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In this paper, we quantify the asymptotic limit of collective behaviorkinetic equations arising in mathematical biology modeled by Vlasov-typeequations with nonlocal interaction forces and alignment. More precisely, weinvestigate the hydrodynamic limit of a kinetic Cucker--Smale flocking modelwith confinement, nonlocal interaction, and local alignment forces, lineardamping and diffusion in velocity. We first discuss the hydrodynamic limit ofour main equation under strong local alignment and diffusion regime, and werigorously derive the isothermal Euler equations with nonlocal forces. We alsoanalyze the hydrodynamic limit corresponding to strong local alignment withoutdiffusion. In this case, the limiting system is pressureless Euler-typeequations. Our analysis includes the Coulombian interaction potential for bothcases and explicit estimates on the distance towards the limiting hydrodynamicequations. The relative entropy method is the crucial technology in our mainresults, however, for the case without diffusion, we combine a modulatedmacroscopic kinetic energy with the bounded Lipschitz distance to deal with thenonlocality in the interaction forces. For the sake of completeness, theexistence of weak and strong solutions to the kinetic and fluid equations arealso established.

Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forces