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In this note we establish hypocoercivity and exponential relaxation to theMaxwellian for a class of kinetic Fokker-Planck-Alignment equations arising inthe studies of collective behavior. Unlike previously known results in thisdirection that focus on convergence near Maxwellian, our result is global forhydrodynamically dense flocks, which has several consequences. In particular,if communication is long-range, the convergence is unconditional. Ifcommunication is local then all nearly aligned flocks quantified by smallnessof the Fisher information relax to the Maxwellian. In the latter case the classof initial data is stable under the vanishing noise limit, i.e.\ it reduces toa non-trivial and natural class of traveling wave solutions to the noiselessVlasov-Alignment equation.  The main novelty in our approach is the adaptation of a mollified Favrefiltration of the macroscopic momentum into the communication protocol. Suchfiltration has been used previously in large eddy simulations of compressibleturbulence and its new variant appeared in the proof of the Onsager conjecturefor inhomogeneous Navier-Stokes system. A rigorous treatment of well-posednessfor smooth solutions is provided. Lastly, we prove that in the limit of strongnoise and local alignment solutions to the Fokker-Planck-Alignment equationMaxwellialize to solutions of the macroscopic hydrodynamic system with theisothermal pressure.

Global hypocoercivity of kinetic Fokker-Planck-Alignment equations