Hydrodynamic alignment with pressure II. Multi-species

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 flocks 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 (see E. Tadmor [Bull. Amer. Math. Soc. (2023), to appear]) and driven by fractional p p -alignment. In particular, it follows that such multi-species hydrodynamics approaches a mono-kinetic description. This generalizes the mono-kinetic, “pressure-less” study by He and Tadmor [Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021), pp. 1031–1053].