Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

Abstract

We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing 2-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy type arguments combined with the 2-Wasserstein distance. This together with the quantitative error estimate between the pressureless Euler system and the aggregation equation in 2-Wasserstein distance in [Commun. Math. Phys, 365, (2019), 329-361] establishes the quantitative bounds on the error between the kinetic equation and the aggregation equation.

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

“…Hydrodynamic limits towards macroscopic limits have been derived in 87,108 . Other corrections of (5.55) that smooth the momentum term have been studied in 36,86 Regarding the aggregation equation (5.3), there is a huge literature, see 35,44,54,167,166,205,206 and related references. Indeed, similar estimates to those in Theorem 5.15 have been proved for such family of gradient-flow systems.…”

Mean-Field Limits: From Particle Descriptions to Macroscopic Equations

“…The limiting nonlinearly coupled aggregation equations (1.4)-(1.5) have been recently studied in [39,40]. Several authors have studied particular choices of interactions V, W and comunication functions ψ for some of the connecting asymptotic limits from the kinetic description (1.10) with/without noise to the hydrodynamic system (1.11) in [8,11,42,57], from the hydrodynamic system (1.11) to the aggregation equation (1.4)- (1.5) in [23,59,60], and for the direct limit from the kinetic equation to the aggregation equation (1.4)- (1.5) in [8,53].…”

“…Furthermore, its relative energy/entropy has no strict convexity in terms of density variable due to the lack of pressure term. In order to overcome these difficulties, ideas of combining the modulated macro energy and the first or second order Wasserstein distance have been recently proposed in [8,11,32] quantifying the hydrodynamic limit from kinetic equation to the pressureless Euler type system. More recently, in [24], a general theory providing some relation between a modulated macro energy-type function and p-Wasserstein distance is also developed.…”

Large friction-high force fields limit for the nonlinear Vlasov–Poisson–Fokker–Planck system

“…In [22,25], types of compactness arguments are employed, and thus the convergences of solutions are only qualitatively investigated. On the other hand, in a recent paper [3] a new idea of establishing the large friction limit of kinetic equation to (7) is proposed. Introducing an intermediate system, which is pressureless Euler-type system, and employing the second order Wasserstein distance between ρ and ρ, a quantified high force-field limit for the equation (1) with m e = ε, τ e = ε, and σ e = 0 is provided.…”