Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

Roman Shvydkoy

doi:10.3934/krm.2022005

Article Contents

2022, Volume 15, Issue 2: 213-237. Doi: 10.3934/krm.2022005

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Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

Roman Shvydkoy^,

851 S Morgan St, M/C 249, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607

^* Corresponding author: Roman Shvydkoy
^* Corresponding author: Roman Shvydkoy

Received: July 2021

Revised: January 2022

Early access: January 2022

Published: April 2022

This work was supported in part by NSF grants DMS-1813351 and DMS-2107956.

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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.

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Mathematics Subject Classification: 35Q84, 35Q35, 92D25.

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