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- Title
Local existence of weak solutions to kinetic Cucker‐Smale‐Fokker‐Planck equation with singular commutation weights.
- Authors
Li, Donghao; Ma, Yaxian; Zhang, Xianwen
- Abstract
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)$$ O\left({\left&amp;amp;#x0007C;x\right&amp;amp;#x0007C;}&amp;amp;#x0005E;{-\alpha}\right) $$ as x→0$$ x\to 0 $$. First, local existence of weak solutions with finite kinetic energy is established for 0<α<1ifd=1$$ 0&amp;lt;\alpha &amp;lt;1\kern0.1em \mathrm{if}\kern0.3em d&amp;amp;#x0003D;1 $$ and 0<α<2ifd≥2$$ 0&amp;lt;\alpha &amp;lt;2\kern0.1em \mathrm{if}\kern0.3em d\ge 2 $$. Second, for 0<α<d$$ 0&amp;lt;\alpha &amp;lt;d $$, we show that any initial datum with finite velocity moment of order ≥d$$ \ge d $$ launches a weak solution that propagates the initial velocity moment.
- Subjects
KINETIC energy; CAUCHY problem; EQUATIONS; VELOCITY
- Publication
Mathematical Methods in the Applied Sciences, 2023, Vol 46, Issue 9, p9902
- ISSN
0170-4214
- Publication type
Article
- DOI
10.1002/mma.9092