On long-time asymptotics for viscous hydrodynamic models of collective behavior with damping and nonlocal interactions

Hydrodynamic systems arising in swarming modeling include nonlocal forces in the form of attractive-repulsive potentials as well as pressure terms modeling strong local repulsion. We focus on the case where there is a balance between nonlocal attraction and local pressure in presence of confinement in the whole space. Under suitable assumptions on the potentials and the pressure functions, we show the global existence of weak solutions for the hydrodynamic model with viscosity and linear damping. By introducing linear damping in the system, we ensure the existence and uniqueness of stationary solutions with compactly supported density, fixed mass and center of mass. The associated velocity field is zero in the support of the density. Moreover, we show that global weak solutions converge for large times to the set of these stationary solutions in a suitable sense. In particular cases, we can identify the limiting density uniquely as the global minimizer of the free energy with the right mass and center of mass.