Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

TL;DR

This paper proves global hypocoercivity and exponential relaxation to Maxwellian for kinetic Fokker-Planck-Alignment equations, with results applicable to both long-range and local communication scenarios, and introduces a novel mollified Favre filtration approach.

Contribution

It establishes the first global hypocoercivity results for these equations, extending convergence beyond near-Maxwellian states and adapting advanced filtration techniques.

Findings

Unconditional convergence for long-range communication.

Relaxation to Maxwellian for nearly aligned flocks with small Fisher information.

Stability of initial data under vanishing noise limit.

Abstract

In this note we establish hypocoercivity and exponential relaxation to the Maxwellian for a class of kinetic Fokker-Planck-Alignment equations arising in the studies of collective behavior. Unlike previously known results in this direction that focus on convergence near Maxwellian, our result is global for hydrodynamically dense flocks, which has several consequences. In particular, if communication is long-range, the convergence is unconditional. If communication is local then all nearly aligned flocks quantified by smallness of the Fisher information relax to the Maxwellian. In the latter case the class of initial data is stable under the vanishing noise limit, i.e.\ it reduces to a non-trivial and natural class of traveling wave solutions to the noiseless Vlasov-Alignment equation. The main novelty in our approach is the adaptation of a mollified Favre filtration of the macroscopic momentum into the communication protocol. Such filtration has been used previously in large eddy simulations of compressible turbulence and its new variant appeared in the proof of the Onsager conjecture for inhomogeneous Navier-Stokes system. A rigorous treatment of well-posedness for smooth solutions is provided. Lastly, we prove that in the limit of strong noise and local alignment solutions to the Fokker-Planck-Alignment equation Maxwellialize to solutions of the macroscopic hydrodynamic system with the isothermal pressure.