Swarming: hydrodynamic alignment with pressure

Tadmor, Eitan

doi:10.1090/bull/1793

Swarming: hydrodynamic alignment with pressure
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by Eitan Tadmor

Bull. Amer. Math. Soc. 60 (2023), 285-325

DOI: https://doi.org/10.1090/bull/1793

Published electronically: April 25, 2023

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Abstract:

We study the swarming behavior of hydrodynamic alignment. Alignment reflects steering toward a weighted average heading. We consider the class of so-called $p$-alignment hydrodynamics, based on $2p$-Laplacians and weighted by a general family of symmetric communication kernels. The main new aspect here is the long-time emergence behavior for a general class of pressure tensors without a closure assumption, beyond the mere requirement that they form an energy dissipative process. We refer to such pressure laws as “entropic”, and prove the flocking of $p$-alignment hydrodynamics, driven by singular kernels with a general class of entropic pressure tensors. These results indicate the rigidity of alignment in driving long-time flocking behavior despite the lack of thermodynamic closure.

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