Hydrodynamic alignment with pressure II. Multispecies

TL;DR

This paper proves long-time flocking behavior in multi-species hydrodynamic systems with complex interactions, extending previous models to include pressure laws and fractional p-alignment without requiring closure assumptions.

Contribution

It introduces a new framework for multi-species flocking that incorporates entropic pressure laws and fractional p-alignment, generalizing prior pressure-less models.

Findings

Flocking occurs under heavy-tailed cross-interactions.

Self-interactions governed by singular kernels lead to flocking.

Multi-species hydrodynamics approach mono-kinetic states.

Abstract

We study the long-time hydrodynamic behavior of systems of multi-species which arise from agent-based description of alignment dynamics. The interaction between species is governed by an array of symmetric communication kernels. We prove that the crowd of different species flock towards the mean velocity if (i) cross-interactions form a heavy-tailed connected array of kernels, while (ii) self-interactions are governed by kernels with singular heads. The main new aspect here, is that flocking behavior holds without closure assumption on the specific form of pressure tensors. Specifically, we prove the long-time flocking behavior for connected arrays of multi-species, with self-interactions governed by entropic pressure laws [E. Tadmor, Swarming: hydrodynamic alignment with pressure, ArXiv 2208.11786, (2022)] and driven by fractional $p$-alignment. In particular, it follows that such multi-species hydrodynamics approaches a mono-kinetic description. This generalizes the mono-kinetic, "pressure-less" study in [S. He and E. Tadmor, A game of alignment: collective behavior of multi-species, AIHP (c) Non Linear Anal. 38(4) (2021) 1031-1053.]