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Hydrodynamic systems arising in swarming modelling include nonlocal forces inthe form of attractive-repulsive potentials as well as pressure terms modellingstrong local repulsion. We focus on the case where there is a balance betweennonlocal attraction and local pressure in presence of confinement in the wholespace. Under suitable assumptions on the potentials and the pressure functions,we show the global existence of weak solutions for the hydrodynamic model withviscosity and linear damping. By introducing linear damping in the system, weensure the existence and uniqueness of stationary solutions with compactlysupported density, fixed mass and center of mass. The associated velocity fieldis zero in the support of the density. Moreover, we show that global weaksolutions converge for large times to the set of these stationary solutions ina suitable sense. In particular cases, we can identify the limiting densityuniquely as the global minimizer of the free energy with the right mass andcenter of mass.

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