Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming

Abstract

We consider a one-dimensional hydrodynamic model featuring nonlocal attraction–repulsion interactions and singular velocity alignment. We introduce a two-velocity reformulation and the corresponding energy-type inequality, in the spirit of the Bresch–Desjardins estimate. We identify a dependence between the communication weight and interaction kernel and between the pressure and viscosity term allowing for this inequality to be uniform in time. It is then used to study long-time asymptotics of solutions.