Local existence of weak solutions to kinetic Cucker‐Smale‐Fokker‐Planck equation with singular commutation weights

摘要

In this paper, we investigate the Cauchy problem of the d$$ d $$ dimensional kinetic Cucker‐Smale‐Fokker‐Planck equation with singular communication weight behaving like O(|x|−α)$$ Oleft({left&amp;amp;amp;#x0007C;xright&amp;amp;amp;#x0007C;}&amp;amp;amp;#x0005E;{-alpha}right) $$ as x→0$$ xto 0 $$. First, local existence of weak solutions with finite kinetic energy is established for 0<α<1ifd=1$$ 0&amp;amp;alpha &amp;amp;1kern0.1em mathrm{if}kern0.3em d&amp;amp;amp;#x0003D;1 $$ and 0<α<2ifd≥2$$ 0&amp;amp;alpha &amp;amp;2kern0.1em mathrm{if}kern0.3em dge 2 $$. Second, for 0<α