Existence and long time behavior of weak solutions to the Fokker-Planck-Alignment models
数学专题报告
报告题目(Title)：Existence and long time behavior of weak solutions to the Fokker-Planck-Alignment models
报告人(Speaker)：Prof. Roman Shvydkoy (University of Illinois Chicago)
地点(Place)：Zoom 会议 ID：811 946 16434, 密码：123456, Zoom Link: https://us02web.zoom.us/j/81194616434
时间(Time)：2025 年 05 月 06 日（周二） 9:00--10:00
邀请人(Inviter)：薛留堂
报告摘要
In this talk we discuss 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 procedure, which circumvents the problem of degenerate diffusion in non-perturbative regime.
