We consider the kinetic Cucker–Smale model with local alignment as a mesoscopic
description for the flocking dynamics. The local alignment was first proposed by
Karper, Mellet and Trivisa (2014), as a singular limit of a normalized nonsymmetric
alignment introduced by Motsch and Tadmor (2011). The existence of weak solutions
to this model was obtained by Karper, Mellet and Trivisa (2014), and in the
same paper they showed the time-asymptotic flocking behavior. Our main
contribution is to provide a rigorous derivation from a mesoscopic to a macroscopic
description for the Cucker–Smale flocking models. More precisely, we prove the
hydrodynamic limit of the kinetic Cucker–Smale model with local alignment
towards the pressureless Euler system with nonlocal alignment, under a
regime of strong local alignment. Based on the relative entropy method,
a main difficulty in our analysis comes from the fact that the entropy of
the limit system has no strict convexity in terms of density variable. To
overcome this, we combine relative entropy quantities with the 2-Wasserstein
distance.
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