2021/07/21 by Roman Shvydkoy, Shvydkoy, Roman · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35Q35 #35Q84 #92D25 #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Stochastic processes and financial applications #math.AP #msc:35Q35 #msc:35Q84 #msc:92D25
paper · pdf · doi:10.48550/arxiv.2107.10322
20 pages
arxiv created 2021/07/21 · openalex publication_date 2021/07/21 · arxiv updated 2021/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.