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We develop tools to construct Lyapunov functionals on the space ofprobability measures in order to investigate the convergence to globalequilibrium of a damped Euler system under the influence of external andinteraction potential forces with respect to the 2-Wasserstein distance. Wealso discuss the overdamped limit to a nonlocal equation used in the modellingof granular media with respect to the 2-Wasserstein distance, and providerigorous proofs for particular examples in one spatial dimension.

Convergence to Equilibrium in Wasserstein distance for damped Euler equations with interaction forces