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We study an asymptotic limit of Vlasov type equation with nonlocalinteraction forces where the friction terms are dominant. We provide aquantitative estimate of this large friction limit from the kinetic equation toa continuity type equation with a nonlocal velocity field, the so-calledaggregation equation, by employing $2$-Wasserstein distance. By introducing anintermediate system, given by the pressureless Euler equations with nonlocalforces, we can quantify the error between the spatial densities of the kineticequation and the pressureless Euler system by means of relative entropy typearguments combined with the $2$-Wasserstein distance. This together with thequantitative error estimate between the pressureless Euler system and theaggregation equation in $2$-Wasserstein distance in [Commun. Math. Phys, 365,(2019), 329--361] establishes the quantitative bounds on the error between thekinetic equation and the aggregation equation.

Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces