Swarming: hydrodynamic alignment with pressure

TL;DR

This paper investigates the long-term flocking behavior in hydrodynamic models of swarming, demonstrating that alignment leads to flocking even with complex pressure laws and singular communication kernels, without requiring thermodynamic closure.

Contribution

It introduces a novel analysis of hydrodynamic alignment with entropic pressure tensors and proves flocking for a broad class of models with singular kernels, extending previous understanding.

Findings

Flocking occurs under general entropic pressure laws.

Alignment persists despite the absence of thermodynamic closure.

Results apply to models with singular communication kernels.

Abstract

We study the swarming behavior of hydrodynamic alignment. Alignment reflects steering towards 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 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.