Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces

TL;DR

This paper establishes the global existence, uniqueness, and decay rates of solutions to the Vlasov-Fokker-Planck equation with local alignment forces, connecting agent-based models to PDE analysis.

Contribution

It proves the global well-posedness and decay properties of solutions near Maxwellian, including exponential decay in periodic domains, using energy and dissipation methods.

Findings

Global existence and uniqueness of classical solutions.

Algebraic decay rate towards equilibrium.

Exponential decay in periodic spatial domains.

Abstract

In this paper, we are concerned with the global well-posedness and time-asymptotic decay of the Vlasov-Fokker-Planck equation with local alignment forces. The equation can be formally derived from an agent-based model for self-organized dynamics which is called Motsch-Tadmor model with noises. We present the global existence and uniqueness of classical solutions to the equation around the global Maxwellian in the whole space. For the large-time behavior, we show the algebraic decay rate of solutions towards the equilibrium under suitable assumptions on the initial data. We also remark that the rate of convergence is exponential when the spatial domain is periodic. The main methods used in this paper are the classical energy estimates combined with hyperbolic-parabolic dissipation arguments.