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We study the mathematical theory of second order systems with two species,arising in the dynamics of interacting particles subject to linear damping, tononlocal forces and to external ones, and resulting into a nonlocal version ofthe compressible Euler system with linear damping. Our results are limited tothe $1$ space dimensional case but allow for initial data taken in aWasserstein space of probability measures. We first consider the case of smoothnonlocal interaction potentials, not subject to any symmetry condition, andprove existence and uniqueness. The concept of solutions relies on a stickinesscondition in case of collisions, in the spirit of previous works in theliterature. The result uses concepts from classical Hilbert space theory ofgradient flows (cf. Brezis [7]) and a trick used in [4]. We then consider alarge-time and large-damping scaled version of our system and prove convergenceto solutions to the corresponding first order system. Finally, we consider thecase of Newtonian potentials -- subject to symmetry of the cross-interactionpotentials -- and external convex potentials. After showing existence in thesticky particles framework in the spirit of [4], we prove convergence for largetimes towards Dirac delta solutions for the two densities. All the resultsshare a common technical framework in that solutions are considered in aLagrangian framework, which allows to estimate the behavior of solutions via$L^2$ estimates of the pseudo-inverse variables corresponding to the twodensities. In particular, due to this technique, the large-damping result holdsunder a rather weak condition on the initial data, which does not requirewell-prepared initial velocities. We complement the results with numericalsimulations.

Second order two-species systems with nonlocal interactions: existence and large damping limits