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First-order aggregation models and zero inertia limits

Banff International Research Station for Mathematical Innovation and Discovery

We consider a first-order aggregation model in both discrete and continuum formulations and show how it can be obtained as zero inertia limits of second-order models. The limiting procedure becomes particularly important when one considers anisotropy in the first-order discrete model, as in that case the model becomes {\em implicit}, and issues such as non-uniqueness and jump discontinuities are being brought up. To extend solutions beyond breakdown we propose a relaxation system containing a small parameter \(\epsilon\), which can be interpreted as a small amount of inertia or response time. We show that the limit \(\epsilon \to 0\) can be used as a jump criterion to select the physically correct velocities. In the continuum setting, the procedure consists in a macroscopic limit, enabling the passage from a kinetic model for aggregation to an evolution equation for the macroscopic density. This is joint work with Joep Evers, Lenya Ryzhik and Weiran Sun.

Mathematics; Partial differential equations; Dynamical systems and ergodic theory

video/mp4

Author affiliation: Simon Fraser University

2017-01-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60060

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/