Supercritical Fokker-Planck equations for consensus dynamics: large-time behaviour and weighted Nash-type inequalities

TL;DR

This paper analyzes a complex Fokker-Planck equation with variable diffusion and nonlinear drift, revealing its long-term behavior and establishing new inequalities relevant for consensus formation in large systems.

Contribution

It introduces novel weighted Nash and Gagliardo-Nirenberg inequalities to study the long-time behavior of a nonlinear Fokker-Planck equation with boundary conditions.

Findings

Propagation of regularity demonstrated

New weighted inequalities established

Steady states similar to Bose-Einstein condensate identified

Abstract

We study the main properties of the solution of a Fokker-Planck equation characterized by a variable diffusion coefficient and a polynomial superlinear drift, modeling the formation of consensus in a large interacting system of individuals. The Fokker-Planck equation is derived from the kinetic description of the dynamics of a quantum particle system, and in presence of a high nonlinearity in the drift operator, mimicking the effects of the mass in the alignment forces, allows for steady states similar to a Bose-Einstein condensate. The main feature of this Fokker-Planck equation is the presence of a variable diffusion coefficient, a nonlinear drift and boundaries, which introduce new challenging mathematical problems in the study of its long-time behavior. In particular, propagation of regularity is shown as a consequence of new weighted Nash and Gagliardo-Nirenberg inequalities.