-range interactions
Abstract
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 geometric 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, non-vacuous solutions of the -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 non-symmetric singular kernels that arise with our topological communication, we develop a new analysis for local fractional elliptic operators, interesting for its own sake, encountered in the construction of our class of models.
Cite
@article{arxiv.1806.01371,
title = {Topologically-based fractional diffusion and emergent dynamics with short-range interactions},
author = {Roman Shvydkoy and Eitan Tadmor},
journal= {arXiv preprint arXiv:1806.01371},
year = {2020}
}