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 global relaxation to Maxwellian 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 energy, $(1+ |v|^2) f_0 \in L^1$, $f_0 \in L^\infty$, and finite higher moment $|v|^q f\in L^2$, $q \gg 2$, 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, which circumvents the problem of degenerate diffusion in non-perturbative regime.