2024/12/28 by Shvydkoy, R. · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.20294
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) f0 ∈ L1, f0 ∈ L^∞, and finite higher moment |v|q f∈ L2, q ≫ 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.