A new understanding of grazing limit

Tong Yang; Yu-Long Zhou

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arXiv · 2308.08375

A new understanding of grazing limit

Tong Yang·Yu-Long Zhou

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Abstract

The grazing limit of the Boltzmann equation to Landau equation is well-known and has been justified by using cutoff near the grazing angle with some suitable scaling. In this paper, we will provide a new understanding by simply applying a natural scaling on the Boltzmann operator without angular cutoff. The proof is based on a new well-posedness theory on the Boltzmann equation without angular cutoff in the regime with optimal ranges of parameters so that the grazing limit can be justified directly for any $\gamma>-5$ that includes the Coulomb potential corresponding to $\gamma=-3$. With this new understanding, the scaled Boltzmann operator in fact can be decomposed into two components. The first one converges to the Landau operator when the singular parameter $s$ of interaction angle tends to $1^{-}$ and the second one vanishes in this limit.

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Tong Yang, Yu-Long Zhou. 2023-08-16. A new understanding of grazing limit. https://arxiv.org/abs/2308.08375

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