Swarming: hydrodynamic alignment with pressure

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.