Topologically-based fractional diffusion and emergent dynamics with short-range interactions

Roman Shvydkoy; Eitan Tadmor

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arXiv · 1806.01371

Topologically-based fractional diffusion and emergent dynamics with short-range interactions

Roman Shvydkoy·Eitan Tadmor

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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 $n$-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 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.

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Roman Shvydkoy, Eitan Tadmor. 2018-06-04. Topologically-based fractional diffusion and emergent dynamics with short-range interactions. https://arxiv.org/abs/1806.01371

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