| Abstract: |
| We present recent results on the existence and long-time asymptotic behavior of solutions to a hydrodynamic model of flocking-type with pressure, within the framework of weak solutions. The analysis is carried out in one spatial dimension, considering both all-to-all interaction kernel and non-constant kernel in a periodic domain. For entropy weak solutions on the torus, we establish exponential decay in time toward a flocking state in the $L^2$ norm, under the assumptions of an integrable interaction kernel and a density uniformly bounded away from vacuum. The approach relies on the front-tracking approximate solutions, the relative entropy and a suitable energy functional. |
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