Global well-posedness and relaxation for solutions of the Fokker-Planck-Alignment equations

TL;DR

This paper proves the global existence, regularization, and exponential relaxation of solutions for a broad class of Fokker-Planck-Alignment models in collective dynamics, even with minimal initial data regularity.

Contribution

It introduces a new thickness-based renormalization method to establish global solutions without regularity or no-vacuum assumptions.

Findings

Global weak solutions exist for broad Fokker-Planck-Alignment models.

Solutions become instantly smooth and satisfy entropy equality.

Solutions relax exponentially fast to equilibrium.

Abstract

In this paper we prove global existence of weak solutions, their regularization, and relaxation for large data for a broad class of Fokker-Planck-Alignment models which appear in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of regularity or no-vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker-Smale model, we demonstrate that any bounded data with finite higher moment, $f_0 \in L^1(1+ |v|^q) \cap L^\infty$, $q \geq n+4$, gives rise to a global instantly smooth solution, satisfying entropy equality and relaxing exponentially fast. The results are achieved through the use of a new thickness-based renormalization procedure, which circumvents the problem of degenerate diffusion in non-perturbative regime.