Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

Abstract

<p style='text-indent:20px;'>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.</p><p style='text-indent:20px;'>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.</p>

Environmental averaging

“…The only restrictive assumption for this model comes from the uniform thickness (150), which translates into ρ φ c 6 . Local existence of such solutions can be shown similar to [Shv22]. Global existence near equilibrium which satisfies the thickness automatically was established in [DFT10].…”

“…Again, because of the roughness of u ε the result had to be cast in the settings of a special class of weak solutions established in [KMT13], see also [KMT14] for the justification of a local alignment limit. The work [Shv22] implemented similar method to prove hydrodynamic limit in the context of the M φ -model. Now, we can cast the Maxwellian limit in the framework of general environmental averaging models with the additional implementation of the mollified local alignment field u ε δ -the same methodology we have been using in the monokinetic regime.…”

“…While Favre filtration is symmetric as an operator over the strength measure κ ρ = ρs ρ , it looses symmetry in the Motsch-Tadmor variant s ρ ≡ 1, labeled M MT . This can be remedied by another version of the Favre averaging introduced in [Shv21,Shv22],…”

“…This results in the Euler-alignment system with isothermal pressure tensor R = ρ Id. The Cucker-Smale model was analyzed in [KMT13, KMT14,KMT15], and M φ was analyzed in [Shv22]. Section 9.2 presents a general result.…”

“…With the nonlinear alignment force the works [DFT10,Cho16] establish relaxation for perturbative solutions near equilibrium in the case of the Cucker-Smale and purely local models, respectively. The first global result was proved in [Shv22] in the context of the M φ -model, where linear technique was adapted to the nonlinear problem enabled by special cancelations in the alignment forcing.…”