Global Well‐Posedness and Relaxation for Solutions of the Fokker–Planck–Alignment Equations

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, , , 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 nonperturbative regime.